Range and Richness of Vascular Land Plants: The Role of Variable Light

P1: JZP Trim: 7in × 10in Top: 42.5pt Gutter: 78ptAGUB001-FM AGU001/Eagleson December 15, 2009 14:2

Range and Richness of VascularLand Plants:The Role of Variable Light

Peter S. Eagleson

American Geophysical UnionWashington, DC


Published under the aegis of the AGU Books Board

Kenneth R. Minschwaner, Chair; Gray E. Bebout, Joseph E. Borovsky, Kenneth H. Brink, RalfR. Haese, Robert B. Jackson, W. Berry Lyons, Thomas Nicholson, Andrew Nyblade, Nancy N.Rabalais, A. Surjalal Sharma, Darrell Strobel, Chunzai Wang, and Paul David Williams, members.

Library of Congress Cataloging-in-Publication Data

Eagleson, Peter S.Range and richness of vascular land plants : the role of variable

light / Peter S. Eagleson.p. cm.

Includes bibliographical references and index.ISBN 978-0-87590-732-1 (alk. paper)1. Phytogeography—Climatic factors. 2. Plants—Effect of solar

radiation on. 3. Plant species diversity. I. Title.QK754.5.E17 2009581.7—dc22 2009048108

ISBN: 978-0-87590-732-1

Book doi:10.1029/061SP

Copyright 2009 by the American Geophysical Union2000 Florida Avenue, NWWashington, DC 20009

Front cover: Spong trees moving toward the light at the ruins of Ta Prohm, Cambodia. Film imagecourtesy of Beverly G. Eagleson. Digital image by James M. Long of the Massachusetts Instituteof Technology.

Figures, tables, and short excerpts may be reprinted in scientific books and journals if the source isproperly cited.

Authorization to photocopy items for internal or personal use, or the internal or personal use ofspecific clients, is granted by the American Geophysical Union for libraries and other usersregistered with the Copyright Clearance Center (CCC) Transactional Reporting Service, providedthat the base fee of $1.50 per copy plus $0.35 per page is paid directly to CCC, 222 Rosewood Dr.,Danvers, MA 01923. 978-0-87590-732-1/09/$1.50 + 0.35.

This consent does not extend to other kinds of copying, such as copying for creating new collectiveworks or for resale. The reproduction of multiple copies and the use of full articles or the useof extracts, including figures and tables, for commercial purposes requires permission from theAmerican Geophysical Union.

Printed in the United States of America


To my dearest Bev, who has taught me how to live and to love and, in so

doing, has inspired my work and enriched my life beyond measure


In Memoriam

Helen Sturges Eagleson (1900–1989), mother, binder of childhood wounds,

cultivator of intellect, supporter of ambitious dreams, guide through the minefields

of male adolescence, and setter of the standards for life, who, through continuing

personal sacrifice, single-handedly prepared her children for early and productive

independence.

Arthur Thomas Ippen (1907–1974), teacher, advisor, advocate, professional ex-

emplar, colleague, surrogate father, and dear friend, whose unfailing confidence and

support placed a Massachusetts Institute of Technology career within the author’s

grasp and whose foresight, in the early 1960s, directed that career toward develop-

ment of the neglected hydrologic sciences.


Ecosystem Research Needs

We lack a robust theoretical basis for linking ecological diversity to ecosystem

dynamics. . . .

Carpenter et al. [2006, p. 257]


Estimated global numbers of vascular land plant species: The key to analytical formulation of localspecies range and richness as a function solely of incident light lies in finding a robust one-to-one connection between species and a biologically optimum value of intercepted shortwave solarradiation. Such a connection exists at the intersection of the asymptotes of the photosynthetic-capacity curve of the leaves of C 3 vascular land plants, and this illustration demonstrates theglobal dominance of this photosynthetic pathway. Keyed letters indicate the following Websites: a, http://www.bio.umass.edu/biology/conn.river/photosyn.html; b, http://en.wikipedia.org/wiki/Bromeliaceae; c, http://en.wikipedia.org/wiki/Orchidaceae; d, http://en.wikipedia.org/wiki/Succulent plant; e, http://science.jrank.org/pages/6418/Spurge-Family.html; f, http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/C/C4plants.html; g, http://en.wikipedia.org/wiki/Ferns; h,http://en.wikipedia.org/wiki/Lycopodiophyta; and i, http://www.discoverlife.org/20/q?search=Bryophyta.


Contents

Foreword xi

Preface xiii

Acknowledgments xvii

Part I: Overview 1

Chapter 1: Introduction 3

Historical summary 3

Modeling philosophy 5

Bioclimatic basis for local community structure 7

Range 9

Richness 13

Major simplifications 14

Principal assumptions 15

Principal findings 15

Part II: Local Species Range and Richness 17

Chapter 2: Local Climate: Observations and Assessments 19

Major biomes of North America 19

Growing season 19

Solar radiation 20

Zonal homogeneity 27

Looking ahead 29

vii


viii R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

Chapter 3: Mean Latitudinal Range of Local Species: Prediction

Versus Observation 31

Introduction and definitions 31

Range of local mean species as determined bylocal distributions about the mean 32

Theoretical estimation of the range with climaticforcing by SW flux only 36

Range of local modal species versus mean of localspecies’ ranges 39

Probability mass of the distribution of observedlocal species 42

Analytical summary for climatic forcing by SW fluxonly 43

Point-by-point estimation of range versusobservation for North America 45

A thought experiment on the variation of SW fluxin an isotropic atmosphere 49

Range of modal species at maxima and minima ofthe SW flux 51

Gradient estimation of range versus observationfor North America 52

Point-by-point estimation of range versusobservation for the Northern Hemisphere 55

Gradient estimation of range versus observationfor the Northern Hemisphere 60

Low-latitude smoothing of range by latitudinalaveraging of the growing season 62

Range as a reflection of the bioclimatic dispersionof species 63

A high-latitude shift in bioclimatic control fromlight to heat? 65

Extension of these range forecasts by use ofmultiple forcing variables 68

A look ahead 68

Chapter 4: Richness of Local Species: Prediction Versus Observation 69

Introduction 69

From continuous to discrete distribution of localspecies 72

Local SW flux as a stationary Poisson stochasticprocess 73

Distribution of C 3 species–supporting radiationintercepted in a growing season 75


C O N T E N T S ix

Moments of C 3 species–supporting radiationintercepted in a growing season 77

Moments of the number of C 3 species–supportingcloud events in a growing season 78

From climatic disturbance to C 3 speciesgermination 79

Parameter estimation 80

Predicted potential richness versus observedrichness 82

The theoretical tie between range and richness 84

Part III: Recapitulation 85

Chapter 5: Summary and Conclusions 87

Precis 87

Mathematical approximations in range calculation 89

Evaluation of range prediction 90

Evaluation of richness prediction 92

Finis 93

Part IV: Appendices: Reductionist Darwinian Modeling ofthe Bioclimatic Function for C3 Plant Species 95

AppendixA: The Individual C 3 Leaf 97

Photosynthetic capacity of the C 3 leaf 97

Mass transfer from free atmosphere to chloroplasts 99

Assimilation modulation by leaf temperature andambient CO2 concentration 104

Exponential approximation to the C 3

photosynthetic capacity curve 104

Potential assimilation efficiency of C 3 leaves 105

The state of stress 107

Darwinian operating state of the individual C 3 leaf 107

The univariate bioclimatic function at leaf scale 108

AppendixB: The Homogenous C 3 Canopy 111

Idealized geometry of the leaf layer 111

Darwinian heat proposition 113

Vertical flux of radiation in a closed canopy 113

C 3 species parameters 116

Bioclimatic function at canopy scale 117

Local evolutionary equilibrium: An hypothesis 118


x R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

AppendixC: Evaluation of the Evolutionary Equilibrium

Hypothesis 121

The equilibrium hypothesis at leaf scale 121

The equilibrium hypothesis at local canopy scale 121

Summary 125

Notation 127

Glossary 137

Bibliography 141

Additional Reading 147

Author Index 149

Subject Index 151


Foreword

This immensely creative and original book addresses one of the most important

problems in evolutionary biology and ecological theory, namely, the observed

decrease of species richness with increasing latitude and the accompanying increase

of the latitudinal range of individual species. Professor Eagleson starts from the

hypothesis that climate is the key conditioning of the above two gradients and that

the answer for a theoretically solid explanation of the variability of species range and

richness may lie in their links with the spatial and temporal variability of climate. Thus

the ambitious goal of this book is to establish the bioclimatic basis of local community

structure. This is indeed a challenging objective that may resist a generally applicable

explanation to specific situations because of the infinite variety of conditions that

may affect a particular species. Recognizing this, Eagleson focuses on the magnitude

and gradient of the maximum possible local species richness: an equally challenging

goal, which if solved, will bring to light a number of patterns found embedded in

immensely complex ecological systems.

Focusing on the forests of the middle and high latitudes, whose growth is basically

limited by light, Eagleson develops a theoretical, analytical, bioclimatic explanation

of the variability of species range and richness over the midlatitudes. This book

presents a theory and framework of analysis that provides synthesis and promotes

understanding of the structure and diversity of ecological communities.

Local climate experiences fluctuations throughout time and acts as a causative

agent for a succession of optimally supported species. From a bioclimatic function

relating a key plant characteristic, the projected leaf area index, to the controlling

climate variable, shortwave radiative flux, Eagleson proceeds to derive a theoretical

prediction of the range of C3 plants as a function of latitude that agrees extremely

well with the observations available from the North America continent.

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xii R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

The maximum possible local species richness is assumed to be controlled by the

local disturbances of shortwave radiative flux, which are, in turn, estimated by Eagle-

son via the statistical structure of local cloud arrivals and their shortwave interception.

Again, the theoretical maximum thus estimated compares very well with the zonal

richness observed for C3 plants in North America.

In summary, the author provides compelling evidence that the biogeography of

plants over middle and high latitudes can be theoretically explained by the space-time

patterns of the shortwave radiative flux. Professor Eagleson’s book is a most original

and exciting monograph that comprehensively explains an extremely important and

challenging problem of ecosystem science.

The approach and style of the book is one based on the best tradition of scientific

research. The enormous complexity of the problem does not distract the author from

his goal of finding an explanation founded in solid theoretical principles. Eagleson

is not afraid of making simplifying assumptions that will then allow for analytical

constructs leading to quantitative understanding of a general type. The assumptions

are carefully stated, and the results are thoroughly tested against large amounts of

data.

Professor Eagleson has written a book whose influence will only increase with the

passage of time. This monumental work will forever change the way that ecologists,

hydrologists, climatologists, and geographers study a set of fundamental phenomena

lying at the intersection of their sciences. Researchers in all those disciplines will be

at the same time challenged and inspired by the search for quantitative explanation

and by the creativity continuously displayed throughout the book. The beauty of the

analysis is probably its greatest intellectual appeal.

Peter S. Eagleson has continuously led hydrology into new and exciting territories

throughout the last 50 years. He has eloquently said:

We need to get away from a view of hydrology as a purely physical science. Life on

earth also has to be a self-evident part of the discipline. In particular, I’m thinking of

vegetation and its powerful interactive relationship with the atmosphere, at both a local

and a global level. In attempting to get the full picture, we must not be afraid to express

the role of plants in our mathematical equations [Hanneberg, 2000].

This wonderful book is science at its best: It attempts to get the full picture and

succeeds beautifully in this effort! It is for me a privilege to introduce it to the scientific

community.

Ignacio Rodrıguez-Iturbe

James S. McDonnell Distinguished University Professor

of Civil and Environmental Engineering

Princeton University


Preface

This is a research monograph and not a textbook. Here I demonstrate analytically

how the observed, opposing, latitudinal gradients in the average range and richness

of local vascular land plant species are (outside the moist-tropical zone, at least) driven

primarily by the local temporal and spatial variability of shortwave radiative flux at

the canopy top. (The term “richness” as used here means the local number of different

vascular land plant species unlimited by the size of the area sampled.) The hypotheses

are simplistic but are nevertheless convincingly accurate in extratropical latitudes

when tested against observations over the continental land surfaces of the Northern

Hemisphere, the only areas tested here.

Species geographical range and local richness lie at the interface of two complex

sciences, biology and geophysics, each having its own established techniques and

traditions of analysis. A rigorous, general explanation of range and richness covering

all the many microclimates of Earth and the myriad species evolved in accommodation

thereto seems impossible at this time; the number of variables is daunting, and the

necessary observational detail is unavailable. This is, or at least was, in earlier years, a

common situation in many branches of engineering, and a variety of useful approaches

exist to deal with such complexity. We must first agree to seek a limited rather than

generalized solution; that is, ask a different and less demanding question! Here I

will then need to limit the independent variables (climate and soil variables, in this

case) to the one or two reasoned to be most important and be willing to accept

the resulting restricted accuracy and/or geographical applicability of the findings.

We shall see in chapter 1 that if the fundamental biophysical relation between the

observable independent (climate) variable(s) and the dependent (species) variable is

locally quasi-linear, then we need know neither its sense nor its true mathematical

form; we can derive an approximate probability distribution of the local species

xiii


xiv R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

and proceed to an approximate and restricted solution of the original problem. This

process is an example of “reductionism” (see the epigraphs on the section I, II, and

III opening pages) and forms the basis for the work described herein.

This volume contains a substantive section (section II) preceded by an overview

(section I) and followed by both a recapitulation (section III) and a set of supportive

appendixes (section IV). Because it is a research monograph rather than a textbook,

the volume more or less follows the path of discovery, describing what does not work

as well as what does, and why, for the failures are often as instructive as the successes.

Section II begins with the presentation, in chapter 2, of latitudinal distributions of

the mean, variance, and latitudinal gradient of the annual zonal SW flux at canopy

top during the growing season, for continental land surfaces in both North America

and in the entire Northern Hemisphere, as derived from NASA satellite observations

and generously prepared for use here by my longtime Massachusetts Institute of

Technology colleague and friend, Dara Entekhabi.

In chapter 3, I employ a local linearization of the bioclimatic function (derived in

the appendixes from simplified biological behaviors) relating a physical property of

separate C3 species to their saturating SW flux. This permits derivation of the standard

deviation of the local frequency distribution of species as being directly proportional

to the standard deviation of the local annual SW flux and thus, from local flux ob-

servations, to the associated “standard deviation of latitude,” as measured in degrees.

These transformations provide the scale by which to estimate local range. Latitudinal

oscillations in both the mean and variance of the observed local seasonal SW flux give

“point-by-point” predictions of range that are wildly oscillating. However, elimina-

tion of these local flux oscillations in favor of flux gradients reveals underlying linear

trends and range gradients, yielding close agreement, in both North America and the

Northern Hemisphere, with the widely referenced North American observations of

Brockman [1968] over their full span of 41◦N latitude.

Chapter 4 employs the role that ground-level SW flux variations play in both seed

germination [Pickett and White, 1985] and the follow-on stressing of the emergent

species to estimate the potential number of local species, acknowledging that the actual

number of local species will be less than the potential by virtue of that unknown

(and/or unaccounted for) myriad of special local conditions referred to earlier. I

derive this potential from local temporal variations in the pixel-scale atmospheric

interception of solar radiation (and hence in the heat) during the growing season,

when represented as a stationary time series of independent and Poisson-distributed

arrivals of cloudy periods. Assuming the total energy intercepted annually by the

random number of annual cloud events to be gamma distributed (this assumption

does not weaken the analysis substantially as the gamma distribution can represent

a variety of shapes), the shape parameter, κ , of the latter must be estimated. I do so

from existing similar analyses of local North American rainstorms and, with it, obtain


P R E F A C E xv

the first two moments of the cloud disturbance frequency as an inverse function of the

variance of the local annual SW flux. From these moments, I estimate the maximum

number of (assumed normally distributed) local annual stressful disturbances to be

approximated as their mean plus (at 99% probability mass) 2.5 standard deviations

therefrom. This formulation predicts quite closely the maximum envelope of the

observed number of local vascular plant species over the 48◦ of latitude in North

America encompassed by the work of Reid and Miller [1989]. The theoretical relation

of local range to local richness is found to be inverse through the derived nature of their

separate dependencies on the variance of local annual SW flux, thereby corroborating

the observation of Rapoport [1975].

Chapter 5 presents a set of paired summaries of the major issues considered along

with the associated conclusions derived herein, plus mention of a few promising

related, but unresolved, problems.

The appendixes are devoted to reductionist modeling of the bioclimatic process by

which radiation drives the conversion of carbon dioxide into solid plant matter. Be-

cause of their predominance, at least in the humid and shady habitats [e.g., Ehleringer

and Cerling, 2002], I consider only vascular plants having the C3 photosynthetic

pathway and examine their behavior at two scales: individual leaf (Appendix A) and

homogeneous canopy (Appendix B). It is in Appendix A that I draw heavily on my

previous hypotheses [Eagleson, 2002]. There I (1) review the generalized geometry

of the classic leaf-scale C3 photosynthetic capacity curve, (2) identify the principal

species variable to be the projected leaf area index and the principal climatic forcing

to be incident SW radiation, and (3) arrive at a generalized bioclimatic function at

leaf scale that relates local C3 species to average local incident SW radiation in the

growing season such as to maximize unstressed productivity. Appendix B expands

the leaf-scale development to the full homogeneous canopy.

In Appendix C, I find and verify, using a small sample of data from the literature,

that the leaf-scale bioclimatic function is applicable across both of the considered

scales, provided that the CO2 supply and demand are both maximized and equal.

I call this the “evolutionary equilibrium hypothesis” and suggest it as a possible

quantification (only for the case of C3 plants, of course) of so-called punctuated

equilibrium [Eldredge and Gould, 1972; Gould and Eldredge, 1977]. Except for

Appendix A, the monograph is new work.

My interest in the geographical distributions of species range and richness was

stimulated by the writings of Stevens [1989] and Wilson [1992], who left me with

their sense that the problems were related, were among the great theoretical problems

of evolutionary biology, and at those times, were unsolved. Accepting this as a personal

challenge, I began this work in 2002 and was delighted to find them still unsolved as

late as 2006, at least [Carpenter et al., 2006]. With this monograph, I hope to convince

the reader that, at least for C3 plants at North American latitudes, this is no longer the


xvi R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

case. I also hope to convince the reader that the science of ecology, lying as it does at

the interface of biology and Earth science, has much to gain from practitioners skilled

in mathematics and physics (and from their cousins in engineering science) as well

as in the usual chemistry and biology.

My apologies for the difficult (if not impenetrable) notations brought on at least in

part by the need to average in four dimensions.

Peter S. Eagleson

Cambridge, Massachusetts


Acknowledgments

W ithout the continuing love and unselfish personal sacrifice of my wife, Beverly,

this work would never have been completed. She must share whatever credit

ensues, while I alone, of course, am responsible for the inevitable errors and omissions.

I wish to thank four friends and colleagues for their unheralded contributions

to this work: Dara Entekhabi (Massachusetts Institute of Technology), for his most

generous donation of time and effort in providing the reduced satellite data used

herein as well as frequent advice on how to use them; Ignacio Rodrıguez-Iturbe

(Princeton University), for being a valued sounding board for my ideas, my guide to

the important people and ideas of modern ecology, and, as my closest friend for almost

40 years, a constant source of advice, encouragement, and inspiration; the late C. Allin

Cornell (Stanford University), for long ago making the power of probability-based

decision accessible to me through both personal tutelage and the clarity of his classic

textbook [Benjamin and Cornell, 1970]; and finally, John MacFarlane (Massachusetts

Institute of Technology), for providing the beautiful line drawings that are critical to

the transmission of these ideas.

I must also thank the anonymous reviewers of the manuscript, whose thoughtful

comments, corrections, and suggestions have improved the finished product measur-

ably.

Finally, I am indebted to the Massachusetts Institute of Technology Department

of Civil and Environmental Engineering, for generous financial assistance with

manuscript preparation through resources of the Edmund K. Turner Professorship,

and to the students and faculty of the department’s Parsons Laboratory for Environ-

mental Science and Engineering, who have graciously tolerated me “hanging around”

after the ball was over.

xvii

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P A R T I

OVERVIEW

Newtonian-Darwinian Synthesis

I suggest that particularity and contingency, which characterize the ecologicalsciences, and generality and simplicity, which characterize the physical sciences,are miscible, and indeed necessary, ingredients in the quest to understand hu-mankind’s home in the universe.

Harte [2002, p. 34]

Universal Laws of Life?

. . . it is reasonable to conjecture that the coarse-grained behavior of living systemsmight obey quantifiable universal laws that capture the system’s essential features.

West and Brown [2004, p. 36]

1


C H A P T E R 1

Introduction

Historical summary

In 1975, Eduardo Rapoport summarized and analyzed the observed geographical

patterns in species’ distribution of both plants and animals. Among other findings,

he reported that with increasing latitude, the richness (also often referred to as di-

versity, which is “richness” in the number of different species, with each species

weighted by the number of like individuals present in the area) of species decreases,

while the latitudinal range of individual species increases. Using a series of simple

ecophysiological models, Woodward [1987] explored his own conclusion that climate

exerts principal control on the distribution of major vegetation types but arrived at

no sense of which climate variable was dominant. Finally, in their assessment of

biodiversity in a warmer world, Svenning and Condit [2008] found that little direct

evidence of what causes range limits had, at that date, been incorporated into models

of the impacts of global warming.

Matching observed exceptions to Rapoport’s [1975] separate latitudinal gradients

of richness and range for common taxa, Stevens [1989] posited an ecological con-

nection between the two gradients. He observed the correlation between north-south

range and latitude to hold for a wide variety of taxa and therefore to be the fundamen-

tal, independent relationship. He gave it the name “Rapoport’s rule.” Using trees as an

example (see Figure 1.1a), Stevens reasoned that their tolerance of variable climatic

conditions (he considered only precipitation and temperature) had to span the sea-

sonal climatic variations experienced in their habitat and that therefore, to paraphrase

him, the large latitudinal extent of high-latitude organisms (i.e., their “range”) results

from the selective advantage to those individuals having the wide climatic tolerances

needed for success in a particular high-latitude location. Stevens [1989] traced the

3


4 R A N G E A N D R I C H N E S S O F V A S C U L A R L A N D P L A N T S

FIGURE 1.1 Observed Latitudinal Gradients of Tree Species’ Range and Diversity: a. Ranges ofLocal Tree Species in North America [Stevens, 1989]; Error Bars Define ±1 Std. Error of the MeanLocal Range, N = Number of Sites (data from Brockman [1968]); With permission of The Universityof Chicago Press: AMERICAN NATURALIST, vol. 133, issue 2, February, 1989, pp. 240–256, Fig. 1 (topleft): c© 1989 The University of Chicago Press: b. Local Diversity of Tree Species [Enquist and Niklas,2001] (from Global Data of Gentry [1988, 1995]); Adapted by permission fromMacmillan PublishersLtd: NATURE, vol. 410, 5 April 2001, pp. 655–660, Fig. 1a: c© 2001.

finding of the latitudinal trend in species’ richness to the 1878 work of Wallace and

pointed out its later observational confirmation by a host of others. For a more recent

example, see Figure 1.1b, reproduced from Enquist and Niklas [2001], who used

the extensive data for trees compiled earlier by Gentry [1988, 1995]. However, there

remains continuing lack of agreement on the cause of the latitudinal trend in richness

[Roy, 2001]. For example, Fischer [1960] found species richness to be inversely re-

lated to local seasonal climate variability; Wright [1983] found that richness followed

the amount of energy available; Currie and Paquin [1987] concluded not only that


C H A P T E R 1 • I N T R O D U C T I O N 5

species richness is controlled by the total available energy, but also that seasonal cli-

matic variability has no effect; and Scheiner and Rey-Benayas [1994] found species

richness and climate variability to be directly (as opposed to inversely) related.

Addressing the species richness gradient, or, as he called it, “tropical preeminence,”

Wilson [1992, p. 199] found the cause to be “one of the great theoretical problems

of evolutionary biology,” noting that “many have called the problem intractable,” and

attributed its likely cause to geographic variations in productivity. For a probable basis

for Rapoport’s rule, he, too, pointed to the local climatic variability introduced by the

seasons.

Huston [1994] presented an exhaustive review of published work on biological

richness (approximately 2000 references covering the vast research literature of the

20th century), in which he also sought to explain [Huston, 1994, p. 2] “the regulation

of species diversity and why the number of co-occurring species varies under different

conditions.” He postulated the total species diversity of a local community to be given

by the sum of the diversities of separate classes of species present, in which case, the

same total diversity could be obtained by different combinations of the classes, and

there would be no universal explanation of species diversity.

A major advance in the theory of biodiversity came in 2001 in the form of unified

neutral theory [Hubbell, 2001] (hereinafter referred to as the Neutral Theory, when

capitalized thusly), which determines, from generalized population statistics, the rich-

ness and abundance of species in a single metacommunity. Assuming that nutritional

(i.e., “trophic”) similarity among members of a particular ecological metacommunity

makes other differences among them irrelevant to their presence, Neutral Theory

predicts the richness and abundance of all species in that metacommunity given a

single observation from the same metacommunity of (for example) the abundance of

a single species.

Finally, the Millennium Ecosystem Assessment [Carpenter et al., 2006, p. 257–

258] finds that “we lack a robust theoretical basis for linking ecological diversity to

ecosystem dynamics.”

Modeling philosophy

We propose here that, to the zeroth order, it is the species dependence of the energy

needed for seed germination and (as we shall see) for maximum unstressed produc-

tivity that locally governs both the richness and range of species due to the local and

spatial variability of incident radiation during the growing season. Local variation

in the availability of water and/or nutrients is assumed to be reflected in the local

standing biomass but, to the order of these approximations, not in the selection of the

species present.



FIGURE 1.2 Typical Photosynthetic Be-haviors (see Figure A.4).

We choose for analysis a single vegetation class (i.e., a “functional type” in Huston’s

[1994] classification), namely, vascular land plants, which comprise approximately

98% of all extant land plants, as is shown in the frontispiece. This should include the

predominant trees of at least the middle and high latitudes (i.e., the temperate and

boreal forests), for which observations are plentiful and thus can provide a meaningful

test of our proposition. In a further restriction of this functional type, we consider

only the so-called C3 class of vascular vegetation because the class constitutes about

93% of all living vascular land plants (see the frontispiece). Plants utilizing the C3

(i.e., Calvin cycle) photosynthetic pathway predominate in humid and shady habitats

in the form of deciduous trees and shrubs [Ehleringer and Cerling, 2002], and they

dominate almost exclusively in alpine and cold regions as evergreen trees and shrubs

[Li et al., 2004]. The C3 plants also predominate in submerged habitats, where they

have as great a diversity as in the terrestrial environment [Keely, 1999]. Plants utilizing

the C4 (i.e., Hatch-Slack) pathway dominate in dry and sunny habitats as grasses and

sedges. Finally, plants utilizing the CAM (i.e., Crassulacean acid metabolism) pathway

dominate in very arid regions as succulents, and in low light as epiphytes, but are not

an appreciable part of the global carbon cycle [Ehleringer and Cerling, 2002].

Fortunately for the current purposes, each C3 species has a distinctive, saturating,

leaf photosynthetic capacity function defining, to zeroth order, a Darwinian optimum

state at the function’s asymptote intersection, which is at once unique to that species,

stressless, maximally productive, and maximally efficient (see Figure A3). The rising

asymptote is common to all C3 species and thus, containing all the optima, serves

as our basis for competitive natural selection among other C3, and hence as func-

tionally analogous, local species. This saturating photosynthetic capacity function is

illustrated, along with its (dashed) asymptotes, in the sketch of Figure 1.2, where it is

contrasted with the C4 and CAM classes of species (see earlier discussion), far less

common at these latitudes, and most of which do not saturate. We return to this figure

later in order to illustrate our model of the selection process. Normalization of the C3



photosynthetic capacity curve is carried out in Appendix A and is shown graphically

in Figure A4.

Within these restrictions, we seek primarily to provide a theoretical, analytical,

bioclimatic explanation of species range and richness over the extratropical latitudes.

In their full bioclimatic detail, these problems are dauntingly complex, so we consider

instead a highly idealized and reduced bioclimatic system whose average biophysi-

cal processes obey Darwinian imperatives. By concentrating on capturing the sense

and form of the gradients, rather than their precise magnitude, we admit additional

corresponding mathematical approximations such as space and time averaging, lin-

earization, and order-of-magnitude analysis.

Horn [1971, p. 121] pointed out that “a frontal assault on the first factor in a

multidimensional problem may show that many of the presently known patterns

can be understood in terms of that factor alone.” Wilson [1965, p. 59] defined “the

search strategy employed to find points of entry into otherwise impenetrably complex

systems” as reductionism, and as such, “reductionism is the primary and essential

activity of science.” The reductionist approach is common to physics and engineering

[Harte, 2002] but is anathema to many biologists [e.g., Anderson, 1972]. Prominent

among the latter was the pioneering evolutionary synthesist Ernst Mayr (1904–2005),

noted also for his criticism of reductionists, who tried to analyze biology in the manner

of physics. This issue has resurfaced with the growth of interest in Earth system

science, which, in the words of Harte [2002, p. 29], “seeks no less than a predictive

understanding of the complex system comprising organisms, atmosphere, fresh water,

oceans, soil, and human society.” To find a useful way through this overwhelming

complexity, Harte [2002] calls for the development of simple, mechanistic models

that capture the essence of the problem but not all the details. West and Brown [2004,

p. 36] agree that “such idealized constructs would provide a zeroth-order point of

departure for quantitatively understanding real bioclimatic systems,” and we subscribe

to this viewpoint herein, adopting their use of “zeroth order” as broadly descriptive

of our level of approximation. According to MacArthur [1972, p. 127], “the ranges

of single species would seem to be the basic unit of biogeography,” and hence we will

begin there.

Bioclimatic basis for local community structure

Tilman [1982, 1988] suggested that the particular local species having the lowest re-

quirement among multiple local species for a single, common limiting resource (such

as light, water, nitrates, or phosphates) will always be competitively dominant locally

and that the local community structure results from one or more such competitive

interactions. Stevens [1989] was perhaps the first to reason that certain key aspects of

community structure, namely, the local variability of species range and richness, may



result from spatial and temporal variability of the local climate. We follow these leads

in this work. In keeping with a zeroth-order approach, our use of the term “climate”

refers to the statistics of the single most important external influence on local vege-

tation growth from among such factors as the availability, over the growing season,

of light, CO2, heat, water, or nutrients. (The concentration of CO2 at the canopy top

is not the same everywhere, being distinctly seasonal in the Northern Hemisphere

and essentially without seasonality in the Southern Hemisphere, but having a diurnal

fluctuation in both hemispheres [Bonan, 2002]. The concentration falls steeply within

the canopy from a maximum in the free atmosphere, and at least in tropical forests,

from decay at the forest floor, to a minimum at the lowest leaf, or at some internal

level in the tropics.) It is our view that over much, if not most, of Earth’s surface, this

is clearly the shortwave radiative flux (i.e., “light”) due weakly to its selective germi-

nation of species by production of heat on absorption by viable seed, and strongly to

its subsequent support of stable emergent plant matter through C3 photosynthesis (see

Figures 1.2 and Appendices A–C). We assume that to this order, these processes are

modulated, rather than controlled, by any local unavailability of the other resources

listed previously.

In our zeroth-order approximation of the C3 species, as is shown in Figure 1.2,

we replace the actual photosynthetic capacity curve of species C23 by its asymptotes,

thereby fixing the optimum operating state for species C23 at the shortwave (SW)

flux, corresponding to the asymptotic intersection. If this flux is the long-term (i.e.,

multiseasonal) local average, I0, then C23 will be the modal species at that location,

and in Figure 1.2, we refer to this flux as I 20 .

In any year, the single-season average SW flux, I0, may be, at this same location,

either larger or smaller than I 20 , thereby optimally supporting species C3

3 or C13 in

that year, respectively. However, in the long-term average at location 2, only C23

can be optimally supported at the average SW flux I 20 . Species 1 will be unstressed

(triangle) and thus stable at location 2 but underproductive compared to many other

stable species there, and species 3 will be stressed (diamond) and thus absent at

location 2.

With such reasoning, we arrive at a one-sided distribution of stable species, sup-

ported by all I0 ≤ I0 at each location, which proves to be key to our predictions of both

range and richness. (It seems appropriate at this juncture to point out that our sim-

plifying omission of respiration from the photosynthetic capacity function prevents

identifying other C3 species having the same productivity but differing respiration.

Also, because we are interested in the meridional variation of range and richness, our

selection of C3 species behavior for our model on the basis of their global predom-

inance will overlook the very large numbers of C4 and CAM plants in the tropical

latitudes.)



FIGURE 1.3 Taylor Se-ries Approximation to theBioclimatic Function.

Range

In the case of species range, we assume that the controlling mechanism is this “in-

terannual” variability (about a constant long-term mean value) of light during the

growing season, and we begin by using the statistical technique of “derived distri-

butions” to estimate an approximate relationship between the (causative) mean and

variance of this local light (i.e., “climate”) and the (resulting) mean and variance of

stable, local species. This approximation replaces the mean of a function of random

variables by the same function of their means and is valid for small coefficients of

variation of those random variables and for small nonlinearity of the function [see,

e.g., Benjamin and Cornell, 1970]. We apply this Taylor series approximation with

respect to independent variations in climate as illustrated in Figure 1.3. Its analytical

basis follows.

We let c represent the randomly annually variable local light as averaged over

each annual growing season (i.e., c ≡ I0), and we let s be a continuously distributed,

single-valued, numerical representation of the resulting optimally supported local

C3 species in each season. These stressless species are identified from the normal-

ized C3 photosynthetic capacity curve as averaged over the depth of the canopy

[Eagleson, 2002, Appendix H] (Appendices A and B) by their surrogates, the projected

leaf-area indices (i.e., s ≡ βLt ), producing saturation at each of the local seasonal-

average SW fluxes, I0. We call them the “optimally supported” species at these fluxes



and relate them through the bioclimatic function

s = g(c). (1.1)

As described earlier, in any year, c has a single value locally, and thus only one local

species is being supported optimally. Among species already existing there, those for

which the saturating SW flux is greater than the local annual SW flux will be stressed

and thus unstable there this year, while those for which the saturating flux is smaller

than the local annual flux will be unstressed and thus stable but underproductive there

now (see Figures 3.2 and 3.4a). Actually, of course, the local species will be discretely

distributed in s, but we postpone that consideration until it becomes necessary for

counting purposes, when dealing with the species richness issue. At that time, we

must also consider the geographical scale of the species richness count because (all

else remaining constant) species count is observed to rise and eventually saturate with

increasing area of observation [Huston, 1994].

In the next growing season, when the climate takes on a new value, a new species

will be optimally supported. Every new growing season having a previously expe-

rienced climate will support no new optimum species (provided that species still

survives), and any prior season’s optimum species may or may not survive to the

present time. In this way, we can imagine a stable distribution of local species evolv-

ing over the years, which reflects the characteristic annual variability of local climate

(see Figure 3.4a). Issues of species hardiness in the face of stress will control surviv-

ability and hence the presence or absence of certain predicted off-mean local species

at a given time, particularly in the tails of the distribution. We will discuss this issue

further when interpreting our results.

Referring again to Figure 1.3, if the coefficient of variation, CV, of c, that is,

CV(c) ≡ [VAR(c)]1/2/E(c), is small enough at any given location, c is likely to lie

close to its long-term mean value, E(c) ≡ c, there. We may then expand g(c) in a

Taylor series about this mean climatic state to obtain [see, e.g., Benjamin and Cornell,

1970]

g (c) = g (c) + (c − c)dg (c)

dc

∣∣∣∣c+ (c − c)2

2

d2g (c)

dc2

∣∣∣∣∣c

+ . . . . (1.2)

If, as we assume, the curvature of g(c) at c (i.e., d2g(c)/dc2|c) is small, the third

and higher terms of equation (1.2) may be neglected. Since the expected value of the

second term of equation (1.2) is identically zero, the approximate first moment of

equation (1.1) is then

s ≡ E [s] ≈ g (c) , (1.3)

in which s is the local community-average species, demonstrating that under these

approximations, the mean of the bioclimatic function is equal to the same function of



its means. Similarly, because the variance of the first term of equation (1.2) is zero,

the approximate second moment of equation (1.1) becomes

σ 2s (s) ≈ σ 2

c

[dg(c)

dc

∣∣∣∣c

]2

. (1.4)

with c always close to c locally, as we have assumed,

dg(c)

dc

∣∣∣∣c≈ dg(c)

dc. (1.5)

It is important to note that equations (1.3) and (1.4) are independent of the sign

of dg (c)/

dc, leaving the geometric form of g (c) unconstrained at this level of

approximation. Because VAR [aY ] ≡ a2VAR [Y ] [see Benjamin and Cornell, 1970],

equations (1.4) and (1.5) allow estimation of the standard deviation of local species

to be

σs (s) ≈ σc

∣∣∣∣

ds

dc

∣∣∣∣, (1.6)

in which σs (s) is the standard deviation of the local species given in species units.

As we shall see in chapter 2, in our idealized, unchanging, and zonally homogeneous

world, there is also a one-to-one relationship between the zonal average climate, c, and

the associated zonal latitude, �, and hence, given equation (1.3), there is a one-to-one

relationship, s = h (�), between the zonal average species, s, and �. This allows us

to change the variable in terms of which σs is expressed from σs (s) to σs (�), which

is quite convenient for the current purposes. With local linear approximation, this is

written, as for equation (1.6),

σs (�) ≈ σs (s)∣∣∣

dsd�

∣∣∣

= σs (s)∣∣∣

dsdc

∣∣∣

∣∣ dc

d�

∣∣. (1.7)

Finally, combining equations (1.6) and (1.7) yields the zeroth-order approximation

σs (�) ≈ σc∣∣dc

/

d�∣∣, (1.8)

which forms the basis for our estimation of the range of the mean species at latitude,

�0. Without considering species stability, the local distribution of species is double

sided, resulting in this range being formed as shown by R s|�0 in Figure 1.4, and which

we note to be independent of the form of the bioclimatic function, s = g (c). While not

needed here for our zeroth-order theoretical estimation of species range and richness,

we include identification of the full bioclimatic function in the appendices. There we

derive the optimal form of g (c) from a proposition that equates the maximums of

plant CO2 supply and demand in a temporary state we call “evolutionary equilibrium.”

In its simplest form, this results in the zeroth-order bioclimatic function

βLt = g (I0) , (1.9)



FIGURE 1.4 Idealized Range ofMean Local Species (for the Case I 0↓as �↑, and without consideration ofspecies stability).

which is the theoretically defined form of equation (1.1), and its approximate local

average

βLt ≈ g(

I0)

(1.10)

is the theoretically defined form of equation (1.3). Once again, in these equations,

βLt is the (dimensionless) species-defining, total horizontal leaf-area index of the

particular local species that is optimally supported by the climate-defining local

seasonal shortwave radiative flux at the canopy -top, I0 (β is the cosine of the canopy-

average leaf angle, and Lt is the canopy leaf-area index, i.e., the total single-side leaf

area per unit of projected canopy area). The local average of the species variable is

βLt , and the local multiseason average of the climate variable is I0.

This model is essentially an expression of Neutral Theory [Hubbell, 2001] in that

it implicitly assumes the equivalent per capita fitness for all local species unstressed

on the local average. We refer to our model as a neutral theory (lower case intended)

in that, contrary to Hubbell [2001], its basis for prediction of local species richness

is local observations of light variability, rather than vegetation observations at a

different scale. We must remember that our bioclimatic function is single valued in

the assumed species-defining βLt , while in reality, it is likely that multiple species

share the same leaf-area index and instead are differentiated productively by their

superior utilization of resources neglected here such as water or nitrogen. Kraft et al.

[2008] present evidence supporting a nonneutral view of tropical forest dynamics in

which co-occurring species display differing ecological strategies.



FIGURE 1.5 Range and Richness of Vascular Land Plants on the Continents. Range: Theory isfor C 3 vascular land plants in N.A.; observations are for all trees (open circles) in N.A. [Brockman,1968]. Richness: Theory is for C 3 vascular land plants in the N.H.; observations are for all vascularland plants (solid circles) in the W.H. as presented by Huston [1994, Figure 2.1, p. 20]† based uponReid and Miller [1989]†† and for all trees (pluses) in the N.H. [Gentry, 1988, 1995] as scaled in Fig-ure 4.1. †Reprinted with the permission of Cambridge University Press. ††Walter V. Reid and KentonR. Miller, 1989, Keeping Options Alive: The Scientific Basis for Conserving Biodiversity, World Re-sources Institute, Washington, D.C., using data from Davis et al. [1986] and WRI//IIED [1988], bothunavailable to the author. With kind permission of the World Resources Institute.

For the particular case in which the right-hand side of equation (1.7) increases

monotonically with s, we illustrate in Figure 1.4 (lower abscissa) the use of σs (s) to

estimate Rs|�◦ (s), which is the range, in species units, of the mean species, s, to be

expected at latitude �◦. In using this range of the mean to compare with Brockman’s

[1968] observed mean of the ranges (see Figure 1.1a), we assume zonal homogeneity

of climate. We note here that Figure 1.4 is idealized for illustrative purposes in its

use of normal distributions of local species, truncated everywhere at ±ns standard

deviations, σs (s), from the local mean, s. Actually, as we have discussed earlier,

species optimally supported by I0 > I0 locally will be stressed on average and thus

assumed absent, leaving the distribution of observed s as single sided. To estimate

Rs|�◦ (�), which has the same range as Rs|�◦ (s) but is measured in units of � (upper

abscissa in Figure 1.4), we use the transformation of the independent variable from

s → �, as given by equation (1.7) and embodied in equation (1.8). Note also that as

a result of equation (1.9), σc ≡ σI0 .

We compare our theoretical prediction of range with the Brockman [1968] obser-

vations in advance in Figure 1.5.

Richness

It has long been recognized that local intraseasonal disturbances in light and hence

heat play an important role (among many other factors) in the local germination of



terrestrial plant seedlings [see, e.g., Tilman, 1982; Larcher, 1983]. Following our

apparent success in chapter 3 when identifying thriving species with time-average

stresslessness, we assume here as well that only those light pulses i0 for which i0 ≤ I0

will germinate and support stable seedlings leading to countable local species. We

thus assume the maximum possible zonal species richness, max �s , to be equal to

the zonal-average maximum number, νmax, of those particular, independent, discrete

pulses, i0 ≤ I0, in the local shortwave radiative flux occurring during that basic unit

of ecological time, the growing season. Defining the light pulses as a continuous

series of supportive i0 ≤ I0, followed by an unsuppportive i0 > I0, there will be, on

average, an equal number of each in a season. This number is estimated (chapter 4)

assuming a Poisson distribution of independent local i0 ≤ I0 arrivals and a gamma

distribution of their seasonal shortwave interception to be

max �s∼= mν + niσν = ( I� − I0)2

σ 2I0

[

1 + 1

κ

]

+ ni( I� − I0)

σI0

[

1 + 1

κ

]1/2

,

(1.11)

which is termed “maximum” due to the possible presence of some serial depen-

dencies and biologically insufficient strengths among the pulses. I� and I0 are the

growing season–average, top-of-the-atmosphere and canopy-top shortwave fluxes,

respectively; κ is the shape parameter of the gamma distribution of seasonal short-

wave interception by the individual cloud events; and ni is the number of standard

deviations of this distribution incorporating the desired probability mass. This is an

inversion of a successful existing model for predicting annual local rainfall statistics,

given the observed frequency and properties of individual local storms [Eagleson,

1978]; here we know the statistics of the observed SW flux and seek the maximum

frequency, mν + niσν , of its seasonal fluctuations. We compare this theoretical max-

imum with the zonal richness observed by Gentry [1988, 1995], as summarized in

advance here in Figure 1.5.

Once again, we must remember that our assumed single-valued relationship be-

tween light and species will cause us to misrepresent the number of local species

wherever the supply of water and/or nutrients controls productivity, which happens

in the tropics, as has been shown by Kraft et al. [2008].

Note in equation (1.11) the inverse relationship of σI0 to the local limit of species

richness in contrast to its direct relationship to the range (see equation (1.8), in which

σc ≡ σI0 ). Therein lies the theoretical basis for the opposing latitudinal gradients of

range and richness previously observed by Stevens [1989] and others.

Major simplifications

Our reductionist approach to the biophysics of these problems invokes many ideal-

izations in addition to the mathematical approximations introduced earlier. Principal



among these physical simplifications are the following: (1) species interactions, in

which the analysis allows for polycultures but neglects both the competitive interac-

tions that may occur between different species and the pervasive “more is different”

effect [Anderson, 1972] of multicultural symbiosis; (2) predator neglect, which omits

the effects of insects and other animals, including man, acting largely to reduce

theoretical productivity; (3) disease and fire neglect, thereby further overpredicting

productivity; (4) light as the limiting resource, which restricts concern to forest sys-

tems in which the local availability of water, nutrients, heat, and carbon dioxide is not

limiting and assumes the canopy-top atmosphere to be an effectively infinite reservoir

of CO2 at a concentration that is constant in both space and time; (5) a neutrally stable

atmosphere, which omits buoyant convection; (6) lateral advection of energy neglect,

which assumes only vertical local exchanges with the atmosphere; (7) a climate un-

affected by vegetation, which omits feedback from the surface; and (8) a spatially

homogeneous canopy structure, in which biophysical relations are developed for ad-

jacent leaf layers and applied without modification throughout monocultural canopies

in terms of spatially averaged crown structure and shade-induced variations in leaf

photochemistry.

Principal assumptions

Principal assumptions include the following: (1) maximization of net primary produc-

tivity, in which the governing selection mechanism is assumed to be a maximization of

the probability of reproductive success, as expressed through the surrogate maximiza-

tion of biomass, and hence seed, productivity at optimum average leaf temperature

and with adequate water and nutrients as well as negligible respiration; (2) bioclimatic

function, whereby the governing bioclimatic relation is derived for an assumed stress-

less, productivity-maximizing steady state, which yields a single-sided distribution

of stable local C3 species when forced by a normally distributed annual SW flux (it

is considered to be single valued and linear over the local range and only its sense

need be known); (3) range, in which the coefficient of variation of the local range is

small; (4) richness, whereby the number of local seasonal SW flux pulses, i0 ≤ I0,

sets the maximum number of local C3 species through their stimulation of selective

germination and stressless follow-on support of the struggling emergent plant matter;

and (5) flux pulses, which are intraseasonal flux pulses of intensity i0 ≤ I0 that arrive

locally at Poisson-distributed intervals and with gamma-distributed energy.

Principal findings

Regardless of our many approximations and unverified assumptions, we will con-

firm in conclusion that, at least within the latitudinal range 25◦N ≤ � ≤ 60◦N of



continental North America, both range and richness owe their latitudinal gradients

to the local variability (both temporally and latitudinally) in shortwave radiative flux,

produced by transient local cloud events and solar altitude, as outlined above, respec-

tively. Our conclusive demonstration of this is previewed here by our very favorable

comparison of theory and observation for species range and richness over extratropi-

cal latitudes, as presented in Figure 1.5. It seems from this work that the spatial and

temporal variabilities in shortwave flux may be the true basis for the biogeography of

plants over at least the extratropical fraction of Earth’s vegetated land surface.


P A R T I I

Local Species Range and Richness

“Zeroth-Order” Analysis

A frontal assault on the first factor in a multidimensional problem may show thatmany of the presently known patterns can be understood in terms of that factoralone.

Horn [1971, p. 121]

Such ideal constructs would provide a zeroth-order point of departure for quan-titatively understanding real biological systems . . .

West and Brown [2004, p. 36]

17


C H A P T E R 2

Local Climate: Observations and Assessments

Major biomes of North America

Figure 2.1 sketches the approximate boundaries of the major biomes of North Amer-

ica, as adapted from maps presented by Bailey [1997]. This figure makes qualitatively

apparent the zonal heterogeneity of the actual bioclimate owing to such irregularly

distributed influences as land surface topography and land-sea interactions. Never-

theless, to enable our zeroth-order analysis to go forward, we represent these biomes

as zonally homogeneous, with the approximate latitudinal boundaries listed in Ta-

ble 2.1 and shown as dashed lines in Figure 2.1. After presenting the observations

of pixel climate, we will make a more quantitative assessment of the actual zonal

homogeneity.

Growing season

As just stated, behavior of the land surface is idealized to be independent of longitude

in this work. Accordingly, we estimate the distribution of a zonally homogeneous, but

meridionally variable, nominal growing-season length, mτ , from the map of Trewartha

[1954, p. 46]. We present those estimates here in Table 2.2, centered commonly on the

summer solstice, Julian day 173 (22 June). Alternatively, when focusing our attention

on the warmer latitudes, say, below 35◦, we may use the summer solstice as the

centering date, but for the colder latitudes, say, above 35◦, the autumnal equinox (22

September) because by that time of year, the local ground temperatures will be more

supportive of growth in vascular plants.

19



FIGURE 2.1 Major biomesofNorthAmerica, as adapted fromBailey [1997]. Dashed lines boundapproximately zonally homogeneous biomes.

Solar radiation

To implement the zeroth-order estimation of species range, as outlined in chapter 1,

we select the incident shortwave radiation, I0, at canopy top during the growing season

as the single climatic forcing variable (c in equation (1.1)). This choice is supported

theoretically in the appendices through derivation of the “zeroth-order” bioclimatic

TABLE 2.1 Latitudinal Boundaries of North American Forest Biomesa

Forest Biome Latitude (◦N)

Tundra Northward of 60◦

Boreal 52◦–60◦

Humid temperate 24◦–52◦

Humid tropical 0◦–24◦

aApproximated from Figure 2.1.


C H A P T E R 2 • L O C A L C L I M A T E : O B S E R V A T I O N S A N D A S S E S S M E N T S 21

TABLE 2.2 Estimated Growing Season

Growing Season

Latitude (◦N) Nominal Lengthamτ (days) Periodb(Julian days)

0 365 1–3655 365 1–365

10 365 1–36515 330 8–33820 330 8–33825 200 73–27330 200 73–27335 150 98–24840 150 98–24845 105 120–22550 105 120–22555 75 136–21160 75 136–21165 35 155–19070 35 155–190

aEstimated from Trewartha [1954, Figure 1.35].bCentered on summer solstice (Julian day 173).

function (see equation (C.2)). Satellite remotely sensed solar radiation data (NASA–

Goddard Institute for Space Studies (GISS) International Satellite Cloud Climatology

Project (ISCCP) data set, with modeled modifications) have been reduced (D. En-

tekhabi, personal communication, 2005) to yield global values of annual average

surface all-sky daytime shortwave flux, I0, at the surface (i.e., canopy top), for each

land surface pixel over its associated nominal zonal growing season, mτ (see Table 2.2)

and for each of the 17 years (1984–2000) of this record. The pixels are of equal area

(77,312 km2) and are aligned in 2.5◦ zonal bands, giving a global total of 6596 land-

only pixels distributed latitudinally, as illustrated in Figure 2.2a. Distribution of the

number of land-only pixels in the Western Hemisphere is shown in Figure 2.2b. A

mixture of geostationary and polar-orbiting satellites provides global coverage every

3 hours [Pinker and Laszlo, 1992]. D. Entekhabi (personal communication, 2005)

used these annual pixel fluxes to calculate growing-season values of the following

climatic parameters of interest in this estimation of species range and diversity.

1. The first climatic variable is global zonal average, 〈I0〉, of the annual land

surface (i.e., canopy top) pixel shortwave radiative flux, I0 (hereinafter “SW flux”

or simply “light”). As an example, this is plotted in Figure 2.3, in watts-total (i.e.,

including UV as well as photosynthetically active radiation) per projected square

meter (Wtot m−2, or simply Wm−2), at all latitudes for growing season days 8–338.

Although there is a separate value of 〈I0〉 for each sample year of record at each



(a)

(b)

FIGURE 2.2 (a) Global number of land-only pixels in a zonal band. From NASA–Goddard Insti-tute for Space Studies (GISS) International Satellite Cloud Climatology Project (ISCCP) data set. (b)Number of land-only pixels in a zonal band in the Western Hemisphere. From NASA-GISS ISCCPdata set.



FIGURE 2.3 Global zonal average seasonal canopy-top, pixel, SW flux, 〈 I 0〉 (land only; daytime;growing season days 8–338). From NASA-GISS ISCCP data set, 1984–2000.

latitude, for clarity, in Figure 2.3, we show only bounding values at each �. The

latitudes having this particular nominal growing season (see Table 2.2) are indicated

in this and subsequent figures by the solid vertical lines at � =15◦ and 20◦. A similar

figure (not shown here) has been prepared for each of the separate growing seasons,

mτ , associated with the latitudes indicated in Table 2.2. The temporal sample mean

of the zonal average annual growing-season, pixel, canopy-top, SW flux at each � is

〈I0〉 and is given, for the Northern Hemisphere, in column 4 of Table 2.3. We note

that the mean of the average is identical to the average of the mean, 〈I0〉 ≡ ⟨

I0⟩

.

Also shown in Figure 2.3 are the extensions, to the equator, of the upper-latitude

gradients of⟨

I0⟩

. The significance of their intersection there is important to this work

and will be discussed in chapter 3.

2. The second climatic variable is global zonal average,⟨

σI0

⟩

, of the standard

deviation (over time), σI0 , of the seasonal, pixel, canopy-top, SW flux, I0 (watts-total

per meters squared). As an example,⟨

σI0

⟩

is plotted at all latitudes for growing season

days 8–338 in Figure 2.4. The latitudes having this estimated growing season (Table

2.2) are again indicated by the dashed vertical lines at � = 15◦ and 20◦, and the

values of⟨

σI0

⟩

are given, for the Northern Hemisphere latitudes, �, in column 5 of



TABLE

2.3

Zona

lAverage

ofObserved

PixelC

limatein

theNorthernHem

isphe

rea

Estim

ated

Growing

Num

ber

of〈 I 0

〉 ≡⟨ I 0

⟩ d⟨ σ

I 0

⟩σI 0

∣ ∣⟨ dI 0

/d�

⟩∣ ∣∣ ∣ ∣�

⟨ I 0⟩ / �

�

∣ ∣ ∣

�(◦N)

Season

b(Julianda

ys)

Land

Pixelsc

(Wtotm

−2)

(Wtotm

−2)

(Wtotm

−2)

(Wtotm

−2de

g−1)

(Wtotm

−2de

g−1)

σI 0

/⟨ I 0

⟩⟨ σ

I 0

⟩/⟨ I 0

⟩

01–

365

440.0

9.23

51–

365

450.0

8.57

3.5

101–

365

3347

5.0

8.00

18.73

5.1

4.4

0.03

90.01

715

8–33

837

494.0

8.70

30.23

0.0

1.9

0.06

10.01

820

8–33

843

493.8

9.17

42.20

3.8

1.0

0.08

50.01

925

73–2

7348

484.4

11.95

70.09

3.1

1.3

0.14

50.02

530

73–2

7354

481.3

12.60

63.93

1.7

1.5

0.13

30.02

635

98–2

4845

469.0

13.88

47.43

5.4

4.3

0.10

10.03

040

98–2

4850

438.5

12.85

34.50

8.0

6.5

0.07

90.02

945

120–

225

4940

4.0

12.56

29.38

7.2

7.7

0.07

30.03

150

120–

225

5636

1.5

14.35

22.37

7.8

6.9

0.06

20.04

055

136–

211

5033

5.0

17.46

15.39

5.2

5.0

0.04

60.05

257

.513

6–21

141

324.0

17.12

11.33

5.6

3.9

0.03

50.05

360

136–

211

4431

5.4

16.39

11.06

5.7

7.3

0.03

50.05

265

155–

190

4626

9.2

17.58

16.62

6.9

5.5

0.06

20.06

5

aNASA

-GISSISCC

Pda

taset,19

84–2

000,

land

surfaceon

ly.D

atasetredu

cedby

D.Entekha

bi(persona

lcom

mun

ication,

2005

).Bo

ldfacedvalues

arefrom

Figu

res2.3–

2.6

inclusive.Re

maining

values

inthesecolumns

arefrom

similarfi

guresno

trep

rodu

cedhe

re.

bTable2.2.

c Figure2.2a.

dTimeaverag

eof

Figu

re2.3.



FIGURE 2.4 Global zonal average of the standard deviation of seasonal canopy-top, pixel, SWflux, 〈σ I 0 〉 (land only; daytime; growing season days 8–338). From NASA-GISS ISCCP data set, 1984–2000.

Table 2.3. In Figures 2.4–2.7, the North American value of the climate variable for the

same growing season (assumed to be at the same �) is indicated by the plotted circle.

Note (Figure 2.2b) that at this latitude, there are only three land pixels to average, and

the oceanic influence is therefore large.

FIGURE 2.5 Global standard deviation across longitudes of the average annual seasonalcanopy-top, pixel, SW flux, σ I 0 (land only; daytime; growing season days 8–338). From NASA-GISSISCCP data set, 1984–2000.



FIGURE 2.6 Global zonal average of the meridional gradient of the average annual seasonalcanopy-top, pixel, SW flux, 〈d I 0/d�〉 (land only; daytime; growing season days 8–338). FromNASA-GISS ISCCP data set, 1984–2000.

3. The third climatic variable is global zonal standard deviation (across all pix-

els in the common zone), σI0, of the mean seasonal, pixel, canopy-top, SW flux, I0

(in watts-total per meters squared). This standard deviation is plotted for Northern

Hemisphere growing season days 8–338 in Figure 2.5 and is given at each Northern

FIGURE 2.7 Global zonal average of the daytime average SW flux at the top-of-the-atmospherefor June–September inclusive, I � . From NASA-GISS ISCCP data set, 1984–2000.



Hemisphere � in column 6 of Table 2.3. Note, in Figure 2.5, the large difference

between σI0at 17.5◦for North America compared to the entire Northern Hemi-

sphere, suggesting, at least for this latitude, the relative climatic homogeneity of the

former.

4. The fourth climatic variable is the global zonal average,⟨

d I0/

d�⟩

, of the

latitudinal gradient of the mean annual (i.e., growing season) pixel, canopy-top,

SW flux, I0 (in watts-total per meters squared). This is plotted for growing season

days 8–338 in Figure 2.6, and its absolute value is given at each � in column 7 of

Table 2.3. Note that with constant pixel size, the meridional pixel boundaries are not

common from pole to pole, and hence calculation of meridional spatial gradients of

pixel quantities introduces unnatural noise. This “operational” noise can be reduced

by replacing the average of the gradient,⟨

d I0/

d�⟩

, by the equivalent gradient of the

average, �⟨

I0⟩/

��, as is done in column 8 of Table 2.3. Note that �� carries a

sign, and hence, for ease in comparison of Northern Hemisphere and Southern Hemi-

sphere values of this gradient, the sign of the Northern Hemisphere values should be

reversed.

5. The fifth climatic variable is the Northern Hemisphere top-of-the-atmosphere

SW flux, I� (in watts-total per meters squared), time averaged for the June–September

season (presented in Figure 2.7). Because there are no clouds or other atmospheric

content affecting these top-of-the-atmosphere values, we may confidently attribute the

(approximately) 25 Wtotm−2 difference in I� between the equator and latitude 15◦N

to the average solar declination over the short, 122 day season for which this figure

was prepared. Referring back to Figure 2.3, we note that the 330 day average canopy-

top SW flux, 〈I0〉, displays an (approximately) 50 Wtotm−2 difference over the same

latitudinal interval. It follows, then, that the difference, �〈I0〉, over this same ��, as

shown in Figure 2.3, is likely the result of approximately equal parts solar declination

and an internal atmospheric effect such as radiation attenuation by the tropical cloud

cover.

Zonal homogeneity

The coefficient of spatial (i.e., zonal) variation, CVI0(�), of the temporal mean,

seasonal, pixel, canopy-top, SW flux, I0, at latitude � is written

CVI0≡ σI0

⟨

I0⟩ (2.1)

and measures the normalized longitudinal variability of pixel I0 at given �. It is

listed for the Northern Hemisphere in column 9 of Table 2.3 and is plotted versus

latitude in Figure 2.8. The heterogeneity of I0 is indicated by the magnitude of CVI0,



FIGURE 2.8 Assessment of climate zonal homogeneity in theNorthernHemisphere, CVI 0 . Plot-ted points are from Table 2.3, column 9.

with CV ≤ 0.1 representing a “good” degree of constancy [Benjamin and Cornell,

1970, p. 139]. The observed coefficient of longitudinal variation of the mean annual

pixel, canopy-top, SW flux fails this goodness test (and thus demonstrates a strong

longitudinal heterogeneity) only at those middle latitudes containing “dry” local

climates, as can be seen in Figure 2.8. Coupled with the North American versus

Northern Hemisphere disparity shown at latitude 17.5◦ in Figure 2.5, the preceding

“failure” gives only modest license to our zeroth-order zonal homogeneity assumption

for North America. In these middle latitudes, C3 plants are still plentiful but may be

limited more by water and heat availability than by light, suggesting that to isolate the

climate control of the moist forests, which are the subject of this limited work, and

TABLE 2.4 Zonal Average of Observed Pixel Climate in North America

Estimated Growing Season 〈 I 0〉 ≡ ⟨I 0

⟩b ⟨σ I 0

⟩b ∣∣�⟨I 0

⟩/��

∣∣c

�(◦N) (Julian days)a (Wtotm−2) (Wtotm−2) (Wtotm−2 deg−1)

26 73–273 497.2 11.931 73–273 473.7 9.9 4.636 98–248 451.3 10.9 5.841 98–248 416.0 10.8 6.346 120–225 388.5 11.9 6.351 120–225 353.0 12.9 6.156 136–211 327.9 12.9 5.461 136–211 299.2 12.3

aFrom Table 2.2.bData reduced by D. Entekhabi (personal communication 2005) from NASA–Goddard Institute for Space Studies

International Satellite Cloud Climatology Project data set (http://isccp.giss.nasa.gov/projects/flux.html).cDifferentials from columns 1 and 3.



thus more fully legitimize the assumption of zonal homogeneity, the dry-biome pixels

may need to be eliminated from the data sample. However, such sample selection is

not performed in this work.

Looking ahead

Looking forward to our estimation (in chapter 3) of species range for comparison

with the observations in North America (see Figure 1.1a) by Brockman [1968], we

add here Table 2.4, which contains the necessary North American climate estimates

as called for in equation (1.8). Owing to the shape of the North American continent,

there are five or more pixels per zone only for 25◦N ≤ � ≤ 70◦N (see Figure 2.2b);

thus, to reduce the inevitable noise introduced by small samples, we confine our

analysis to this latitudinal range, which coincides with that explored observationally

by Brockman [1968].


C H A P T E R 3

Mean Latitudinal Range of Local Species:Prediction Versus Observation

It is suggested that the greater annual range of climatic conditions to whichindividuals in high-latitude environments are exposed relative to what low-latitudeorganisms face has favored the evolution of broad climate tolerances in high-latitude species. This broad tolerance of individuals from high latitudes has ledto wider latitudinal extent in the geographical range of high-latitude species thanof lower-latitude species.

Stevens [1989, p. 253]

Introduction and definitions

The data of Brockman [1968] presented here as the circles and bars in Figure 1.1a (as

adapted from Stevens [1989]) represent the mean (circles) and ±1 standard error of

the mean (bars) of the latitudinal ranges, in degrees of latitude, of the various species

found in N samples from within separate 5◦ zonal (i.e., latitudinal) bands across North

America. (To quote Stevens [1989, p. 240], “This pattern can be found by rounding

to the nearest 5◦ the northernmost and southernmost extremes of the geographical

ranges of individual species and then calculating the average north-to-south extent of

species found at each 5◦ band of latitude.”) The northerly and southerly latitudinal

extremes of location defining a given species’ range are not necessarily found at the

same longitude (i.e., not necessarily in the same remotely sensed pixel). As noted

by Svenning and Condit [2008], little direct evidence of what causes the limits of

range exists. We seek here to demonstrate the climatic basis for at least the observed

latitudinal trend in (if not the magnitude of) the mean latitudinal range of these zonal

species, but we first need to establish a clear (and regrettably complex) notation for the

several variables of importance. While not necessary, the reader may find it helpful

to read Appendices A–C before continuing.

31



The local (i.e., pixel) “climate,” and hence the associated pixel species, varies in

both time and space. By virtue of its temporal variability from growing season to

growing season, each pixel climate, c, will produce a probability distribution (i.e., a

probability density function (pdf)) of species, s, in that pixel. The local means of c (i.e.,

c) and s (i.e., s) are related by the bioclimatic function s = g (c). However, we find here

that, to zeroth order, the range is independent of the shape of this function. By virtue

of inevitable longitudinal inhomogeneities of climate, each latitude demonstrates a

zonal array of pixel climate means, c, and variances, σ 2c , resulting in a corresponding

zonal array of pixel species means, s, and variances, σ 2s .

With the primary species variable being its projected leaf area or radiation inter-

ception index, βLt (see equation (B.30)), we have s ≡ βLt and s ≡ βLt . (Note that to

limit the symbol size, we use the double overbar here, rather than the “hat,” to indicate

the local average species.) The zonal average of these pixel means is indicated by the

brackets, 〈· · ·〉, giving the zonal average climate, 〈c〉, and the zonal average species,

〈s〉 ≡ 〈βLt 〉. In summary, we define the following:

βLt ≡ individual species in a local pixel corresponding to the climate in that pixel

during a particular growing season

βLt ≡ mean of the local pixel species distribution corresponding to the time-averaged

local pixel climate over the average local growing season

〈βLt 〉 ≡ zonal average of the mean species, βLt , for all pixels at latitude �

σs ≡ standard deviation of local pixel species

〈σs|� 〉 ≡ zonal average of the standard deviation of local pixel species, σs , at �

Rs|�◦(�) ≡ range in degrees latitude of species, s ≡ βLt , at a local site having

� = �◦

Rs|�◦(�) ≡ range in degrees latitude of the mean species, s ≡ βLt , at a local site of

latitude, �◦

Rs|�◦(�) ≡ range in degrees latitude of the modal species, s, at a local site of latitude,

�◦

Rs|�◦(�) = mean of the ranges in degrees of all species, βLt , at a local site at �◦

R〈s〉|�◦(�) ≡ range in degrees of the zonal average local mean species, 〈s〉, at �◦

〈Rs|�◦(�)〉 ≡ zonal average in degrees of the mean of the ranges of all species at �◦

m2p = horizontal projection of canopied area, m2

Range of local mean species as determined by local distributionsabout the mean

The cause of, and the latitudinal change in, the range of local mean species has been

illustrated earlier in Figure 1.3, in which the ordinate, local mean species, s, is plotted


C H A P T E R 3 • M E A N L A T I T U D I N A L R A N G E O F L O C A L S P E C I E S 33

FIGURE 3.1 Idealized range of the mean local species (for the case I 0↓ as �↑).

against the statistically distributed local species, s, on the lower abscissa. This il-

lustration, repeated here for convenience as Figure 3.1, is idealized in that the local

species are shown to be distributed symmetrically about their mean to an effective

limit of ±ns standard deviations. The mean local species, s ≡ βLt , is reflected onto

the species scale of the lower abscissa of Figure 3.1 by the 1 to 1 rising straight line.

An example of bioclimatic function is derived in Appendix C as equation (C.16).

There maximally productive unstressed local average species is related to the local

average climate by the bioclimatic function

〈 I0〉⎡

⎣1 − e−⟨

βLt

⟩

1

⎤

⎦ = E1 = 0.62M Jpar m−2 h−1 = 172 Wtot m−2,

(3.1)

which is applicable for⟨

I0⟩

> E1. This zeroth-order mean value function specifies

s to be an inverse function of the local mean growing-season SW flux, which, in

turn, is determined by latitude � (see Figure 2.3), and with it we can transform

s (nonlinearly) into �, as is indicated, for the predominant case of I0 varying in-

versely with �, by the upper abscissa of Figure 3.1. To find the idealized ±nsσs



FIGURE 3.2 The physical basisof local C 3 species distribution. Fig-ure 3.2b adapted from Eagleson[2002, Figure 8.11a]. Copyright c©2002 Cambridge University Press.Reprinted with permission.

range of the mean species at latitude �◦, we take the difference, �+ − �−. To be

realizable, however, a species must be stable, that is, unstressed, but not necessarily

maximally productive, when the SW flux is other than that for optimal productivity.

We illustrate the consequences of this requirement in Figure 3.2 and in the following

paragraphs.

For a zeroth-order estimate of the distribution of local species resulting from

variable local annual SW flux, we note first that⟨

σI0

⟩

/⟨

I0⟩ ∼= O(10−2), as was shown

in column 10 of Table 2.3. This small value justifies a Taylor expansion of the

bioclimatic function about a local mean value, as was shown in equation (1.2).

For small curvatures of the bioclimatic function, this allows use of the form of



equation (3.1) in the vicinity of the mean to estimate the bioclimatic function as being

βLt = f (I0) ≈ �n[

1 − 172

I0

]−1

, I0 > 172 Wtot m−2. (3.2)

Equation (3.2) is sketched in Figure 3.2a. We see there that when the annual growing-

season SW flux, I0, is less than its local temporal average, I0, the species compatible

with I0 through equation (3.2) is greater than the local mean, βLt . In Figure 3.2b, we

present the potential assimilation efficiency (see Figure A.5) for the average leaves

of C3 trees in which the point labeled “1” represents the optimal operating point

[Eagleson, 2002] for the mean species, βLt . (The potential assimilation efficiency

(sometimes referred to as the “climatic assimilation potential”) represents the (zeroth-

order) locus of maximally efficient CO2 assimilation, for all species of C3 plants. We

restrict our consideration to these plants because of their global predominance coupled

with the strong species selection provided by this saturation efficiency maximization

mechanism.) The associated SW flux, Is�(βLt ), on the average leaf corresponds to the

mean canopy SW flux, I0, isolated in Figure 3.2a. Referring to Appendix A and the

discussion of Figure A.4, we can see in Figure 3.2b that a fluctuating annual leaf SW

flux (point 2) greater than that at point 1 will support a species, βLt < βLt , in that

year, which will be stressed under the average light at point 1, while those years of

less than average SW flux will introduce species, βLt > βLt , that remain unstressed

(point 3) under average conditions. Owing to their average condition of stress, the

species βLt < βLt are assumed to be absent from the local distribution over the long

term. This circumstance is summarized perhaps more clearly in Figure 3.3, where

the species assumed absent (and their associated SW fluxes) are shaded. Of course,

some of these stressed species may be present during the relatively short term of a

field observation program, thereby giving the observed distribution at any time an

attenuated left-hand (i.e., βLt < βLt ) branch.

The nonlinear form of equation (3.2), as plotted in Figure 3.2a, assures that an

assumed symmetrical distribution of annual SW fluxes will produce a distribution

of species, βLt , having positive skew (i.e., asymmetry about the mean) as shown in

Figure 3.3. At least for low curvatures of βLt = f (I0), we judge this asymmetry re-

finement to be inconsistent with the zeroth-order analysis adopted herein and proceed

using, at all I0, the idealized distribution shown in Figure 3.4a, along with the corre-

sponding cumulative distribution of stresslessness. Assuming species which would be

stressed on the average to be unstable and hence locally absent at all times, we adopt

the stress-constrained species distribution illustrated in Figure 3.4b. Furthermore, in

the same spirit of simplification through approximation, we take the mode, s, of the

truncated and skewed stress-free distribution to be the same as the mean, s, of the

idealized symmetrical distribution. How do we estimate the standard deviation, σs ,



FIGURE 3.3 Biological transformation of local distributions: SW flux⇒ C3 species (for the caseI 0↓ as �↑).

of this distribution, for in Figure 3.1, we have seen this second moment of equation

(3.2) to be a predictor of range?

Theoretical estimation of the range with climatic forcingby SW flux only

Writing the Taylor series approximation of equation (1.3) in terms of the primary

bioclimatic variables, we have, at arbitrary I0 [Benjamin and Cornell, 1970],

βLt = f (I0) ∼= βLt + (I0 − I0) dβLt

d I0

∣∣∣∣∣

I0

+ · · ·. (3.3)

For linear f (I0), or with only small variations of I0 for nonlinear f (I0), taking the

expected value and variance of both sides of equation (3.3) gives, respectively,

s ∼= βLt ≈ f ( I0), (3.4)

which we have found as equation (3.1), and

σ 2s (βLt ) = σ 2

I0

[

dβLt

d I0

∣∣∣∣∣

I0

]2

(3.5)



FIGURE 3.4 Assumed frequen-cy distribution of local species. (a)Idealized distribution of speciesand stress. (b) Assumed stress-constrained local distribution ofC 3 species.

or,

σs(βLt ) = σI0

∣∣∣∣∣

dβLt

d I0

∣∣∣∣∣

I0

∣∣∣∣∣≈ σI0

∣∣∣∣∣∣

dβLt

d I0

∣∣∣∣∣∣

, (3.6)

in which σs(βLt ) is the standard deviation of the local species, s, as given in species

units, βLt . As was seen in Figure 3.1, σs is a fundamental determinant of range.

However, for comparison with Brockman [1968], we want σs and hence the range

to be measured in units of latitude, �, rather than of species, and will so define

it as σs(�). This is illustrated for the range of the modal (i.e., the most frequent)

species, s, at latitude � = �0 in Figure 3.5 for the case in which I0↓ and hence s↑(equation (3.2)) as �↑. To obtain the desired variable transformation, βLt → �, we

again employ the first term of a Taylor series expansion, this time of βLt = h(�) at

� = �−, as shown in Figure 3.5. For all latitudes at which species are determined



FIGURE 3.5 Constrained range of the modal local C 3 species (for the case I 0↓ as �↑).

“solely” by SW flux, this local linear approximation is written

σs(�) ≈ σs(βLt )∣∣∣∣

dβLt

d�

∣∣∣∣

= σs(βLt )∣∣∣∣

dβLt

d I0

∣∣∣∣·∣∣∣

d I0d�

∣∣∣

, � = �−. (3.7)

Eliminating σs(βLt ) between equations (3.6) and (3.7) gives, finally,

σs(�) ≈ σs(βLt )∣∣∣∣

dβLt

d I0

∣∣∣∣·∣∣∣

d I0d�

∣∣∣

≈σI0

∣∣∣∣

dβLt

d I0

∣∣∣∣

∣∣∣∣

dβLt

d I0

∣∣∣∣·∣∣∣

d I0d�

∣∣∣

= σI0∣∣d I0

/

d�∣∣, � = �−. (3.8)

As shown in Figure 3.5, the range, in degrees latitude, of the modal local species

at � = �◦ is then

Rs|�◦(�) = nsσs|�−(�) ≡ ns ·⎡

⎣σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⎤

⎦

∣∣∣∣∣∣�−

, (3.9)

where, introducing the simplified notation σs|�−(�) ≡ σs(�−),

�◦ ≡ �− + nsσs(�−), (3.10)

and ns , the number of local species standard deviations at � = �−, reflects the

observational completeness of the species identification.



It is important to note that to the zeroth order, this estimate of the range of the local

modal species at � = �◦ is independent of any specific structure of the bioclimatic

function such as is proposed in equation (3.2). It requires only that we recognize

species determination to be solely by variation of the SW flux (specifically, the local

temporal, σI0 , and latitudinal spatial,∣∣d I0

/

d�∣∣, variabilities of the local SW flux at

� = �− or �+). In Figure 3.5, the truncated Taylor expansion of the bioclimatic

function used at � = �− in this approximation overpredicts the actual range by the

amount �R(�), as is indicated by the short-dashed line at angle α with the horizontal.

Considering the form of the bioclimatic function used here (equation 3.2), this error in

predicted range is largest at low latitudes, which is where we will find equation (3.9)

to break down.

Were I0 an increasing (i.e., direct) function of �, as is seen (using equation (3.2)

and Figure 2.3) to be the case for low latitudes, equation (3.10) would be replaced

by

�◦ ≡ �+ − nsσs(�+). (3.11)

Range of local modal species versus mean of local species’ ranges

To obtain the data plotted as the circles with error bars in Figure 1.1a, Brockman [1968]

averaged the observed ranges, Rs|�◦(�), in degrees latitude, �, of the different species,

s, found at each of N sample sites in a common zone, �0, to obtain the “sample”

(superscript “s”) zonal average range,⟨

Rss|�◦(�)

⟩

, and the sample variance, σ 2Rs

s|�◦ (�),

in that zone. Graphically scaling Brockman’s [1968] plotted ranges to obtain σRss|�◦ (�)

and⟨

Rss|�◦(�)

⟩

, we estimate the standard error of estimate, SE, of the zonal average

range,

SE(⟨

Rs|�◦(�)⟩) =

σRss|�◦ (�)

⟨

Rss|�◦(�)

⟩ , (3.12)

to be of order 10–2 to 10–1, depending on zonal latitude. This small variability of

average range over the sample longitudes in the same zone supports our assumption,

in chapter 1, of reasonable zonal homogeneity in the causative climate, at least at

those North American latitudes studied by Brockman [1968]. Thus, in keeping with

our zeroth-order approximation, we assume the interannual pixel climatic variability,

and hence species variability, to have common statistics for all pixels in the same zone

on any land surface, making σs|�◦(�) ∼= constant from pixel to pixel, and s ∼= 〈s〉 at

any common �◦.

Furthermore, assuming the functional bioclimatic relation between range and

species to be linear over the span �� of the ranges Rs|�◦(�) at �◦, the zonal



average range,⟨

Rs|�◦(�)⟩

, which is the quantity measured by Brockman [1968] and

displayed here in Figure 1.1a, may be approximated by the range of the zonally av-

eraged species, R〈s〉|�◦(�). This, in turn, is equal to the range of the zonal average

of the local average species, 〈s〉. Zonal homogeneity further provides that the zonal

modal species is identical to the zonal average of the local modal species, 〈s〉. These

approximations give

⟨

Rs|�◦(�)⟩ ≈ R〈s〉|�◦(�) ≈ R〈s〉|�◦(�), (3.13)

and therefore we seek, for zeroth-order comparison with Brockman’s [1968] obser-

vations, the theoretical range of the zonal average of the modal pixel species at the

given latitude. Note that we have assumed the mode and mean of the local species

distribution to be equal, which is exact for a “complete” (i.e., double-sided normal

distribution) but only approximate for the “single-sided” (and skewed) distributions

considered stable herein. With the approximation of the standard deviation of local

species in latitude units, as given by equation (3.8), our estimator of the n–standard

deviation range of the zonal average local modal species at � = �◦ becomes (see

Figure 3.5)

⟨

Rs|�◦(�)⟩ ≈ ⟨

Rs|�◦(�)⟩ = ns ·

⟨

σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

�−= ns · ⟨σs(�−)

⟩

, (3.14)

in which [Benjamin and Cornell, 1970]

⟨

σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

�−=⎡

⎣⟨

σI0

⟩ ·⟨∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩⎤

⎦

�−

+ COVz

⎡

⎣σI0,

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⎤

⎦

�−

, (3.15)

where COVz is the zonal covariance at � = �−, as given by

COVz

⎡

⎣σI0,

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⎤

⎦

�−

≡⟨⎡

⎣(

σI0 − ⟨σI0

⟩) ·⎛

⎝

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1

−⟨∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩⎞

⎠

⎤

⎦

�−

⟩

.

(3.16)

Calculation of these covariances is noisy due to the necessary inversion of the

latitudinal gradient of local I0, and this gradient must be estimated using observational

pixels from adjacent zones that, due to identical pixel area at all latitudes, are not

aligned on common meridians. D. Entekhabi (personal communication, 2007) has

estimated the terms of equation (3.15) globally using the 22 year NASA–Goddard

Institute for Space Studies (GISS) International Satellite Cloud Climatology Project

(ISCCP) data set, and a relevant summary is given here in Table 3.1, from the last



TABLE 3.1 Observed Covariance of σ I 0 and d�/d I 0a

�(◦N)⟨σ I 0d�

/d I 0

⟩(◦N) COV

[σ I 0 , d�

/d I 0

](◦N) Column 3/Column 2

15, 20 23 2 0.0925, 30 30 0 035, 40 60 0 045, 50 80 2 0.0355, 60 130 0 0

aFrom 22 year NASA–Goddard Institute for Space Studies (GISS) International Satellite

Cloud Climatology Project (ISCCP) data set (daytime, growing season, land surface only,

Northern Hemisphere). Calculations are by D. Entekhabi (personal communication, 2007).

column of which we note that the covariance term of equation (3.15) may be neglected,

leaving

⟨

σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

�−≈⎡

⎣⟨

σI0

⟩ ·⟨∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩⎤

⎦

�−

. (3.17)

This observation confirms our earlier assumption (see the paragraph following equa-

tion (3.12)) that the zonal climate is sufficiently homogeneous that at least its first two

moments are essentially the same for all pixels in a common zone, thereby ensuring

that the covariances of equation (3.16) are identically zero and, furthermore, that

⟨∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

≈∣∣∣∣∣

d⟨

I0⟩

d�

∣∣∣∣∣

−1

. (3.18)

Abbreviating the range notation, as we have done earlier for the variance, and remem-

bering that (in this example) s is an inverse function of I0, equation (3.14) is, finally,

for negative d I0/d�,

⟨

Rs|�◦(�)⟩ ≈

for notational simplicity︷︸︸︷⟨

Rs|�◦(�)⟩ ≡ 〈Rs (�◦)〉 ≈ ns

⎡

⎣⟨

σI0

⟩ ·∣∣∣∣∣

d⟨

I0⟩

d�

∣∣∣∣∣

−1⎤

⎦

�−︸︷︷︸

〈σs (�−)〉�◦ ≡ �− + ns

⟨

σs(�−)⟩

⎫

⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

, (3.19)

and for positive d I0/d�,

⟨

Rs|�◦(�)⟩ ≈

for notational simplicity︷︸︸︷⟨

Rs|�◦(�)⟩ ≡ 〈Rs(�◦)〉 ≈ ns

⎡

⎣⟨

σI0

⟩ ·∣∣∣∣∣

d⟨

I0⟩

d�

∣∣∣∣∣

−1⎤

⎦

�+︸︷︷︸

〈σs (�+)〉�◦ ≡ �+ − ns

⟨

σs(�+)⟩

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (3.20)



FIGURE 3.6 Cumulative probability under the single-sided standardized normal probabilitydensity function. Adapted from Mosteller et al. [1961] and Benjamin and Cornell [1970].

Probability mass of the distribution of observed local species

If we knew the form of the pdf of the local species at a given latitude, and the

completeness (i.e., “percentage mass”) of Brockman’s [1968] identification of local

species there, we could specify the number, ns , of local species standard deviations

(from the mean species) within which this mass lies, a necessary and crucial factor of

equations (3.19) and (3.20). However, we do not have this information and must infer

it from the observed distribution of the local causative climate variable(s), which are

assumed to be limited to the SW flux in this work.

Assuming the bioclimatic function s = f (I0), as given by equation (3.2), to define

a locally linear and one-to-one transformation over the range of latitude implied by

the local fluctuations of I0 (see Figure 3.3), the pdf of s will be of the same type as

that of I0, although perhaps expanded, shrunken, or shifted, depending on what the

particular linear transformation f (I0) calls for at a given �. In such cases, letting nI

be the number of standard deviations of the I0 fluctuations (i.e., nI = (I0 − I0)/σI ),

if the probability distribution of I0 at a given latitude contains x% probability mass

at the distance nI σI0 from its mean, I0, then the probability distribution of s at that

latitude also has x% probability mass at the distance nsσs from the species mean, s. In

the special case of normal distributions (see Figure 3.6), a particular probability mass

is defined solely by n and therefore, by the preceding argument, a linear, one-to-one

transformation yields

ns ≡ nI . (3.21)



Actually, ns is likely less than nI . The weakest of the I0 fluctuations may produce

species that are unstable for reasons unconsidered herein.

We continue our assumption of zonal homogeneity to assemble a large sample

of observed zonal fluctuations in annual SW flux, �I0 ≡ I0 − I0, by using the �I0

of each zonal land surface pixel during the daytime growing season of each of the

years of the satellite record. Histograms of these fluctuations at various latitudes in

North America are presented in Figure 3.7 as a function of nI , as determined by D.

Entekhabi (personal communication, 2007) at a later date, when a larger, 22 year

sample became available from the NASA-GISS ISCCP data set. Sample size for

30◦ ≤ � ≤ 70◦ in North America is thus between 220 and 110 (see Figure 2.2b).

Except at desert latitudes, � = 35◦ and � = 45◦, the histograms in Figure 3.7 are

sufficiently symmetrical about a central modal value to justify considering them, at

the current level of approximation, to be normal distributions albeit with truncated

tails. We indicate at each latitude in Figure 3.7 the approximate truncation value of nI

for the negative (i.e., left hand) side of the distribution, which we have assumed here

(see Figure 3.2) to be the fluctuations responsible for supporting the stable (and thus

observable) species on a one-to-one basis. With assumed normality and locally-linear

s = f (I0), ns ≡ nI , and we have all we need to estimate the range using equations

(3.19) and (3.20).

Analytical summary for climatic forcing by SW flux only

We have just seen (equations (3.19)–(3.20)) that the range of the modal species

at latitude �0 depends on the breadth of the species frequency distribution at �−

(or �+), which has been estimated using a series of linearization approximations

developed in earlier portions of this chapter. It is helpful to review the nature of these

approximations before testing their utility against the field observations of Brockman

[1968]. For species whose local existence is a function solely of the SW flux, and for

conditions of perfect zonal homogeneity (i.e., zonal averaging notation omitted), the

preceding analysis is summarized for negative d I0/d�,

Rs|�0 (�) = nsσ s|�−(�) = nsσs|�− (βLt )∣∣∣�βLt

��

∣∣∣

∣∣∣�0−�−

︸︷︷︸

a

≈nsσI0

∣∣∣

dβLt

d I0

∣∣∣�−

∣∣∣

dβLt

d I0

∣∣∣�−

·∣∣∣

d I0d�

∣∣∣�−

︸︷︷︸

b

= nsσI0∣∣∣

d I0d�

∣∣∣�−

≈ nI σI0∣∣∣

d I0d�

∣∣∣�−

︸︷︷︸

c

(3.22)

(note that (a) exact variable transformation, σs(βLt ) → σs(�), requires knowledge

of the “one-to-one” bioclimatic function, βLt (I0); �◦ − �− = nsσs(�−); (b) both



FIGURE 3.7 Histograms of observedpixel annual shortwave fluctuations in theNorthernHemi-sphere Americas, nI = (

I 0 − I 0)/σ I 0 (daytime, seasonal, land surface only; 22 year NASA–Goddard

Institute for Space Studies (GISS) International Satellite Cloud Climatology Project (ISCCP) data set;D. Entekhabi, personal communication, 2007).



numerator and denominator assume βLt (I0) is linear over the local range �0 − �−,

and the denominator assumes I0(�) to be linear over �0 − �− also; and (c) this

assumes normal distribution of I0, making ns = nI at common percentage mass due

to the linearizations of b), and for positive d I0/d�,

Rs|�0 (�) = nsσ s|�+(�) = nsσs|�+ (βLt )∣∣∣�βLt

��

∣∣∣

∣∣∣�+−�0

︸︷︷︸

a

≈nsσI0

∣∣∣

dβLt

d I0

∣∣∣�+

∣∣∣

dβLt

d I0

∣∣∣�+

·∣∣∣

d I0d�

∣∣∣�+

︸︷︷︸

b

= nsσI0∣∣∣

d I0d�

∣∣∣�+

≈ nI σI0∣∣∣

d I0d�

∣∣∣�+

︸︷︷︸

c

(3.23)

(note that (a) exact variable transformation, σs(βLt ) → σs(�), requires knowledge

of the “one-to-one” bioclimatic function, βLt (I0); �+ − �0 = nsσs(

�+); (b) both

numerator and denominator assume βLt (I0) to be linear over the local range, �+ −�0, and the denominator assumes I0(�) to be linear over �+ − �0 also; and (c)

this assumes normal distribution of I0, making ns = nI at common percentage mass

due to the linearizations of b). The case of d I0/

d� = 0 will be considered later (in

Figure 3.16) for both maxima and minima of I0.

Point-by-point estimation of range versus observationfor North America

In Figure 3.8, we plot, for the land surface pixels of North America, the ob-

served functions⟨

I0⟩ = f I (�) and

⟨

σI0

⟩ = gσI (�), along with (from Figure 3.7)

nI ≈ ns = hn(�), all as determined by D. Entekhabi (personal communication, 2007)

from the 17 year (22 year for nI ) NASA-GISS satellite data set. Note in Figure 3.8 that

for all North American latitudes, d I0/

d� is negative, and with increasing latitude,

all three of the plotted climate variables display observed “wavelike” oscillations.

We will demonstrate the necessity of filtering these oscillations to obtain the mono-

tonic increase of range with latitude displayed in Figure 1.1a by the observations of

Brockman [1968].

Letting � = �−, the plotted values of⟨

I0⟩

and⟨

σI0

⟩

from Figure 3.8 are listed

in columns 2 and 3 of Table 3.2. The absolute value of the latitudinal gradient of⟨

I0⟩

is given in column 4, as determined by differentials from columns 1 and 2, and

the standard deviation of local species,⟨

σs(�−)⟩

, is found through equation (3.8).

Referring to Table 3.2, we now estimate the local ranges by two methods.



FIGURE 3.8 Piecewise latitudinal linearization of the components of local species range inNorth America (17 year NASA-GISS ISCCP data set; D. Entekhabi, personal communication, 2007).

1. The first method is point-by-point for all variables. With nI = ns , we interpolate

values of nI (column 6) at the desired values of �− (column 1) and calculate the

modal range at �◦ (column 8) using equation (3.19). The value of �◦ is then given

in column 10 from column 1 plus column 8, as called for in the second part of

equation (3.19). Finally, these modal ranges are plotted at their latitudinal locations

in Figure 3.9 using the open diamond symbol, where they may be compared with the

observations of Brockman [1968] shown by solid dots with error bars. Note that north

of the desert latitudes, they show the range gradient to be of the proper sign and the

range magnitudes to be within about ±20% of the observed.

2. The second method is point-by-point for all but nI . Recognizing that nI proba-

bly contains the most error due to our assumption of zonal homogeneity, we remove

its oscillations, seen in Figure 3.8, by using the average value, n I = 2.9, listed at

all latitudes in column 7 of Table 3.2 and corresponding to a normal probability

mass of 99.5%, as shown in Figure 3.6. (Note in this figure that for a normal prob-

ability mass of 99.7%, nI = ns = 4.0, illustrating the exquisite sensitivity of the


TABLE

3.2

Estim

ationof

theLatitud

inalRa

ngeof

theLo

calM

odalSp

eciesin

North

America(North

American

Land

SurfaceZo

nalA

verage

From

the

NASA

-GISSISCC

PDataSe

t;17

Yearsof

Record,196

4–20

00)a

�−

⟨ I 0⟩ b

⟨ σI 0

⟩ b∣ ∣ �

⟨ I 0⟩/

��

∣ ∣c⟨ σ

s( �

−)⟩ d

⟨ Rs| �

0(�

)⟩ g⟨ R

s| �0(�

)⟩ h�

◦i�

◦j

(◦N)

(Wtotm

−2)

(Wtotm

−2)

(Wtotm

−2de

g−1)

(deg

)nIe

nf I

(deg

)(deg

)(◦N)

(◦N)

12

34

56

78

910

11

2649

7.2

11.9

3.4

2.9

3147

3.7

9.9

4.59

2.16

2.8

2.9

6.0

6.3

37.0

37.3

3645

1.3

10.9

5.77

1.89

2.6

2.9

4.9

5.5

40.9

41.5

4141

6.0

10.8

6.28

1.72

2.3

2.9

4.0

5.0

45.0

46.0

4638

8.5

11.9

6.30

1.89

2.4

2.9

4.5

5.5

50.5

51.5

5135

3.0

12.9

6.06

2.13

3.2

2.9

6.8

6.2

57.8

57.2

5632

7.9

12.9

5.38

2.39

3.8

2.9

9.1

6.9

65.1

62.9

6129

9.2

12.3

3.5

aGrowingseason

asestim

ated

inTable

2.2foreach

latitud

e.Po

int-by

-point

andaverag

ed-n

Icurves

areplotted

byop

endiam

onds

andop

en

circlesin

Figu

re3.9,respectiv

ely.

bFrom

observations

(Tab

le2.4).

c Differen

cesfrom

columns

1an

d2.

dEq

uatio

n(3.8):column3

÷column4.

e Point-by-point,Figures

3.7an

d3.8.

f Average

dn l:Figures

3.7an

d3.8.

gPo

int-by

-point

with

interpolated

n l:colum

n5

×column6.

h Average

dn l:colum

n5

×column7.

i Point-by-point

with

interpolated

n l:colum

n1

+column8.

j Average

dn l:colum

n1

+column9.

47



FIGURE 3.9 Latitudinal distribution of the mean latitudinal range of local species in NorthAmerica: point-by-point estimation.

estimated range to the estimated value of this parameter.) With this sole change,

we repeat the calculation of modal range and its location, as listed in columns 9

and 11 of Table 3.2. These are plotted in Figure 3.9 using the open circle symbol,

where they are numbered 1–6 and may be compared with both the observations and

with the (completely) point-by-point values of the open diamond symbol. Note the

similar results in the desert latitudes but, now, how nearly perfectly points 3–6 of the

predictions (open circles), using nI = n I , track the observations over the latitudes

46◦N ≤ � ≤ 63◦N.

We note the following from Figure 3.9.

1. For latitudes between 46◦N and (at least) 63◦N, our “zeroth-order,” point-by-

point estimator of local ranges is quite accurate in North America, provided we use

the average of the estimated local nI .

2. For latitudes 46◦N ≤ � < 63◦N (at least) in North America, the point-by-point

estimates of range (using average nI ) appear to lie on a straight line (short dashes),



which, when fitted by linear “least squares,” has a slope d Rs/d� = 0.11 and passes

very close to the origin, Rs = 0, � = 0, as it intercepts the zero-range axis at � < 1◦.

In the next section, we will conduct a Gedankenexperiment, which leads to a physical

requirement that the linear projection of high-latitude species range pass precisely

through the range-latitude origin, (0,0).

3. For latitudes 37◦N ≤ � < 46◦N (at least) in North America, either estimator

seriously overestimates the range. This may be due to our assumption in arriving

at equation (3.19) that the range is controlled solely by SW flux. Indeed, we note

in Figure 3.9 that the latitudes between about 22◦N and 45◦N contain the North

American deserts [Strahler, 1971], where we expect the availability of water to be

vegetation limiting, and we have made no attempt here to include this variable in the

analysis.

4. We have demonstrated, through smoothing of the wavelike latitudinal variations

of nI , that our estimate of range is dramatically improved north of the desert region

in North America. This suggests that some sort of meridional smoothing of all the

variables composing the range estimate may reduce (or even eliminate) the excursion

of range we predict between 37◦N and 46◦N from the point-by-point analysis made

using equation (3.19). We explore this tactic in a later section of this chapter.

5. There are insufficient land surface pixels at the lower latitudes of North America

to generate stable theoretical ranges at �◦ ≤ 30◦N, and there are no Brockman [1968]

observations there either. Were there adequate numbers of land surface pixels equator-

ward of the deserts, it would seem reasonable to expect our SW-flux-based estimator

to regain its accuracy near the equator only if we used a higher-order approximation

that accounts for the curvature of the bioclimatic function.

6. If the other Northern Hemisphere land surfaces behave at least qualitatively in

the same manner as those of North America, we may gain insight into the latitu-

dinal variations of range at �◦ < 46◦N through a similar analysis of the Northern

Hemisphere climate data. We explore this also in a later section of this chapter.

A thought experiment on the variation of SW fluxin an isotropic atmosphere

Imagine a world in which vegetation responds everywhere solely to the SW flux

and that (to the zeroth order) the average growing-season value of this flux has

the value at high latitude associated there with the autumnal equinox in the real

world. Imagine further that the atmosphere in this world has everywhere the same

atmospheric moisture content (i.e., specific humidity) found at the higher latitudes

of the real world [Peixoto and Oort, 1992, Figure 12.6]. We might then expect a

surface SW flux that increases linearly with falling latitude only at high latitudes on



FIGURE 3.10 Global pixel SW flux in an imaginary, low-moisture atmosphere: a“thought experiment” (daytime at autumnal equinox; 22 year NASA-GISS ISCCP data set;D. Entekhabi, personal communication, 2007).

real-world land surfaces to be found at all latitudes on land surfaces in our imaginary

world. Furthermore, at the autumnal equinox, we might expect this SW flux to be

symmetrical about the equator in this imaginary world.

Turn now to Figure 3.10, in which these real and imaginary conditions are illus-

trated: the highest, continuous, “sinusoidal” curve gives the SW flux at the top of the

atmosphere (identical in both the real and imaginary worlds), while the lower contin-

uous curve (the Americas) and dashed curve (global) give the SW flux at the surface

of the real world, all at the autumnal equinox, as given in the 22 year NASA-GISS

ISCCP data set (D. Entekhabi, personal communication, 2007). The SW flux at the

surface in our imaginary, “dry atmosphere” world is set, as a boundary condition of

this Gedankenexperiment, to be identical to that of the real world at high latitudes,

which is seen by the fitted dash-dotted line in Figure 3.10, to fall linearly with increas-

ing latitude above about 40◦N and 40◦S. Note that the values of 〈 I0〉 at the surface are

a decreasing percentage of those at the top of the atmosphere with increasing latitude.

This is due both to the increasing percentage of temporal-average cloud cover (see

Peixoto and Oort [1992, Figure 7.29], using data from Berliand and Strokina [1980])

and to the increasing length of the SW flux path in the atmosphere, for this season

above about 35◦N and 35◦S. Below this latitude, in both hemispheres, the real-world

surface SW flux is relatively constant between 400 and 500 Wtot m−2. This is due



to the high cloud albedo resulting from the high atmospheric moisture content that

increases toward the equator [Peixoto and Oort, 1992, Figure 12.6a], aided by the

intense (and “homogenizing”) convective vertical mixing at these latitudes. It seems

reasonable to assume that with the imaginary atmosphere being dry globally, the

imaginary surface SW flux would continue to rise linearly for � < 40◦N and 40◦S

at the autumnal equinox. Accordingly, we project the dash-dotted line of Figure 3.10

to the equator in each hemisphere at the slope established at � < 40◦N and 40◦S by

the real-world dry atmosphere at these latitudes. Note that at the autumnal equinox,

these projections meet almost exactly at the equator (as they should), giving a clear

maximum SW flux there of about 850 Wtot m−2 and a pleasing symmetry to our

imaginary world at this season.

To draw from this thought experiment a conclusion helpful to our understanding of

species range in the real world, we look back at Figure 2.3 to see that the gradients of

〈I0〉 from the land surfaces in the higher latitudes of both hemispheres, when averaged

both zonally and temporally over essentially the entire year (i.e., days 8 to 338) and

projected back to the equator, also intersect there. Can there be a connection between

this observation and our theoretical finding (Figure 3.9) that the linear least squares

fit of predicted point-by-point ranges over these same latitudes also projects back to

zero at the equator? To answer this, we look ahead to Figure 3.16 and equation (3.41).

Range of modal species at maxima andminimaof the SW flux

Skipping ahead to Figure 3.16 for the moment (and neglecting the numbers thereon

until later), we consider what maxima and minima of I0(�) imply for species range.

Look first at the maximum of I0(�), as sketched in Figure 3.16b, where the species-

supportive half of the I0 distribution (refer to Figure 3.3 or 3.4) is sketched at latitudes

below, at, and above that maximum. Note that there is no latitude on either the rising

or falling limb of this curve at which the distribution of I0 will contain a value equal to

or larger than the modal value in the distribution at the maximum of I0(�). Referring

to Figure 3.16b, this means that the range of all peak I0 is identically zero.

A numerically similar situation applies with respect to the modal species, s, which

we have seen (equation (3.2)) to rise monotonically with falling I0. This means that

there is no latitude on either the falling or rising limb of the cup-shaped s(�) curve

(i.e., the inverse of I0(�)) at which the distribution of s will contain a stable value

equal to or less than the smallest value, s, at �oo. The (continuous) range of the modal

stable species associated with this maximum of I0(�) is therefore always zero. On the

other hand, as is shown clearly in Figure 3.16a, using similar reasoning, the sketched

distributions allow for finite continuous range of the modal species around a local

minimum of I0(�).



FIGURE 3.11 Latitudinal distribution of the mean latitudinal range of local species in NorthAmerica: mean gradient estimation.

We have seen that I0 in the high, “dry-atmosphere” latitudes, when averaged over

either the single month surrounding the autumnal equinox (Figure 3.10) or over

essentially the full year (Figure 2.3), projects to a maximum at the equator. In light

of the preceding, we now have a theoretical basis for projecting the theoretical range

gradient at high (and dry) latitudes in the real world back through the range-latitude

origin, (0,0), and thereby fixing the gradient in Rs, � space.

Gradient estimation of range versus observation for North America

The quasi-linear variation of range with latitude displayed by Brockman’s [1968] ob-

servations, as reproduced again in Figure 3.11, suggests that estimating the gradient of

range, rather than the point-by-point ranges, may provide additional helpful smooth-

ing of the observed oscillations in the independent parameters seen in Figure 3.8.



This gradient is obtained from the derivative of the final form of equation (3.19) as

d Rs|�0 (�)

d�=∣∣∣

d I0d�

∣∣∣ d(

nI⟨

σI0

⟩)/

d� − nI⟨

σI0

⟩

d∣∣∣

d I0d�

∣∣∣

/

d�∣∣∣

d I0d�

∣∣∣

2 , (3.24)

in which (see Figure 3.8) d∣∣d⟨

I0⟩/

d�∣∣/

d� ∼= 0 for � ≥ 26◦N. Note that the gra-

dient of range is independent of the sense of βLt ( I0). Over these latitudes, equation

(3.24) is then

d Rs|�0 (�)

d�= nI d

⟨

σI0

⟩

/d� + ⟨σI0

⟩

dnI /d�∣∣∣

d I0d�

∣∣∣

, � ≥ 26◦N. (3.25)

As a further smoothing, we average this gradient of range over � ≥ 26◦N, while

neglecting the resulting covariances as being small, giving the approximation

[d Rs|�0 (�)

d�

]

∼= n I d⟨

σI0

⟩

/d� + ⟨σI0

⟩

dnI /d�∣∣d I0

/

d�∣∣

, � ≥ 26◦N, (3.26)

in which, as used here, the double overbar signifies averaging over latitude. From

the latitudinal variations in nI and⟨

σI0

⟩

seen in Figure 3.8, we see that the range

� ≥ 26◦N should be broken up into two continuous segments, 26◦ ≤ � ≤ 35◦ and

35◦ ≤ � ≤ 65◦, and we apply equation (3.26) separately to each of these.

We evaluate the mean gradients in equation (3.26) using a linear least squares

fitting to the observed climate variables, as shown by the dashed straight lines in

Figure 3.8, obtaining

North America:

⎧

⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

∣∣d⟨

I0⟩

/d�∣∣ = 5.82

d⟨

σI0

⟩

/d� = 0.082

dnI /d� = 0.031

⎫

⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

, � ≥ 35◦, (3.27)

and by the solid straight lines, obtaining

North America:

⎧

⎪⎨

⎪⎩

d⟨

σI0

⟩/

d� = 0

dnI/

d� = 0

⎫

⎪⎬

⎪⎭

, 26◦ ≤ � ≤ 35◦. (3.28)

The mean point values (solid lines in Figure 3.8) give

North America:

⎧

⎨

⎩

n I = 2.93

⟨

σI0

⟩ = 11.95

⎫

⎬

⎭, � ≥ 35◦ (3.29)



and (dashed lines for σI0 ; solid line for nI )

North America:

⎧

⎪⎨

⎪⎩

d⟨

σI0

⟩/

d� = 0

dnI/

d� = 0

⎫

⎪⎬

⎪⎭

, 26◦ ≤ � ≤ 35◦. (3.30)

Using equations (3.25) and (3.26), equation (3.24) yields, for North America,

[d Rs|�0 (�)

d�

]

≈ (2.93) (0.082) + (11.95) (0.031)

(5.82)= 0.105, � ≥ 35◦, (3.31)

which is close to the same value, 0.11, found in Figure 3.9 from linear least squares

fitting of the point-by-point-calculated ranges numbered 3–6. To make use of the

preceding result for estimating the mean range at a given latitude, we need to locate

d R/

d� vertically in the space of Figure 3.11, to which end we use the results of our

thought experiment:

R ps|�0 (� = 0) = 0, (3.32)

which is a point on the projection of the latitudinal gradient of range in the high-

latitude region of low atmospheric moisture (estimated here to be � ≥ 45◦) and is

plotted as the open triangle at the origin in Figure 3.11.

We now use equations (3.31) and (3.32) to plot our a priori, gradient-based estimate

of high-latitude range as the solid line for � ≥ 45◦ in Figure 3.11, in comparison with

Brockman’s [1968] North American observations for 29◦ ≤ � ≤ 70◦. We see there, a

posteriori, that equations (3.31) and (3.32) are actually quite accurate over the larger

span 35◦ ≤ � ≤ 70◦, and we extend the solid line accordingly. Furthermore, using

equations (3.26) and (3.28), the range gradient is zero for 26◦ ≤ � ≤ 35◦, and when

matched to the range given by equation (3.31) at � = 35◦, the forecast range picks up

the Brockman-observed flattening in the middle latitudes and thereby demonstrates its

utility over the full range of vegetated land surface in North America, 26◦ ≤ � ≤ 70◦.

The remarkable ability of this gradient analysis (Figure 3.11) to predict not only

the gradient, but also the magnitude of range over essentially the entire span of

vegetated North American latitudes, while the point-by-point predictions (Figure 3.9)

diverge sharply from observation for � < 45◦, suggests that the form of the point-by-

point estimator (equation (3.22)) at these lower latitudes may be incorrect. Perhaps

it is dependent on a different primary forcing variable (available moisture at these

latitudes?) but independent of latitude. We will investigate this issue further as we

consider the entire Northern Hemisphere, where we have more low-latitude land

surface.

We now perform the same point-by-point and gradient estimates of range over the

entire Northern Hemisphere, although we have no Northern Hemisphere observations

outside of North America with which to compare the estimates.



FIGURE 3.12 Histograms of observed pixel annual shortwave fluctuations in the NorthernHemisphere, nI = (

I 0 − I 0)/σ I 0 (daytime, seasonal, land surface only; 22 year NASA-GISS ISCCP

data set; D. Entekhabi, personal communication, 2007).

Point-by-point estimation of range versus observationfor the Northern Hemisphere

The histograms of land surface pixel fluctuations of SW flux in six latitudinal zones of

the Northern Hemisphere are presented in Figure 3.12, as prepared by D. Entekhabi

(personal communication, 2007), again from the 22 year NASA-GISS data set. Note



TABLE 3.3 Estimation of the Latitudinal Range of the Local Modal Species in theNorthern Hemisphere (Northern Hemisphere Land Surface Zonal Average From theNASA-GISS Radiative Flux Data Set; 17 Years of Record, 1984–2000)

�− 〈 I 0〉 ≡ ⟨I 0

⟩a ∣∣�⟨I 0

⟩/��

∣∣b ⟨σ I 0

⟩c ⟨σs

(�−)⟩d 〈Rs (�◦)〉 f �◦g

(◦N) (Wtot m−2) (Wtot m−2 deg−1) (Wtot m−2) (deg) ni = ns e (deg) (deg)1 2 3 4 5 6 7 8

22.5 1.3 10.6 8.15 (3.0) 24.5 47.025 484.4 3.027.5 3.1 12.2 3.9 (3.0) 11.7 39.230 481.3 2.8 11.9 4.25 (3.0) 12.8 42.832.5 3.8 12.3 3.2 (3.0) 9.6 42.135 469.0 5.0 12.7 2.54 3.2 8.1 43.137.5 6.0 13.2 2.2 (4.0) 8.8 46.340 438.5 (4.0)42.5 6.6 14.0 2.09 (4.0) 8.4 50.945 404.0 4.047.5 6.2 14.9 2.40 (4.0) 9.6 57.150 361.5 (4.0)52.5 5.2 15.7 3.01 (4.0) 12.0 64.555 335.0 4.6 16.2 3.51 4.0 14.0 69.057.5 324.0 4.5 16.6 3.69 (4.0)60 315.4 6.6 17.0 2.58 (4.0) 10.3 70.365 269.2 4.0

aFrom observations, Table 2.3.bDifferences from columns 1 and 2.cFrom smoothed observations; Figure 3.13. Growing season as estimated in Table 2.2 for each latitude.dFrom equation (3.8).eFrom Figure 3.12. Parentheses indicate interpolation.f Column 5 × column 6.gColumn 1 + column 7.

the significant departures of these distributions from normality in the desert latitudes,

15◦N < � < 45◦N, which violate an important assumption of equation (3.21). For

North America (Figure 3.7), we have seen this violation to be much less severe and over

a more northerly range of the desert latitudes there (25◦N < � < 55◦N). However,

having noted this problem, we disregard it at the present order of approximation, and

we indicate our associated estimates of nI ≈ ns for the Northern Hemisphere in Table

3.3 and Figure 3.12.

The remaining parameters of equations (3.19) and (3.20) are presented in Fig-

ure 3.13. There we plot, for the land surface pixels of the entire Northern Hemi-

sphere, the observed functions⟨

I0⟩ = f I0 (�) and

⟨

σI0

⟩ = gσI (�), as determined by

D. Entekhabi (personal communication, 2007) from the 17 year NASA-GISS data

set, along with nI ≈ ns = hn(�) from Figure 3.12.



FIGURE 3.13 Piecewise latitudinal linearization of the components of local species range in theNorthern Hemisphere (22 year NASA-GISS ISCCP data set; D. Entekhabi, personal communication,2007).

Letting � = �−, the plotted values of⟨

I0⟩

,⟨

σI0

⟩

, and nI ≈ ns are listed in columns

2, 4, and 6 of Table 3.3. The estimated zonal average modal species range, 〈Rs|�0 (�)〉,and latitude �0 for latitudes � ≥ 35◦ in the Northern Hemisphere are given in columns

7 and 8 of Table 3.3 and are plotted as the open circles in Figure 3.14, where they

are compared with Brockman’s [1968] observations in North America. The points

labeled a1, a2, b, c, and d, plotted using the symbol ⊕ in Figure 3.14, are special cases

resulting from the presence of zero gradients of the SW flux function,⟨

I0⟩ = f I0 (�)

(Figure 3.14), which dominate the formation of continuous range at all latitudes

� < 35◦. Because of the changing gradients and overlapping influences in these

latitudes, calculation of the ranges there does not lend itself to tabular presentation

and is therefore missing from Table 3.3. Instead, using observed values from Figure

3.13, the ranges are reasoned from reference to the generalizations of both maxima



FIGURE 3.14 Latitudinal distribution of themean latitudinal range of local species in theNorth-ern Hemisphere: point-by-point estimation.

and minima presented in Figure 3.15 and then are plotted directly in Figure 3.14.

Beginning at the equator, we describe their estimation as follows.

For ⊕a1—on either side of a minimum in the local mean SW flux, such as at

the equator in Figure 3.13 (enlarged but not to scale in Figure 3.16a), the mean SW

flux rises. According to the fundamental assumption of this work, as summarized

in Figure 3.4, only the species that are larger than the local mean species are stable

locally and hence are available to be observed and counted. Such “stable” species are

supported by those fluctuating annual local SW fluxes that are smaller than the local

mean SW flux. We thus show in Figure 3.16a the range-determining portion of the

local SW flux distribution as extending below the function⟨

I0⟩ = f I0 (�). With our

central assumptions, equal species are found at equal I0, so the mean species at the

equator (� = �0 = 0) are found within the I0 distributions at all latitudes between

the equator and � = �+, the latitude at which I +0 − nI σI +

0= I 0

0 , a trial solution.

The continuous range of the modal species at the equator will have branches on both



FIGURE 3.15 Latitudinal distribution of themean latitudinal range of local species in theNorth-ern Hemisphere: mean gradient estimation.

sides of � = �0 = 0, but here we consider only that portion, RN (�0), that would be

observed in the northern latitudes. The successful trial solution for this portion of the

equatorial range is shown in Figure 3.16a (top) as RN (�0) = �+ − 0 = 9◦.

For ⊕a2—between � = 0◦ and � = 9◦, all the constituents of range nI , σI0 , and∣∣d I0/d�

∣∣ are approximately constant (see Figure 3.13); thus we expect the range to

be constant over this latitudinal span.

For ⊕c—on either side of a maximum in the local mean SW flux, such as at

� = �00 ≈ 17.5◦N in Figure 3.13 (enlarged but not to scale in Figure 3.16b), the

mean SW flux falls. Once again, our basic assumptions have stable local species

supported by those fluctuations in the local annual SW flux that are smaller than the

local mean SW flux, and we show, in Figure 3.16b, that side of the local distribution

of fluctuations in SW flux at �00, where the mean SW flux is at its maximum, I 0

0 , and

at the two flanking latitudes, �0L and �0

R , where the smaller mean SW flux is identical

and equal to I 00 − nI σI 0

0. It follows that the range of the modal species at a maximum

is zero.

For ⊕b, d—there will be equal peaks in the continuous range at the two latitudes

�0L and �0

R (⊕b and ⊕d, respectively) flanking �00 (at ⊕c). To evaluate �0

L and �0R ,



we first calculate nI σI 00

= 26 Wtot m−2 and subtract it from I 00 = 496 Wtot m−2 to get

I L0 = I R

0 = 470 Wtot m−2, which are located in latitude (from Figure 3.13) at 9◦ and

35◦, respectively. The continuous range at these limiting latitudes is given by their

difference, 26◦, and is plotted accordingly in Figure 3.14.

Point-by-point estimates of the ranges at latitudes � ≥ 35◦ in the Northern Hemi-

sphere are estimated as before for North America, and that process can be followed

in Table 3.3. The resulting ranges are plotted in Figure 3.14 using open circles,

and they show discontinuities as well as a wild oscillation apparently forced by the

corresponding large oscillations in⟨

σI0

⟩

seen in Figure 3.13.

Gradient estimation of range versus observationfor the Northern Hemisphere

Beginning as in North America and noting (Figure 3.13) that in the Northern Hemi-

sphere, for � ≥ 35◦, d∣∣d⟨

I0⟩/

d�∣∣/

d� ∼= 0 and dnI /d� = 0, equation (3.26) re-

duces to

[d Rs|�0 (�)

d�

]

≈ n I d⟨

σI0

⟩

/d�∣∣d I0

/

d�∣∣

, � ≥ 35◦N. (3.33)

We fit the⟨

I0⟩

and⟨

σI0

⟩

observations of Figure 3.13 using the linear least squares

method, and the nI by simple averaging, to obtain

Northern Hemisphere:

⎧

⎪⎪⎪⎨

⎪⎪⎪⎩

n I∼= 4.0

d⟨

σI0

⟩

/d� = 0.17

∣∣d⟨

I0⟩

/d�∣∣ = 6.55

⎫

⎪⎪⎪⎬

⎪⎪⎪⎭

, � ≥ 35◦N , (3.34)

which, with equation (3.33), gives

[d Rs|�0 (�)

d�

]

≈ (4.0)(0.17)

(6.55)= 0.104, � ≥ 35◦N. (3.35)

Repeating the previous thought experiment, equation (3.35) must pass through

the origin of Rs|�0 (�), as indicated by the open triangle in Figure 3.15, and we find

equation (3.35) to describe Brockman’s [1968] observations of the mean local species

ranges for � ≥ 35◦ in North America with reasonable accuracy. We note the closeness

of the Northern Hemisphere gradient of range, 0.104, to the 0.105 of North America

(equation (3.31)) and will return to this finding in the next section.



FIGURE 3.16 Estimation of the SW flux-dependent species range at maxima and minima ofI 0(�).

Continuing the gradient analysis for � < 35◦, we have

Northern Hemisphere:

⎧

⎪⎪⎪⎨

⎪⎪⎪⎩

n I = 3.0

d⟨

σI0

⟩

/d� = 0.17

∣∣d⟨

I0⟩

/d�∣∣ = 1.52

⎫

⎪⎪⎪⎬

⎪⎪⎪⎭

, 17.5◦ ≤ � < 35◦ , (3.36)


[d Rs|�0 (�)

d�

]

≈ (3.0)(0.17)

(1.52)= 0.34, 17.5◦ ≤ � < 35◦, (3.37)

which, assuming continuous R, makes Rs|�0 (�) = 0 at

� = 35◦ − Rs (35◦) /0.34 = 35◦ − 3.6/0.34 = 24.4◦.

For 0◦ ≤ � ≤ 17.5◦, we see in Figure 3.13 that d⟨

σI0

⟩

/d� ∼= 0, and although

we have only one observation of nI in this region, comparison with observations at

higher � suggest that it is reasonable to assume that dnI /d� = 0 also over this range

of low �. Hence, over these low latitudes, it appears as though the gradient as given

by equation (3.29) maintains Rs|�0 (�) = 0. To emphasize the uncertainty in these



reasonings, the two low-latitude gradients are plotted in Figure 3.15 as the (short)

dashed lines.

The similarity of the estimated range gradients for North America (Figure 3.9) and

for the whole Northern Hemisphere (Figure 3.11) at � ≥ 35◦ lead us to conclude that

the species ranges in the Northern Hemisphere are relatively insensitive to longitude

over these latitudes. However, the point-by-point estimates of range do not match the

gradient estimates for � ≥ 35◦ in the Northern Hemisphere, as they did in North

America (see Figure 3.9, as compared with Figure 3.14), and neither the point-by-

point nor the gradient estimates appear adequate at � < 35◦ with SW flux as the sole

climatic forcing. We conclude that for � < 35◦, the Northern Hemisphere ranges

vary in an as yet unexplained manner, which may be due to the unaccounted for local

nonlinearity there and/or the influence, beyond that of insolation, of one or more

additional forcing variables.

Low-latitude smoothing of range by latitudinal averagingof the growing season

We empirically discovered, through our thought experiment in Figure 3.10, that lat-

itudinally averaging the surface shortwave flux during the growing season to every-

where be that at autumnal equinox (22 September) explained well the positioning of

the observed gradient of range at high latitude, � ≥ 35◦N, as shown by the latter’s

imaginary projection through Rs = 0, � = 0 in Figures 3.11 (North America) and

3.15 (Northern Hemisphere). We now extend this use of the “equinoctial growing

season” to the low latitudes, � < 35◦N, where the gradient appears to be flattening,

although we have only one guiding range observation. However, Figure 3.10 shows

the surface shortwave flux to be “noisily” variable about an approximately constant

value of I0 ≈ 480 Wtot m−2 over the latitudes 0◦ ≤ � ≤ 35◦N, as is indicated by the

horizontal dashed line at those latitudes in that figure. What does this mean for the

continuous range of particular species over these latitudes?

In the appendices, we develop the form of the bioclimatic function βLt = g(

I0)

(for primary canopies), relating local average species to local average surface short-

wave flux. Because the local seasonal-average shortwave flux is a unique function

of local latitude, so then will the local average species be a unique function of local

latitude. Over the latitudes of present concern, 0◦ ≤ � ≤ 35◦N, the previously noted

small continuous latitudinal oscillation of the equinoctial-average I0 will produce a

small continuous latitudinal oscillation in both the local modal species and its range.

This latitudinal oscillation prevents the local modal species from being identical

across these latitudes.

We propose here that it is physically unrealistic for either the local modal species

or its range to be discontinuous at any latitude, and thus the range at � = 35◦N(−)



must equal that at � = 35◦N(+), with the latter being given by the gradient analysis

in Figure 3.16 to be Rs (35◦N) ∼= 3.6◦. At � = 35◦N (−) and below, we assume a

meridionally homogeneous equinoctial climate, dominated by vertical convective

mixing in which both range-controlling variables, σI0 and I0 are assumed to be

meridionally constant. Consistent with the accuracy of this entire work, the local

range at each latitude 0◦ ≤ � ≤ 35◦N will then be taken as constant at the value

Rs (35◦N) ∼= 3.6◦ and is shown by the horizontal solid line over those latitudes in

Figure 3.16.

Alternatively, we note that at low latitudes, the equinoctial average (Figure 3.10)

gives d I0/d� = 0, and using seasonal averages (Figure 3.13) dnI σI0/d� = 0, equa-

tion (3.25) leads to an indeterminate-range gradient if forced everywhere solely by

SW flux. One is led thereby to the conclusion that other factors may control species

selection in these tropical latitudes, and even that the selection process there is not

neutral [Hubbell, 2001], as we have assumed herein, but instead varies with species

at a given location. In support of this alternative view, Kraft et al. [2008] present

evidence supporting a nonneutral view of tropical forest dynamics, in which co-

occurring species display different ecological strategies. More will be said about such

tropical nonneutrality relative to both range and richness in chapter 4.

Range as a reflection of the bioclimatic dispersion of species

Because the mean light decreases with latitude above 17.5◦N in the Northern Hemi-

sphere, and thus through the bioclimatic function (equation (3.2)), the “size” of the

modal species, βLt , increases over these latitudes (Figure 3.3), and because we rea-

son (Figure 3.4) that only those species larger than the modal will exist locally, the

local modal species at one of these latitudes will be found also and exclusively at all

lesser latitudes southward to the latitude at which that particular species is locally the

“largest,” but given a normal distribution of “sizes,” is the least numerous (Figure 3.5).

This seems to indicate that under the current climate, the seeds of these local modal

species could not have originated at latitudes outside their respective current ranges.

Probabilistically, each species is most likely to have evolved at or near its respective

modal latitude, that is, where the frequency of its occurrence is maximum. Subse-

quently, its seeds have been carried to other latitudes, northward or southward, by

some fluid and/or animal agents. Restricting consideration for the moment to lati-

tudes above 17.5◦N in the Northern Hemisphere, the seeds of modal species carried

northward from their modal latitude into a regime of lower light cannot yield viable

plants, according to the theory put forth here (Figures 3.3 and 3.4). However, those

deposited and germinated southward of their modal latitude, where their seedlings

are “stable” (see Figure 3.4), will mature in decreasing local frequency with distance



from the latitude of their modal rank. This latitudinal span, from being the smallest

but most numerous (i.e., modal) local species at its northernmost location, �0, to

being the largest and least numerous (i.e., extreme) local species at its southernmost

location, �−, is, of course, the modal species’ range as we have defined it earlier in

this chapter.

For latitudes 0◦ ≤ � ≤ 17.5◦N, and using the latitudinally variable season length

of Table 2.2, I0 increases with � (Figure 3.13), the modal species decreases with �

(Figure 3.4), and the modal species at a given latitude will be dispersed northward

(imagine a mirror image of Figure 3.5).

From the observations available, we have found the extent of the southward latitu-

dinal bioclimatic dispersion of local modal species for � > 17.5◦N (i.e., the “range”

Rs(�)) to increase linearly with increasing latitude at latitudes greater than 35◦N; that

is, from equation (3.26), with the zonal averaging notation omitted for simplicity, the

range gradient, a dimensionless number, is written

d Rs(�)

d�=∣∣∣

d I0d�

∣∣∣ d(

nI σI0

)/

d� − nI σI0 d∣∣∣

d I0d�

∣∣∣

/

d�∣∣∣

d I0d�

∣∣∣

2 . (3.38)

For the set of observed conditions, that is, both constant,∣∣d I0/d�

∣∣, and positive, ��,

equation (3.38) becomes

d Rs(�)

d�= d

(

nI σI0

)/

d�∣∣d I0

/

d�∣∣

=∣∣∣∣∣

d(

nI σI0

)

d I0

∣∣∣∣∣. (3.39)

We note that equation (3.39) is a mixture of local and spatial variabilities of SW

flux and as such displays a (weak) analogy with the dimensionless Schmidt number

found useful in categorizing the physics of mass dispersion in fluids and defined [Bird

et al., 2002] as

Schmidt number = (local) viscous momentum diffusivity

(convective) mass diffusivity. (3.40)

Remembering the direct connection theorized here between SW flux and stable

species, as manifested in equation (3.1) and displayed in Figure 3.5, we use equations

(3.3), (3.4), and (3.6), and note that ns = nI , to rewrite equation (3.39) for small

curvatures of s(�) in its species form:

E = d Rs(�)

d�=∣∣∣∣∣

d(

nI σI0

)

d I0

∣∣∣∣∣

∼=∣∣∣∣

d (nI σs)

ds

∣∣∣∣, (3.41)

to which we assign the symbol E , signifying a dimensionless ecodynamic parameter

governing the climatically induced latitudinal dispersion of C3 plant species due to

variabilities in light. We call E the bioclimatic dispersion parameter.



This raises a question: is there some physical principle operating, perhaps in the

generic concept of such dispersion, to specify a critical and perhaps “universal”

value of this parameter? Such is often the case, of course, in physics, examples

being the value of unity for the Mach and Froude numbers, dividing regimes of

subcritical and supercritical flow, and the value of the Reynolds number, separating

laminar and turbulent flow regimes for particular geometric arrangements. These

examples all describe dynamic similarity of point (i.e., local) variability, whereas

dispersion imposes simultaneous spatial (i.e., convective) variability. This is exactly

what equation (3.41) does, and thus, for this dispersion process to be bioclimatically

similar from place to place, the value of E must be constant. Note once again that

this is not the mass dispersion characterized by equation (3.40); rather, it is species

dispersion caused by their selective forced emergence and stable support under the

combination of locally and latitudinally varying SW flux.

The magnitude of E , as evaluated from the climatic observations, is almost identical

for the continental land surface of North America (E = 0.105) and for the combined

continental land surfaces of the entire Northern Hemisphere (E = 0.104), even though

the components of the gradient (see equation (3.26)) are very different in the two cases

(equations (3.27) and (3.29) for North America; equation (3.34) for the Northern

Hemisphere). In addition, we can conclude from equation (3.41) that (1) if the variance

is the same at all latitudes and the mean light maintains a constant gradient (either

increasing or decreasing northward), the latitudinal gradient of range will vanish and

the range of the local modal species will be identical everywhere; (2) if the mean

light is the same at all latitudes and only the variance varies with latitude, the modal

species will be the same everywhere and its range will everywhere be the full span

of vegetated latitudes; (c) if the mean light decreases northward and the variance also

decreases northward, the range of the local modal species decreases northward; and

(d) if the mean light decreases northward and the variance increases northward, the

range of the local modal species increases northward without apparent limit.

These possible behaviors do not seem to reveal an optimum state and hence a

critical value for E . However, they do seem to indicate a convergence of the bioclimatic

dispersion at 17.5◦N.

A high-latitude shift in bioclimatic control from light to heat?

We now suggest that as the asymptotic limit of the bioclimatic function (equation (3.1))

is approached going northward, the control of 〈βLt〉 may begin to be shared with heat

because the ambient temperature falls below the evolutionary limit of first one and

then another species. We must recognize, then, that

βLt = f(

Io, T0)

, (3.42)



FIGURE 3.17 Latitudinal distribution of atmospheric temperature at the surface.

whereupon, assuming that I0 and T0 are both functions only of latitude, equation

(3.42) yields

dβLt

d�= ∂βLt

∂ I0

d I0

d�+ ∂βLt

∂ T0

dT0

d�. (3.43)

We have assumed in chapter 2 that species selection/adaptation ensures that the

mean leaf temperature, T�, is equal not only to the ambient atmospheric temperature,

T0, but also to the temperature, Tm , at which photosynthesis is maximally efficient.

Larcher [1983, Figure 3.35] shows the temperature dependence of carbon assimilation

for a wide variety of woody C3 plants, from Arctic pine (Pinus cembra), for which

Tm = 15◦C, to an Australian arid bush (Acacia craspedocarpa), for which Tm =37◦C. Assuming both the geographical representativeness of these species and the

completeness of the range reported, we conclude that “full-growth” C3 forest (in the

sense of no temperature limitation to productivity) should be confined globally to

those latitudes at which the zonally averaged mean daylight-hour growing-season

temperature,⟨

T0⟩

, obeys

15◦C ≤ ⟨T0⟩ ≤ 37◦C. (3.44)

We present in Figure 3.17 a composite latitudinal distribution of observed av-

erage atmospheric temperatures at the surface in moist climates of the Northern



Hemisphere, using temperature data of different types and from different sources, as

follows.

1. From 40 North American climate stations above latitude 45◦N [Canadian Cli-

mate Program, 1982, 1984], arranged in 5◦ zonal bands, we determined the variation

with latitude of the zonally averaged mean daily maximum temperature, 〈T0m〉, at the

surface during the growing season. (We use the mean daily maximum, rather than the

mean daylight-hour temperature, due to its ready availability.) These data are plotted

as the circles in Figure 3.17.

2. At intermediate latitudes, we supplement the Canadian data with values of the

average station maximum July surface temperature at scattered U.S. locations, as

given by the U.S. National Weather Service [1974; see Eagleson, 2002, Figure 10.6]

and plotted in Figure 3.17 as the triangular points. Note that these data mesh smoothly

with the Canadian data.

Note by the dashed straight lines between 45◦N and 65◦N, which approximate

the observed temperature distribution in Figure 3.17, that the temperature is nearly

constant at 〈T0m〉 = 14◦C between latitudes 52◦N and 60◦N, which coincide with the

approximate bounds to the boreal forest given by the maps of Bailey [1997], and that

the distribution changes quite abruptly from and to falling temperatures outside these

limits. Rosswall and Heal [1975] state that the warmest annual temperature at 15

sites in the tundra biome is 14◦C. Thus, with 15◦C apparently the lowest optimum

temperature for a C3 plant (Arctic pine) [Larcher, 1983, Figure 3.35], we select the

beginning of the boreal forest at 52◦N as the latitude at which temperature replaces

light in determining the local species of C3 plants and 60◦N as the latitude at which

decreasing temperature forces suboptimal productivity from these limiting Arctic

plants.

We have assumed that I0 ceases to control βLt for � ≥ 52◦N (i.e., ∂βLt/∂ I0 = 0

there), and we have seen in Figure 3.17 that T0(�) ∼= constant between 52◦N and

60◦N (i.e., dT0/d� = 0 there). Thus, from equation (3.43),

dβLt

d�= 0, 52◦N ≤ � ≤ 60◦N. (3.45)

The absence of need for the bioclimatic detail of equation (3.2) to achieve our

present goals suggests that the extension of this simple Taylor expansion technique

to other latitudes may require only substitution of one climatic forcing for another or

the addition of a second (or even third) forcing variable in a multivariable expansion.

Perhaps in the lower latitudes, bioclimatic control shifts to water. It seems important

as a next step to examine, in the same manner, the predictive ability of variations in

both seasonal precipitation and surface temperature.



Extension of these range forecasts by use of multipleforcing variables

Although we do not attempt here an expansion of our formulation to incorporate

one or more additional forcing variables, we note that this can be done without a

great deal of added complexity. Benjamin and Cornell [1970] point out that if the

species-climate relationship, s(c), is multivariate (rather than univariate, as assumed

here), that is, if

s = f (c1, c2, . . . , cn) , (3.46)

then for uncorrelated climate influences, ci , the first-order species variance is

σ 2s ≈

n∑

i=1

(

∂ f

∂ci

∣∣∣∣ci

)2

σ 2ci. (3.47)

We leave the pursuit of this higher-order approximation for others to explore.

A look ahead

As the variance of light increases northward, there is, under the univariate and one-

to-one forcing assumed herein, an increase in the variance of local species in this

direction. However, we cannot draw conclusions from this behavior alone concerning

the observed number (i.e., richness) of species at each latitude. Although it might

seem that richness should rise with the variance of light, we must remember that

to be counted, the species identifier (βLt in this work) must be a discrete, rather

than a continuous, variable. We approach this problem in chapter 4 by defining and

counting the discrete, cloud-forced, intraseasonal light (and hence heat) “pulses” that

we observe serve both as potential germinators of existing seeds of separate species

and as support for emergent plants on an assumed one-to-one basis.


C H A P T E R 4

Richness of Local Species:Prediction Versus Observation

Germination is the process of greatest importance for distribution ecology.Larcher [1983, p. 31]

Introduction

Wallace [1878] was probably the first to report a latitudinal gradient in observed local

richness of plant species (i.e., the number of species in a “community”), which he

noted to decrease by 3 orders of magnitude from tropics to tundra. (Stevens [1989]

reported similar gradients for other taxa.) Other similar observations have followed,

as detailed in chapter 1. Many investigators have postulated the cause of this rich-

ness gradient to be gradients in plant resources such as precipitation, soils, and light,

as summarized by Huston [1994]. Among them, Fischer [1960] suspected gradi-

ents in environmental factors such as temperature and humidity, whereas Currie and

Paquin [1987] as well as Scheiner and Rey-Benayas [1994] pointed to variations

in the biologically usable available energy. However, a theoretical expression con-

necting local richness to ecosystem dynamics is needed to understand, manage, and

prevent loss of biodiversity due to climate change [Carpenter et al., 2006; Weir and

Schluter, 2007; Marshall, 2007; Schluter and Weir, 2007], and no such theory has

been found.

Having had such success here (chapter 3), associating the local statistical distri-

bution of species, and hence their latitudinal range, with the local variations of SW

flux, and noting (in chapter 1) Rapoport’s observation of related range and richness

gradients, we follow Wilson’s [1992] “climate variability” insight and seek to derive

the number (i.e., richness) of local species also in terms of the properties of the dis-

tribution of local SW flux. Before beginning this derivation, it is helpful to see where

69



FIGURE 4.1 Latitudinal envelopes of observed plant richness. Dashed curve: from Reid andMiller [1989], adapted by Huston [1994, Figure 2.1, p. 20]; further adaptation here with permis-sion of the World Resources Institute and Cambridge University Press. Solid curve: data of Gentry[1988, 1995] as presented by Enquist and Niklas [2001] and further adapted here by permissionfrom Macmillan Publishers Ltd: NATURE, vol. 410, p. 656, Fig. 1a, c© 2001.

we are going. Accordingly, two frequently referenced latitudinal gradients of plant

richness are shown in Figure 4.1.

The solid line depicts the maximum envelope of local richness observations for

all tree species in 0.1 ha (103 m2) communities of the Northern Hemisphere. This

envelope derives from a global set of 227 observations of tropical and temperate

closed-canopy forest communities on six continents, which was assembled by Gentry

[1988, 1995] and later presented by Enquist and Niklas [2001, Figure 1a]. The latter

presentation of these data is reproduced entirely as Figure 1.1b and as a maximum

envelope for the Northern Hemisphere as the solid line in Figure 4.1.

The dashed line in Figure 4.1 depicts the maximum envelope of local richness

of all vascular plant species (including trees) observed by Davis et al. [1986] in

globally distributed communities of varying size. These community species counts

were subsequently scaled allometrically, as we discuss later, to a common 106 ha

(1010 m2 or 10,000 km2) area and are presented in bar graph form on a global map

by Reid and Miller [1989, Figure 4], and later as a (scaled) count versus latitude by

Huston [1994, Figure 2.1]. (The property of obeying a power law is called “scaling”

because power laws are a source of self-similarity that reveals the generic properties

of a class of systems without understanding all the details of the underlying processes

[Rodrıguez-Iturbe and Rinaldo, 1997; West et al., 1997; Martın and Goldenfeld,

2006].) The envelope of the scaled data points are shown by the solid circle symbols

and connecting dashed line in Figure 4.1. The scaling process is described as follows.


C H A P T E R 4 • R I C H N E S S O F L O C A L S P E C I E S 71

FIGURE 4.2 Latitudinal distribution of observed richness of vascular plants (continentalWestern Hemisphere). Adapted from Reid and Miller [1989] and Huston [1994] with permissionof the World Resources Institute and Cambridge University Press.

Lomolino [1989] credits Huxley as being first to recognize the so-called species-

area relationship, �s = cAz , scaling the number of separate local species, �s , ac-

cording to the local (“community”) area, A, in which the species are counted, and

with z and c being local constants [see Martın and Goldenfeld, 2006]. The value of z

for plants as found by Williamson [1988] is 0.2 < z < 0.4, whereas in earlier work,

Connor and McCoy [1979] found 0.15 < z < 0.35, in which the lower value best

represented the variability of continental areas and the higher value represented the

homogeneity of islands. It is easy to see that d�s/d A → 0 rapidly as A gets large and

thus that almost all the vascular species will be found in an area very much smaller

than the 106 ha chosen by Reid and Miller [1989] for “homogeneity scaling” of the

Davis et al. [1986] observations.

We restrict our theoretical analysis here to the limiting condition of maximum

species richness at each latitude. Therefore, to continue our (zeroth-order) comparison

of prediction with observation, we select from the literature those species counts

taken from the largest areas having homogeneous climate. In particular, we select

here the Davis et al. [1986] counts of vascular species scaled to 106 ha sampling areas

(presented in their entirety here as a continuous function in Figure 4.2) as representing

the maximum number of separate species that can be germinated and stably supported

(on average) by homogeneous climates at each latitude. This scaling ensures that the

areas sampled are comparable in size to that of the satellite pixel, 77,312 km2 (7.7 ×106 ha), at which the climate is resolved, enabling us to test our proposition that local

species counts are another biological manifestation of the local climatic variations

presented in chapter 2.

It is not clear from study of the open literature whether the Davis et al. [1986]

and Gentry [1988, 1995] databases are totally independent. However, assuming their

independence, we scale the Gentry data (solid curve in Figure 4.1) to 106 ha using

z = 0.25 at all latitudes, an average of the Connor and McCoy [1979] range. The



scaled values are plotted as plusses in Figure 4.1 and can be seen (for whatever

reason) to confirm the Davis data (dashed curve) almost exactly. We now proceed

using the “Davis curve” as the observational basis for evaluating our theory for the

maximum (i.e., “potential”) number of local species.

From continuous to discrete distribution of local species

Population richness is a recognized means of adapting to randomly fluctuating envi-

ronments [e.g., Huston, 1994; Kussell and Leibler, 2005], and we use this connection

here. We assume the species density within our large area to reflect variations in

the variety of separately favorable conditions for seed germination and shoot es-

tablishment experienced locally over the fundamental unit of vegetative time: the

growing season. These germination/establishment conditions are known to include,

among other things, (1) temperature, (2) moisture, (3) preceding period of dormancy,

(4) particular photoperiod, and (5) SW radiative flux.

Assuming the local species, s ≡ βLt (see Appendix B), to be continuously dis-

tributed, we have shown (chapter 3) its mean, s ≡ βLt , to be determined to the zeroth

order in a “neutral” model by the mean local canopy-top SW flux, I0, in the grow-

ing season (equation (3.2)), whereas the local species variance, σ 2s , is fixed by both

the local variance of seasonal SW flux, σ 2I0

, and the local gradient of mean species

with respect to mean seasonal SW flux, dβLt/d I0 (equation (3.6)). We must now

transform this continuously variable measure of local species into the discrete mea-

sure needed for counting the number of species defining local richness. In doing so,

the zeroth-order physical dependency of (continuously variable) species on SW flux,

demonstrated in the previous chapters, must also govern the discretely variable case.

Once again, the complexity of the issue demands approximation, and we extend our

zeroth-order, neutral-model approach of chapter 3 by selecting, as a proxy generator

of potentially enduring local species, the weak alternate land surface heating and

cooling caused by the local time variation in atmospheric interception of incoming

radiation by both transient clouds and the “clear” sky. We begin the development by

deconstructing the observed time series of local growing seasonal, canopy-top SW

flux into a sequence of irregularly sized and spaced, rectangular pulses of alternate

heating and cooling caused by the oscillating atmospheric transmittance due to clear

sky followed by cloud cover. These heating-cooling cycles are propagated to ground

level, where we consider them to be the local environmental “disturbances” that act

to germinate seeds (one of the three germination patterns recognized in the classic

work of Pickett and White [1985]) and to provide stressless initial shoot support

for the variety of local species. We assume that each pulse of canopy-top light

in a single growing season results in the germination and support of a separate

species due to unspecified differences in their photoperiod or in the initial heat and



moisture conditions in the soil when the pulse arrives. During the following local

growing season, the distribution of pulses and their initial conditions will be somewhat

different, germinating some species not present at the start of the season and failing

to support (by stressing) some species present at that time. We also assume that even

weak but repetitive short-term stress can limit the establishment of newly emerged

species. Finally, we assume that these gains and losses of species are equal in number

over time and therefore that we can estimate the local number of species as being

identical to the time average of the local number of light pulses received during the

local average growing season. However, because some local pulses are so small as to

germinate no species or are so similar as to germinate and support the same species,

we will refer to the “one-for-one” population, estimated as just described, as defining

the potential (i.e., maximum possible) number of local species, called herein max �s .

Local SW flux as a stationary Poisson stochastic process

To review briefly, the SW flux data presented in chapter 2 are based on satellite

observations defining canopy-top SW flux, i0 (Wtot m−2), spatially averaged at pixel

(referred to as “local”) scale every 3 hours [Pinker and Laszlo, 1992]. These i0 are

averaged in time over the local growing season for each of the k years of record to

get a time series of annual local averages, I0k (Wtot m−2), which are in turn averaged,

first over the k years to obtain I0, and then zonally at intervals of latitude to obtain

the estimates of zonal climatic mean,⟨

I0⟩

, and variance,⟨

σ 2I0

⟩

(W2tot m−4) reported in

Tables 3.3 and 3.4. Because we have no information about the longitudinal variation in

the richness observations of Davis et al. [1986], we omit the zonal averaging symbol,

〈. . .〉, in the remainder of this chapter and treat both the climate and the resulting

species richness as being zonally homogeneous.

It seems reasonable to assume that it is the larger local SW flux fluctuations,

I0 j > I0, that are responsible for germination and shoot emergence. However, we

have reasoned (see Figures 3.2 and 3.4) that for C3 species, only those supported by

the local annual fluctuations in SW flux satisfying I0 j ≤ I0 are unstressed on average,

and hence stable locally, and therefore survive to be counted in an observational study

of local richness. We begin our theoretical study of local richness by modeling the

instantaneous canopy-top SW flux, i0, at pixel scale during the growing season as

a stationary, Poisson-distributed arrival process of “rectangular pulse” disturbances

(see Figure 4.3), following the generalized method pioneered for “point” rainfall

by Todorovic [1968]. Simplifying the latter work for the stationary case, Eagleson

[1978] derived the first two moments of local annual rainfall, given the observed

properties, including frequency, of the constituent local storms. Here we have the

inverse problem: from the satellite observations (chapter 2), we have the seasonal

statistics of local canopy-top SW flux but not the number of these events due to



FIGURE 4.3 Idealized local time series of species-supporting cloud events.

the 3 hour gap between local revisitations by the satellite, and we seek to derive

the maximum annual number of (potentially) C3 species–supporting subseasonal

local SW flux events. These particular events will be identified by cloudiness, which

produces i0 ≤ I0, because these are associated with the seasonally stable and thus

countable species, βLt ≥ βLt , as we discussed in chapter 3 (Figures 3.2 and 3.4). If

i0 is not equal to or less than I0, it must be greater than I0, and thus the rectangular

pulses of our idealization alternate in time about the value I0, and there will be equal

numbers locally of the pulses, i0 > I0, and i0 ≤ I0. It is therefore immaterial to our

current purpose which class of pulses we choose to count.

Fundamental to this development is the formation of an idealized stationary time

series (see Figure 4.3) consisting of Poisson-distributed local arrivals [Benjamin and

Cornell, 1970] of pairs of alternating rectangular pulse cloud events. First of each

pair is a germination (i.e., warming) event, i0 > I0, of duration tb (hours), followed

immediately by a shoot-support event, i0 ≤ I0 (for C3 species stability, as shown in

Figure 3.2), of duration tc (hours). We seek to estimate the first two moments of υ,

the local number of complete-pair arrivals during a single growing season.

The C3 species–supporting cloud events, i0 ≤ I0, are taken to be independent and

identically distributed and are modeled by the shaded rectangular pulses in Figure 4.3.

We let the length of this time series be τ , the daylight-hour length of the local growing

season, because this is the basic unit of ecological time. The seasonal canopy-top

SW flux, I0 (not shown in Figure 4.3), represents i0 averaged over that time, τ ,

and is a variable from year to year at any given latitude. The SW flux, I�, at the



top of the atmosphere during daylight in the growing season is taken to be a separate,

time-averaged constant, I�, at each latitude, as given in Figure 2.7.

Distribution of C 3 species–supporting radiation interceptedin a growing season

The total shortwave radiant energy, R(ν), intercepted by a random number, ν, of C3

species–supporting cloud events, i0 ≤ I0 (shaded areas in Figure 4.3), each intercept-

ing the random amount of energy, h j , is written

R(ν) =ν∑

j=1

h j (Wtot h m−2), (4.1)

and its probability density function is fR(ν) (r ).

With ω being the seasonal rate of event arrivals, the probability, p|τ (ν), of

obtaining exactly ν arrivals of such cloud events, , in the local annual growing

season, t = τ , is, for small values of the ratio of mean event duration, mtc , to the

mean time between events, mtb , given [Cox and Lewis, 1966; Benjamin and Cornell,

1970] by the Poisson distribution

p|τ (ν) = (ωτ )ν e−ωτ

ν!, ν = 0, 1, 2, . . . , (4.2)

with the mean and variance of the number of arrivals, ν, given by

E [| τ ] ≡ mν = ωτ (4.3)

VAR [| τ ] ≡ σ 2ν = ωτ. (4.4)

We must note that for the radiational series being modeled here, the ratio mtc

/

mtb

will truly be small only in the more cloud-free (often arid) climates, and therefore

the Poisson model is not strictly valid elsewhere (such as in the moist tropics). Here

we assume it to be adequate everywhere at the level of approximation being used.

We note also that since p|τ (ν) < 1, it will take many seasons to thoroughly sample

the full local range of cloudiness, and consequently many seasons to develop the full

local range of stable C3 species.

The probability density of total local shortwave energy, R, available for stable

local speciation during the annual growing season, t = τ , is given by summing the

probability densities, fR(ν)(r ), for each of the mutually exclusive (i.e., different) and

collectively exhaustive (i.e., complete) number of cloud events, , that can together

intercept r species-supporting energy in this time, each weighted by the discrete



FIGURE 4.4 Dimensionless gamma dis-tribution, G (κ, λh). Adapted from Benjaminand Cornell [1970, Figure 3.2.4].

probability, p|τ (ν), that exactly ν C3 species–supporting events will occur. This

gives the compound distribution

fR(τ )(r ) =∞∑

ν=1fR(ν) (r )p|τ (ν), r > 0

p|τ (0) = e−ωτ , r = 0

⎫

⎪⎬

⎪⎭

. (4.5)

To complete this derivation, we need the density function fR(ν)(r ), where R(ν) is

the sum of the random variables, h j , as given by equation (4.1). We assume the h j to be

independent and identically distributed random variables (having units Wtot h m−2),

for which we select the versatile (i.e., fits a variety of distributional shapes) and

analytically tractable gamma distribution [Benjamin and Cornell, 1970]

G (κ, λh) = (λh)κ−1 e−λh

(κ), (4.6)

and its form is illustrated graphically in Figure 4.4. In dimensional form, equa-

tion (4.6) is

fH (h) = λ (λh)κ−1 e−λh

(κ), (4.7)

where κ is the shape parameter (dimensionless) and λ is the scale parameter

(W−1tot h−1 m2). From equation (4.7), the mean energy intercepted by a single species-

supporting event is

mh = κ/λ (Wtot h m−2), (4.8)

and its variance is

σ 2h = κ/λ2

(

W2tot h2 m−4

)

. (4.9)

Or, defining

η ≡ m−1h , (4.10)



we have

λ = ηκ. (4.11)

The gamma distribution is conveniently regenerative, meaning that the sum of ν

independent gamma-distributed variables having common λ and κ is also gamma

distributed, with

κν =ν∑

j=1

κ j = νκ (4.12)

λν = λ. (4.13)

Using the preceding, we can write the dimensionless density function for the total

shortwave energy, R(ν), intercepted by ν C3 species–supporting cloud events, as

G R(ν)(λr ) = (λr )νκ−1 e−λr

(νκ), (4.14)

or, in dimensional form, as

fR(ν)(r ) = λ(λr )νκ−1e−λr

(νκ). (4.15)

The mean of this distribution is

m R(ν) = νκ/λ (Wtot h m−2), (4.16)

and the variance is

σ 2R(ν) = νκ/λ2

(

W2tot h2 m−4

)

. (4.17)

Finally, substituting equations (4.2) and (4.15) into equation (4.5) and using equa-

tion (4.11), we get the compound distribution of cumulative intercepted, C3 species-

supporting, shortwave radiant energy over the growing season, t = τ :

fR(τ )(r ) =∞∑

ν=1

ηκ (ηκr )νκ−1 e−ηκr

(νκ)· (ωτ )ν e−ωτ

ν!, r > 0

pτ(0) = e−ωτ , r = 0

⎫

⎪⎪⎬

⎪⎪⎭

. (4.18)

Moments of C 3 species–supporting radiation interceptedin a growing season

With exactly ν C3 species–supporting cloud events in a growing season, equation

(4.1) gives

Rτ ≡ R(ν) =ν∑

j=1

h j (Wtot h m−2), (4.19)



which, as has been pointed out earlier, is a random number of random numbers. We

now approximate the single-season canopy-top SW flux, I0, as

I0 = I� − Rτ

αcτ(Wtot m−2), (4.20)

in which I�, the seasonally averaged top-of-the-atmosphere “astronomical” SW flux,

varies with latitude only, as shown in Figure 2.7, from D. Entekhabi (personal commu-

nication, 2007), and αc is the fraction of the season experiencing i0 ≤ I0. We assume

both αc and τ also to vary primarily with latitude and not greatly from year to year.

Then, at any given latitude, αc∼= constant and τ ∼= mτ , and equation (4.20) yields

estimates of the first two moments of I0 to be

E[I0] ≡ I0 = I� − m Rτ

αcmτ

(Wtot m−2) (4.21)

VAR [I0] ≡ σ 2I0

= 1

α2c m2

τ

VAR [Rτ ] = σ 2Rτ

/

α2c m2

τ

(

W2tot m−4

)

. (4.22)

From equation (4.19) [see Benjamin and Cornell, 1970],

E[Rτ ] = mνmh (Wtot m−2) (4.23)

VAR[Rτ ] = E[

R2τ

] + E2[Rτ ](

W2tot m−4

)

, (4.24)

or

VAR [Rτ ] = mνσ2h + m2

hσ2ν

(

W2tot m−4

)

. (4.25)

Moments of the number of C 3 species–supporting cloud eventsin a growing season

Using equations (4.3), (4.4), (4.8), (4.9), and (4.11), equation (4.25) becomes

σ 2Rτ

= mν

κ

λ2+ κ2

λ2mν = m2

νm2h

[1

κmν

+ 1

mν

]

= m2Rτ

mν

[

1 + 1

κ

] (

W2tot m−4

)

,

(4.26)

or, rearranging for the immediate purposes,

mν = m2Rτ

σ 2Rτ

[

1 + 1

κ

]

, (4.27)

which represents the mean annual number of separate C3 species–supporting events in

a growing season. We use equations (4.21) and (4.22) to eliminate the dependence of

equation (4.27) on the fractional season length, αcmτ , obtaining, finally, at � = �◦,

mν =(

I� − I0)2

σ 2I0

[

1 + 1

κ

]

, (4.28)



and using equations (4.3) and (4.4),

σν = m1/2ν , (4.29)


σν =(

I� − I0)

σI0

[

1 + 1

κ

]1/2

. (4.30)

From climatic disturbance to C 3 species germination

Remember that our development of equations (4.28) and (4.30) has given us properties

of the local climate, in particular, the first two moments of the distribution of the

numbers of annual SW flux events supporting stable local C3 species. Each of these

support events is paired with a preceding unseasonably “warm” event, i0 > I0, which

(see Figure 3.2) is in a class of annual “disturbances” to the local regime of SW flux.

Considering the growing season to be the basic unit of local ecological time, we

now introduce the proposition that the local biological consequence of these local

seasonal disturbances is the germination and support of a specific set of C3 species.

Certainly, as we pointed out earlier, the full spectrum of potential species implied by

the magnitudes of mν and σν is not realized in a single growing season; many seasons

will be required, and some unconsidered local climatic (such as serial dependence

of pulses) or nutritional shortcoming may intervene at any time. We have assumed

the underlying time series to be stationary, and thus subsequent local seasons will

support the same number of C3 species (statistically speaking), which should be the

same species for the most part. We further assume that over a sufficient number of

seasons, local births (i.e., emergences or speciations) and deaths (or extinctions) will

be equal. We therefore suggest that with increasing time, t , the maximum possible

(i.e., the potential) number of local C3 species that can be realized, max �s , should

approach, but not exceed, the maximum number of local C3 species–enabling SW

flux disturbances, νmax, as given by the moments of the underlying distribution.

Furthermore, the larger νmax is, the larger is that fraction of the ν that is not sufficiently

different from its cohorts such as to germinate different C3 species. For this reason, we

expect the difference νmax − max �s to grow with increasing νmax. Using equations

(4.28) and (4.30), this is written

max �s —→t→∞ νmax = mν + niσν, (4.31)

which is valid at and above the scale of the vegetation community at which the

total number of species supported becomes independent of the supporting area. The

scaled observations of Davis et al. [1986; see Reid and Miller, 1989] are from areas

orders of magnitude larger than those for the data of Gentry [1988, 1995; see Enquist



TABLE 4.1 Estimation of the Gamma Function Shape Factor, κa

Latitudeb (deg) Location κ Reference

10 Kurmuk, Sudan 0.65 El-Hemry [1980]10 Gambela, Sudan 0.41 El-Hemry [1980]10 Chali, Sudan 0.18 El-Hemry [1980]10 Doro, Sudan 0.55 El-Hemry [1980]10 Tabus Bridge, Sudan 1.64 El-Hemry [1980]10 Daga Post, Sudan 0.46 El-Hemry [1980]25 Riyadh, Saudi Arabia 0.33 Eagleson [1981]26 Al Wajh, Saudi Arabia 0.73 Eagleson [1981]27 Qasim, Saudi Arabia 0.40 Caro and Eagleson [1981]26–49 75 first-order stations in continental 0.47 Hawk and Eagleson [1992]c

United Statesc

aOverall average, κ = 0.48.bApproximate.cReproduced by Eagleson [2002, Appendix F].

and Niklas, 2001], and thus the species counts of the former are more likely to be

unconstrained by the area sampled and will be the object of our attempt at prediction

of local C3 species richness.

Parameter estimation

In equation (4.31), ni defines the number of standard deviations above the mean at

which new C3 species can no longer be identified, and both ni and κ are required to

estimate νmax. Both parameters are difficult to specify for prediction purposes, and

we estimate each a priori by the following approximations.

Regarding κ , owing to the large pixel scale (77,312 km2) and the 3 hour pixel

revisitation time, we do not attempt to use the satellite radiation data to isolate series

of individual cloud events for estimation of the necessary values of κ . Instead, we

make use of previous estimates of κ from studies of 84 separate (point) rainstorm

series measured at ground stations in the continental Northern Hemisphere. Sampling

a range of latitudes and climates from arid to moist-tropical, these studies show re-

markable consistency of κ , as summarized in Table 4.1, and yield an average value

κ = 0.48. In Figure 4.5, we plot, as open circles and as a function of latitude, those

estimates of κ deemed most reliable, that is, the 75 estimated [Hawk and Eagleson,

1992] from first-order weather stations in the continental United States and having

an average value at these higher latitudes of κ = 0.47. The average value of these

estimates in each 5◦ zonal band is plotted in Figure 4.5 as a solid circle, from which

we see that these average κs vary latitudinally in a regular manner, which can add



FIGURE 4.5 Latitudinal variation of the gamma distribution shape factor, κ . Estimated [Hawkand Eagleson, 1992] from a series of local rainstorm arrivals from April to September in NorthAmerica, as observed at first-order weather stations by the U.S. Weather Service. Reproduced byEagleson [2002, Appendix F] c© Cambridge University Press. Reprinted with permission.

significantly to the latitudinal variability of the predicted species richness. Never-

theless, in the spirit of our zeroth-order effort, we ignore this variation and use the

average value, κ = 0.48, at all latitudes when evaluating equations (4.28) and (4.30)

in Table 4.2 for later comparison with the Northern Hemisphere observations of Davis

et al. [1986], shown in Figure 4.2. It would be wise to confirm the representativeness

of this rainstorm proxy through generation of a true climatology of local short-term

surface variations in SW flux.

Regarding ni , this multiple of the standard deviation of the number of SW flux

disturbance pairs defines the desired degree of completeness of the probability mass

function and is awkward to compute for the underlying Poisson distribution (equation

(4.2)). Although for small values of the shape parameter, ωτ ≡ mν (equation (4.3)),

say, mν = 5, the Poisson probability mass function is skewed and estimation of ni is

tedious, for larger values, say, greater than mν = 35, the Poisson distribution is closely

normal (see http://en.wikipedia.org/wiki/Poisson distribution), and estimation of ni

is simple using the normal probability mass function presented earlier in Figure 3.6.

Applying equation (4.28) to our Western Hemisphere data (see Table 4.3, column 5)

yields 562 ≤ mν ≤ 5425, justifying our use of the normal approximation, ni = 2.5 at

99% probability mass for the single-sided distribution, i0 ≤ I0. We note in columns

5 and 7 of Table 4.2 that using this approximation, the niσν contribution to νmax is

equal to or less than 10% at all latitudes, and therefore fidelity in estimating ni is

unimportant at the accuracy level pursued herein.



TABLE 4.2 Maximum Richness of Local Species in the ContinentalNorthern Hemispherea

� I b� 〈 I 0〉c 〈σ I 0 〉c

(deg. latitude) (Wtot m−2) (Wtot m−2) (Wtot m−2) mνd,e σν

d, f ni σνg max�s

d,h

0 825 440 9.2 5425 74 185 56105 840 450 8.6 6370 80 200 6570

10 848 475 8.0 6734 82 205 693915 849 494 8.7 5158 72 180 533820 844 494 9.2 4483 67 168 465125 832 484 12.0 2605 51 128 273330 814 481 12.6 2164 47 118 228235 789 469 13.9 1642 41 103 174540 758 439 12.9 1894 44 110 200445 721 404 12.6 1961 44 110 196150 677 362 14.4 1482 39 98 158055 628 335 17.5 868 29 73 94160 572 315 16.4 761 28 70 83165 506 269 17.6 562 24 60 622

aFor predictive comparison with Reid andMiller [1989] data in Figure 4.6.bApril–September Season. Figure 2.7.cTable 2.3.dAt κ = 0.48 (Table 4.1).eEquation (4.28). Mean annual number of local SW flux “pairs” (Figure 4.3).f Equation (4.30). Standard deviation of annual number of local SW flux “pairs” (Figure 4.3).gOr 2.5 × σν . Approximate maximum variation frommean species (ni = 2.5; Figure 3.6 at 99%).hEquation (4.31), columns 5 + 7. Maximum number of local species.

Predicted potential richness versus observed richness

Figure 4.6 shows the results of the comparison of equation (4.31), in which we let

max �s = mν + niσν, (4.32)

with the observations of Davis et al. [1986], as scaled by Reid and Miller [1989]. In

this figure, the local counts of the number of observed separate species of all vascular

plants, as scaled to their maximum allometrically, are plotted as the solid circles at

the latitudes of observation from the equator to 57.5◦ and are connected by a dashed

line, whereas the theoretical maximum number of C3 species, max �s , as given by

equation (4.32) and Table 4.2, is plotted as plus signs connected by a solid line over the

same latitudinal range. Figure 4.6 illustrates how well equation (4.32) represents the

observed local species counts for (1) � ≥ 22◦, where the agreement of theory and ob-

servation defies credulity, as did that for range, over essentially the same latitudes, as

was seen in Figure 3.10 (not only are theory and observation very close, but also, even

the slight “waves” in observed species numbers, the basis for which is visible in the

raw climate data of Figure 3.7, seem to be captured by the theory), and (2) � < 22◦,



FIGURE 4.6 Latitudinal variation of local species richness in the continental Western Hemi-sphere: prediction versus observation for vascular plants. Observations (black dots) of Reid andMiller [1989] as presented by Huston [1994] and adapted here with permission of Cambridge Uni-versity Press.

where, with one exception (� = 5◦), the theory provides a reasonable upper envelope

of the observations. The cause of the oscillating variability of the observed species

numbers at the lower latitudes is unclear, although pronounced oscillations in the

climate variables there are apparent in Figure 3.12. Considering our similar difficulty

with range prediction in the lower latitudes (Figure 3.13), other climate and/or soil

variables and/or unconsidered mechanisms may be exerting their effects. For example,

the anomalously large richness observed at � = 5◦ may be evidence of what Stevens

[1989, p. 253] hypothesized as a “constant input of accidentals” in the tropics, meaning

species continuously dispersed southward from the temperate zone and, while ger-

minated, sprouted and counted in tropical latitudes are not individually stable in the

long term due to the radically reduced intensity and scale of the climate disturbances

there when compared to conditions at the latitude of their modality. Huston [1994]

has recognized this hypothesis of Stevens [1989] as that termed earlier as the “rescue

effect” by Brown and Kodric-Brown [1977] and the “mass effect” by Wilson [1965].

Or, as seems more likely C3 species dominate in the relatively low incident light of

the extratropics, and under the reduced light of the tropical rainforests, they remain

prominent. However, C4 species dominate the tropical grasslands, and within the

tropical rainforests, the vertical variation in microclimate (i.e., light, moisture, and

even plant nutrients due to the tree-clinging body waste of climbing and flying animal

life) establishes a small but effective third spatial dimension to the plant habitat, which

supports extraordinary numbers of the CAM species of epiphytes such as bromeliads

and orchids. Furthermore, the relatively constant local soil temperatures over the year

make their fluctuations a questionable basis for selective plant germination at tropical

latitudes. Thus it is not surprising that the observations of all vascular plants (C3, C4,

and CAM) at about 5◦N latitude in the Western Hemisphere (Figure 4.6) exceed our

predicted numbers of C3 species there by about 40%.



Hubbell et al. [2008], fitting Neutral Theory (which does not posit a specific bio-

climatic forcing, see glossary) to observed abundances of Amazonian tree genera

(without the C3 restriction), and using the commonality of metacommunity dynamics

for species and genera inherent in Neutral Theory, have expanded the genera obser-

vations to species level to predict (for example) a total tree species count (without

the C3 restriction) of 11,210 for the Brazilian Amazon basin, which has an average

latitude of about 5◦S. This estimate is almost twice that of the 6000 C3 species we

predict at 5◦N but is in line with the observations at that latitude for all vascular plants

[Davis et al., 1986] shown in Figure 4.2. We should note that absent an explicit tie to

climate, the Neutral Theory estimate of local species numbers cannot be adjusted a

priori to account for future climate change, as can the neutral theory presented herein.

However, to the extent that κ is dependent on climate over the range of climate

being studied, our neutral theory may lose some of this latter advantage over Hubbell’s

[2001; Hubbell et al., 2008] Neutral Theory. For example, “intertropical convergence”

of surface winds at low latitudes leads to the dominance there of small-scale vertical

transport of moisture and heat through moist convection, which peaks at about 10◦N

and 10◦S. The smaller space and timescales of the resulting moist-convective cloud

there cause the local shape parameter, κ , in the gamma distribution of light interception

events (Figure 4.4) to be smallest at or near these latitudes. Indeed, the decreasing

trend of κ equatorward is indicated clearly in Figure 4.5 by the plotted 2◦ zonal

averages (dark circles) of observations within the United States. Incorporation of

this higher-order scale effect will likely improve the C3 richness predictions at low

latitudes but will not be undertaken in this work.

The theoretical tie between range and richness

Stevens [1989] notes there to be an unspecified “ecological connection” producing an

inverse relation between the geographical range of species and the species richness at

common latitude. Repeated here, equation (3.22) leads to our theoretical forecast of

the range of the mean local species at � = �◦ as

Rs|�0 (�) = nsσs|�−(�) ≈ nI σs|�−(�) = 3.6

⟨

σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

�◦−R�◦

. (4.33)

The species richness at � = �◦ − R�◦ = �−, as approximated by equation (4.28),

also contains the forcing σI0 . Eliminating this factor between equations (4.28) and

(4.33), we obtain

Rs|�◦(�) = 3.6[

1 + 1

κ

]1/2⟨[

I� − I0

ν1/2max

] ∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⟩

�◦−R�◦

, (4.34)

thereby providing an inverse theoretical relation between range and richness, as re-

ported observationally by Stevens [1989].


P A R T I I I

RECAPITULATION

Reductionism

Reductionism . . . is the search strategy employed to find points of entry into other-wise impenetrably complex systems. . . . [It] is the primary and essential activityof science.

Wilson [1999, p. 59]

85


C H A P T E R 5

Summary and Conclusions

Precis

Summary In this work, we have attempted to explain analytically, to “zeroth order”

and in terms of Darwinian interactions of biology and climate, the long-observed,

opposing latitudinal gradients of the range and richness of vascular land plants,

which constitute about 98% of all Earth’s land plant species (see the frontispiece).

Our theoretical model for both range and richness is based on the asymptotic form

of the photosynthetic capacity curve unique to the leaves of C3 plants (Figure 3.2b),

which category embraces the bulk of all vascular land plants (93%, as estimated

in the frontispiece). The intersection of the asymptotes of such saturating carbon

assimilation functions defines, for leaves of each C3 species, the intensity of light that

simultaneously maximizes CO2 assimilation efficiency and minimizes plant stress

(i.e., “instability”) for that species (see Appendix A). We take these intersections

to define the preferred, “Darwinian” operating points of C3 species, which together

constitute a one-to-one relationship between the “climate,” c (i.e., that optimal incident

SW flux), and a common measure, s, of specific species “biology.” Expanded to canopy

scale (see Appendix B), s is shown to be the canopy’s horizontal leaf area index, and

the functional relationship s = g(c) is called herein the “bioclimatic function.” We

show that neither the form nor the sense of this function plays a role in either range

or richness at our level of approximation.

Conclusion Using surface observations at leaf and canopy scales (Appendixes

A–C), with c being the local growing season average SW flux at canopy top, I0

(i.e., the “light”), and the C3 species measure s being the resulting local unstressed

horizontal leaf area index, βLt (i.e., the “species”) [Eagleson, 2002, Appendix H],

87



we find the bioclimatic function s = g(c) to have the alternative zeroth-order forms

Ec =SW flux

︷︸︸︷

I0[

1 − e−βLt] ∼= ε−1

i

CO2 fluxfrom plant biology

︷︸︸︷[

ca

R�cmin

/

βLt

]

Single-species canopy Single-species canopy

EcI = 0.61MJparm−2p h−1 EcC = 0.62MJparm−2

p h−1.

(5.1)

Using the average value εi = 0.81gs MJ−1tot for the potential assimilation efficiency

of intercepted light, as obtained from observations reported in the literature (see

Figure A5), we show in Appendix C that the function Ec represents an equality of the

maximum carbon supply to, and the maximum carbon demand by, the C3 plant when in

a state we call “evolutionary equilibrium.” Using data from the literature, Ec is shown

there to have, at these scales, the nearly constant average value Ec = 172 Wtot m−2,

which completes the zeroth-order bioclimatic function, s = g(c), as

βLt = �n

[

1

1 − 172/

I0

]

, I0 > 172 Wtot m−2, (5.2)

showing that βLt ↑ as I0 ↓ (Figure 3.2).

This bioclimatic function (equation (5.2)) demonstrates, at zeroth order, a one-to-

one relationship between values of local incident SW flux and maximally productive

local C3 species, thereby enabling transformation of a probability density function

(pdf ) of observed local SW flux into an estimate of the pdf of resulting potential

local species. It is shown that those local C3 species having βLt smaller than the local

average, βLt , will be stressed on the average; hence they are unstable in the long

term in that location and are assumed to be missing from the pdf of species existing

there. Similar reasoning shows local C3 species larger than the local average to be

unstressed on average, and thus they are assumed to be stable and present locally

but underproductive there compared with other locally present species (Figures 3.2

and 3.4).

At most latitudes, the pdf of the local annual seasonal-average SW flux, I0, is

shown from observations to be approximately normal about its long-term mean,

I0, although with truncated extremes (Figures 3.7 and 3.12). Although the local

coefficient of variation of I0 is less than 10% (Tables 2.3 and 2.4), the suggested

form of the bioclimatic function (equation (5.2)) is sufficiently nonlinear that the

corresponding coefficient of variation of βLt is high. Nevertheless, the analytical

benefits of linearity are so great, and the true form of the bioclimatic function is

sufficiently uncertain, that we extended our zeroth-order approach to include the


C H A P T E R 5 • S U M M A R Y A N D C O N C L U S I O N S 89

assumption of a “sufficient” degree of local linearity of this function. Major analytical

benefits of this linear approximation include the following: (1) the pdf of local C3

species measure, s, will also be approximately normal about the local average, s,

everywhere with corresponding truncation of the extremes; (2) the local mean values, c

and s, also satisfy the bioclimatic function; and (3) the bioclimatic function, s = g(c),

can be expanded in a useful Taylor series about the local mean.

Mathematical approximations in range calculation

Summary Wherever I0 has a one-to-one relationship with �, the bioclimatic

function of the means can be written βLt = g( I0) = h(�), as is the case for North

America and as is shown in Figure 3.5, along with two sketched distributions of species

about their local mean. Figure 3.5 demonstrates how the modal species at latitude �0

is also found in the (positive) tail of the (one-sided) distributions of stable C3 species

over a continuous span of lesser latitudes, ��−, which defines the range of the modal

C3 species at �0. With the distribution of local I0 being approximated as normal about

the long-term local mean, I0, and the bioclimatic function being locally linear, the

local number of C3 species standard deviations at truncation, ns , equals the observed

number, nI , of I0 standard deviations at truncation for the same probability mass

(Figures 3.7 and 3.12). Except for special cases, estimation of the range nsσs(�)

is then reduced to estimating the C3 species standard deviation in latitude units,

σs(�).

Conclusion The C3 species standard deviation, σs(s), is given (chapter 1) through

a Taylor series expansion of the bioclimatic function about its mean to be σs(s) ∼=σc|ds/dc|. Wherever s is a single-valued and locally linear function of �, this standard

deviation can be rewritten in the desired units of latitude (see chapter 3) as

σs(�) ≈ σc

|dc/d�| . (5.3)

Combining σs(�) with the latitudinal gradient, ds/d�, of the local average species,

we are able to estimate the range of the local modal C3 species, as shown in equation

(3.8) and Figure 3.5.

The predicted range of off-mode species at �0 may be found similarly and averaged

for exact comparison with the reporting of Brockman’s [1968] observations, presented

in Figure 1.1a. However, with small variance of the local I0, we assume small local

variance of βLt and hence compare the theoretical range of the local mode directly

with Brockman’s [1968] observed mean of the local ranges.

Considering the “equinoctial average growing season,” as shown in Figure 3.10,

the denominator of equation (5.3) vanishes over the tropical latitudes, whereas its



numerator (Figure 3.13) appears oscillating, indicating a very large variance of the

continuous variable defining C3 species. It seems reasonable to conclude that this

implies a very large number of discrete C3 species in the tropics, as we find in

chapter 4.

Evaluation of range prediction

Summary For North America, where I0↓ as �↑, the range, R (in units of latitude),

of the modal species, s, at latitude �0 depends on the number, ns ≈ nI , of standard

deviations (in units of latitude) of the species distribution, σs|�−(�), at lower latitude

�−, as given by Figure 3.5, in the form

Rs|�◦(�) = nsσs|�−(�) ∼= nI ·⎡

⎣σI0

∣∣∣∣∣

d I0

d�

∣∣∣∣∣

−1⎤

⎦

∣∣∣∣∣∣�−

. (5.4)

Calculation of R from equation (5.4) at a given �0 is a trial solution, but by choosing

�−, the solution for R is direct.

Conclusion We have used satellite remotely sensed SW flux observations pub-

lished by NASA and reduced for this work by D. Entekhabi (personal communica-

tion, 2007) to evaluate equation (5.4) on a point-by-point (i.e., �− by �−) basis for

North America (Table 3.3). We find (Figure 3.9) that for 46◦ < � = �0 < 63◦, equa-

tion (5.4) reproduces the Brockman [1968] observations with high accuracy in North

America when using a meridional average of nI ; however, below � = 46◦N, equation

(5.4) increasingly overpredicts the range. Perhaps this is due to the estimation error,

�R, introduced by our (increasingly poor) linearization of the bioclimatic function at

these lower latitudes, and to our continued assumption there that ns = nI . Examining

our evaluations of the three SW flux factors of equation (5.4), as plotted versus �

(Figure 3.8), we note oscillations about a linear trend in each. To eliminate any effect

of these oscillations on our range estimates, and noting the closely linear variation

of the observed ranges with latitude, we instead calculate the latitudinal gradient,

d R/d�, of equation (5.4) using the necessary gradients of each factor as determined

from the data by linear least squares fitting. The resulting estimate of d R/d� is almost

exactly that of the Brockman [1968] observations at and above � = 35◦. To compare

R versus �, however, requires locating the gradient in R, � space. We accomplish

this through a thought experiment that imagines an atmosphere as dry globally as the

actual atmosphere is only above 40◦N and 40◦S latitude. The experiment assumes

that at autumnal equinox, the imaginary atmosphere will have the same d I0/d� at

all latitudes, as the real atmosphere has above 40◦. This real gradient of I0 projects

in both hemispheres to a common maximum, I0, at the equator (Figure 3.10) for our

imaginary world, which signifies that the range will be zero at � = 0 (Figure 3.16b).



FIGURE 5.1 Range and richness of vascular land plants on the continents. Range: Theory isfor C 3 vascular land plants in North America; observations are for all trees (open circles) in NorthAmerica [Brockman, 1968]. Richness: Theory is for C 3 vascular land plants in the Northern Hemi-sphere; observations are for all vascular land plants (solid circles) in the Western Hemisphere, aspresented by Huston [1994, Figure 2.1] based on Reid and Miller [1989], reprinted with permissionof Cambridge University Press, and of the World Resources Institute; and for all trees (pluses) in theNorthern Hemisphere [Gentry, 1988, 1995], as scaled in Figure 4.1.

The real-world gradient of R above 40◦ must then project to the 0,0 origin of R versus

�, which the observations verify (Figure 3.11).

It is interesting to note that the point-by-point estimates of ranges for the entire

Northern Hemisphere show no relationship (Figure 3.14) to the North American

observations of Brockman [1968], probably due to the quite prominent oscillations in

σI0 (Figure 3.13). However, the (dimensionless) range gradient, E , calculated using

Northern Hemisphere climate data (Figure 3.13), demonstrates virtually the identical

magnitude (E = 0.104) to that found for North America (E = 0.105). We interpret

this to indicate the same extent of southward latitudinal dispersion of C3 species

on the separate continents of the Northern Hemisphere (at least), and we suggest Eas a possible new dimensionless ecodynamic similarity parameter characterizing a

southward latitudinal dispersion process for vascular land plants.

We have satisfied our goal of explaining theoretically, for latitudes above 29◦N,

the observed rise with latitude of the median range of the local vascular land plant

species using a “neutral theory,” in which that range is dependent on the local temporal

and spatial variability of light, irrespective of species. The comparison of theory and

observation is shown again here in Figure 5.1. Had we used local temperature, rather

than SW flux, as the independent driver of species range in a model similar to that

used here, the constancy of surface temperature observed (Figure 3.17) over 8◦ of

latitude centered at 56◦N would have been reflected in the predicted range at this

latitude. The absence of such an anomaly in observed range lends further weight



to our conclusion that SW flux is the principal determinant of range, at least above

35◦N.

Evaluation of richness prediction

Summary Having noted from the literature [e.g., Rapoport, 1975] observations

of an inverse relationship between the latitudinal gradients of range and richness, and

having found herein that variability of light controls the local statistical distribution

of C3 species, and hence their range (at least above tropical latitudes), it seems

logical that variability of light controls the local number of C3 species as well at

those latitudes. Biology enters once again at this point through species-selective seed

germination (along with subsequent stressless support of the emerging plant) by intra-

annual fluctuations in light. The importance of the germination mechanism has been

pointed out by many investigators, but the words of Larcher [1983] are particularly

relevant here: he notes that a temperature alteration between day and night promotes

germination and that germination is the process of greatest importance to distribution

ecology.

Once again, the formulation of the problem is statistical and is species-neutral in

the Hubbell [2001] sense. We simplify the time series of local, “instantaneous” SW

radiation, i0, into a time series of rectangular pulses, and we count the random number

of times, ϑ , in a single growing season that i0 falls below its long-term time-averaged

value, I0. (Note that in such a model, every “cold” pulse, i0 ≤ I0, is followed by

a “warm” pulse, i0 > I0.) We assume that each of the latter ϑ heat pulses will be

“bioclimatically different” in some way that is important to germination but is not

explicitly defined herein, such as by the heat and/or water content of the soil when

the pulse of heat arrives, and that therefore each pulse is responsible for germination

of a separate C3 species from seeds lying dormant in the soil. Subsequent years

will be more or less different, giving smaller or larger numbers, ϑ , and to the same

degree will germinate some new C3 species on their own, but will also (by virtue of

this difference) fail to support some existing C3 species germinated in prior years.

We assume a steady state system to be reached in the long term, over which the

emergences should balance the deaths and extinctions. Formulating this problem as a

statistical “arrival process” [see, e.g., Eagleson, 1978], we characterize the local C3

species distribution in terms of the first two moments of the seasonal pulse arrivals,

ϑ . We arbitrarily select that ϑ demarcating 99% probability mass as the maximum

possible (i.e., “potential”) number of local species, max s , as being the best measure

to compare with observations. The arbitrariness of this choice of ni affects max s

by less than 10% at all latitudes less than 60◦N (Table 4.2).

Conclusion We repeat here in Figure 5.1 the comparison of the potential number

of local C3 species as derived herein, with the actual numbers found present locally,



both for vascular land plants in the Western Hemisphere [Davis et al., 1986], as

allometrically scaled to (essentially) areal independence at 10,000 km2 [Reid and

Miller, 1989], and for trees in the Northern Hemisphere [Gentry, 1988, 1995], as

scaled to 10,000 km2 in chapter 4. Our theory, as given in equations (4.28), (4.30),

and (4.31), can be seen to closely provide the expected envelope of observed species

numbers at all latitudes above about 20◦N.

For latitudes less than 20◦N, something different appears to be happening, perhaps

significant differences in the ecological strategies of co-occurring species, as has

been suggested at these latitudes by Kraft et al. [2008]. The known tropical presence

of non-C3 plants (see the frontispiece), such as the epiphytes, and particularly the

bromeliads and orchids, having thousands of species each, is suggested here as one

probable difference between our theory and observation at these latitudes. Another

probable difference arises from the shrinking time and space scales of the parent

meteorological events as we move southward from cyclonic to convective latitudes.

The distribution of flux pulses will then approach a “spike,” raising the possibility

that the number of biologically generative pulses may approach twice that of equa-

tion(4.31).

We believe we have shown, quantitatively as well as qualitatively, that along with

species range, species richness is also driven by the local variability of light (in

this case, at least above about 20◦N latitude), and that this common causality is

the bioclimatic basis for Rapoport’s rule [Stevens, 1989]. In the process, we have

confirmed light to be the dominant bioclimatic agent at extratropical latitudes and

that C3 plants must dominate at these latitudes. In the tropics, both range and richness

seem likely to be determined by nonneutral ecological strategies. Forecasts of changes

in the global distribution of plant species due to climate change are enabled by these

findings and should focus on changes in the local variabilities of light.

Finis

Summary The photosynthetically based bioclimatic function (equation (3.2))

demonstrates, to zeroth order, that species are a single-valued function of light, al-

lowing for expression of local species variance (equation (3.8)) and hence both local

average range and richness and local maximum richness, solely in terms of the local

variabilities of light, at least over extratropical latitudes.

Conclusion The results demonstrate (Figure 5.1) that local light alone does pre-

dict range at extratropical latitudes. Local moisture, nutrients, heat, or other forcing

variables alone could not also predict these structures, unless their local and spatial

variabilities were the same as those of light (see equation (3.8)). However, in the

tropics, where light alone fails to predict, joint forcing by more than one variable (in

the manner of equation 3.43) should be attempted.

P1: JZZ Trim: 7in × 10in Top: 42.5pt Gutter: 78ptAGUB001-APXA AGU001/Eagleson December 4, 2009 9:49

P A R T I V

Appendices: Reductionist Darwinian Modelingof the Bioclimatic Function for C3 Plant Species

The Earth system

Life and its environment evolve together as a single system so that not only doesthe species that leaves the most progeny tend to inherit the environment, but alsothe environment that favors the most progeny is itself sustained. What then is themechanism of this geophysiological regulation?

Lovelock [1986, p. 393]

95


A P P E N D I X A

The Individual C3 Leaf†

Photosynthetic capacity of the C 3 leaf

For light-limited vegetation, the principal biophysical control of productivity at a

given temperature and ambient CO2 concentration is the relationship between the

intensity of intercepted radiation and the resulting rate of carbon dioxide assimilation

at the scale of the individual leaf. This relation is known as the leaf photosynthetic

capacity function and follows the classical hyperbolic Michaelis-Menten equation

applicable for enzymatic reactions [e.g., White et al., 1968]. A typical example of this

photosynthetic capacity function for a C3 leaf of given species is shown in Figure A1,

under conditions in which the leaf temperature and ambient CO2 concentration are

both fixed. It has the saturating form, that is, it approaches a photosynthetic maximum

with increasing light [Monteith, 1963; Horn, 1971; Gates, 1980],

Pt = P� + Pr = Ps I�I� + Is�

, (A1)

in which, for an isolated leaf,

Pt total rate of assimilation of CO2 (i.e., the photosynthetic capacity), measured dur-

ing daylight hours and therefore implicitly including all daylight-hour respiration,

in grams CO2 per square meter of projected leaf area per hour;

P� net rate of photosynthesis by the leaf, in grams CO2 per square meter of projected

leaf area per hour;

Ps light-saturated rate of photosynthesis, in grams CO2 per square meter of projected

leaf area per hour;

†Much of this material has appeared earlier [Eagleson, 2002], with somewhat different notation.

97



FIGURE A1 Photosynthetic capacity function for the C3 leaf (fixed temperature and CO2 con-centration).

Pr rate of respiratory release of CO2 during the nighttime destruction of plant sub-

stance (i.e., “dark” respiration) to provide energy for cell metabolism, in grams

CO2 per square meter of projected leaf area per hour;

I� intercepted photosynthetically active “surface” irradiance per unit of horizontal

leaf area, in watts per square meter (Wpar m−2 ≈ 1/2Wtot m−2);

Is� species structural parameter that measures the effectiveness of an isolated leaf in

utilizing PAR [Horn, 1971], in Watts per square meter (Wpar m−2 ≈ 1/2Wtot m−2).

The biochemical structure of the leaf, and hence the three species parameters

Ps, Pr , and Is�, may have alternate values for the same species, depending on whether

their location is near the top (“sun” leaf ) or near the base (“shade” leaf ) of the crown

[Larcher, 1995]. At our level of approximation, we neglect this difference and assume

that all leaves are sun leaves and hence that the three parameters are single-valued for

each species.

Observations of many vegetation types presented by Larcher [1995] show that

the rate of nighttime respiration exceeds 10% of photosynthetic capacity only for

the leaves of those C3 plants having the lowest productivity, such as desert shrubs

and arctic trees, leading us, in the continuing spirit of zeroth-order approximation, to

neglect Pr and thereby reduce the number of parameters to two. We can now write

equation (A1), for leaves of all C3 plants, in the convenient and revealing approximate

form

Pt

Ps

∼= P�

Ps

∼= 1

1 + Is�I�

. (A2)


A P P E N D I X A • T H E I N D I V I D U A L C3 L E A F 99

The power of this approximation, provided by the common saturating property of C3

plants, greatly simplifies work across species, and because the C3 pathway for CO2

assimilation dominates among vascular land plants (see the frontispiece), equation

(A2) becomes the basis for our zeroth-order bioclimatic function, as follows.

For a given C3 species, s, with fixed temperature, T�, and ambient CO2 concen-

tration, c∗a , both Ps and Is� are constant, and we see from equation (A2) that the

relationship between P� and I� for leaves is linear at low light intensities; that is, for

I� � Is�, equation (A2) gives

P�

I�∼= Ps

Is�= εi , (A3)

where we define, with respect to the light intercepted by C3 leaves,

εi = potential efficiency of C3 assimilation of CO2 by intercepted light.

Note that reflected light is charged against this efficiency, as is the daylight-hour

respiration, and thus it cannot be a “lossless” conversion.

At higher light intensities, the C3 photochemical reaction becomes progressively

light saturated, and the efficiency of intercepted light utilization falls off as the CO2

fixation rate approaches its ultimate diffusive limit, Ps , for the given leaf temperature,

T�, and ambient atmospheric CO2 concentration, c∗a . The influence of these two

external modulators, T� and c∗a , on the photosynthetic capacity of a single species will

be discussed after we first follow the physical process of carbon mass transfer from

the free atmosphere to the leaf chloroplasts.

Mass transfer from free atmosphere to chloroplasts

Consider the flux of CO2 from the free atmosphere above the tree to the site of

carbon fixation at the chloroplasts in the interior of a single nonrespiring C3 leaf,

as shown schematically by the resistance path in Figure A2. The flux traverses both

Earth’s atmospheric boundary layer and the imbedded lower-surface boundary layer

of the single leaf, and thence through the leaf stomata to the interior site of fixation

at the chloroplasts. Associated with each of these path segments are the separate flux

resistances (all usually in seconds per centimeter): ra , atmospheric boundary layer

resistance (function of wind speed); rc, canopy (i.e., interleaf ) resistance (species

parameter); r�a , leaf boundary layer resistance (function of wind speed and leaf size,

shape, and texture, and thus largely a species parameter);←→rso , stomatal opening

resistance (function of degree of opening and species); ri , intercellular stomatal

resistance (function of species but small in magnitude with respect to←→rso ); and rm ,



FIGURE A2 CO2 flux resistance of leaf.

mesophyll resistance (function of species and leaf temperature), the first five of which

comprise the diffusive resistance (←→rd ):

←→rd = ra + r�a + rc + ←→

rso + ri = rbl + rc + ←→rs , (A4)

in which the first two diffusive resistances form the boundary layer resistance (rbl),

rbl = ra + r�a, (A5)

and the last two diffusive resistances form the variable stomatal resistance (←→rs ),

←→rs = ←→

rso + ri ≈ ←→rso . (A6)

The interleaf canopy resistance (rc) and the ambient atmospheric boundary layer

resistance (ra) will be omitted from the present analysis of leaf behavior (as indicated

by the bracket spanning them in Figure A2) because observations of leaf photosyn-

thetic capacity and flux resistance are made using the environment local to the leaf.



The diffusive and mesophyll resistances make up the species-dependent, single-leaf

variable resistance (←→R� ),

←→R� =←→

rd + rm = r�a + ←→rs + rm . (A7)

The flux-gradient relationship, Fick’s law, is written for this mass transfer [Legg

and Monteith, 1975] as

Qm = Kmdcm

dx, (A8)

in which Qm ≡ QCO2 is the mass flux density of CO2 in grams per meter squared per

second; Km ≡ KCO2 is the mass diffusivity of CO2 in meters squared per second (in

both of which the area in meters squared refers to an area perpendicular to the local

flux); cm ≡ c is the mass density of the diffusant, CO2, in grams per cubic meter; and

x is the position along the flux path in meters. In the general case, both the diffusivity

and the concentration gradient will be (separate) functions of x , but in a zeroth-order

approximation, we will lump each at its respective spatially constant “average” value,

KCO2 , and �c/�x . Such a linearization is commonly written in the form of Ohm’s

law [Thom, 1975], as we do here for the flux of CO2 from the free atmosphere to

the chloroplasts within a (nonrespiring) leaf. Consistent with the leaf-scale work of

Gates [1980], this becomes

QCO2 = c∗a − cc←→R�

, (A9)

where c∗a and cc are the concentrations of CO2 in the ambient atmosphere above

the isolated leaf and in the (leaf average) chloroplasts, respectively, in grams CO2

per cubic meter, and←→R� ≡ �x/KCO2 is the species-dependent, variable, single-

leaf resistance (s m−1) to diffusion over the path through the leaf boundary layer,

through the stomata, intercellular air spaces, cell walls, cytoplasm, and (finally) into

the chloroplasts of that leaf. Light-driven CO2 assimilation takes place within the

chloroplasts at a rate dependent upon the SW flux, I�, incident on the leaf (as well

as on both the ambient concentration, c∗a , and the leaf temperature, T�, as we will see

later).

In compatible units, QCO2 ≡ P�, and we can write equation (A9) as

P� = c∗a − cc←→R�

. (A10)

Refer now to Figure A3a and, in the manner of Gates [1980], consider the sequence

of events for a C3 leaf of given species starting in darkness with the stomates closed

and CO2 trapped in the intercellular spaces at a concentration equal to that (c∗a) in the

ambient atmosphere outside the leaf. We will follow the process of sunrise using the

asymptotes of this photosynthetic capacity diagram as an aid to easier understanding



FIGURE A3 Structure of the C3 photosynthetic capacity function (for leaf of species s at con-stant c∗

a and T� , with no dark respiration).

of the changing mixture of controlling processes, and for simplicity of the present

argument, we will keep both the leaf temperature and the ambient CO2 concentration

constant.

The sun begins to rise, and as the light intensity (i.e., SW flux), I�, increases

(0 < I� � Is�), it excites the chloroplasts that first fix the trapped CO2. The chloro-

plast CO2 concentration drops quickly, followed by the intercellular CO2 concen-

tration, which, on reaching a critical low level, triggers a stomatal control circuit,

which opens the stomates just enough to admit CO2 at the rate called for by the low

assimilative capacity of this low light intensity and no further, (presumably) to limit

transpirative water loss. As the light intensity continues to increase, and with it the

rate of photochemical carbon assimilation, so do both the stomatal opening (reducing←→rso and hence

←→R� ) and the CO2 concentration in the chloroplasts, until, at I� = Is�,

the stomates are maximally open (as modulated by leaf temperature and ambient CO2



concentration), offering their minimum resistance to flow. With open stomates, the

chloroplast CO2 concentration rises nearly to the ambient (leaf ) value, and the system

becomes “light saturated” in the sense that higher incident light intensity produces

no further increase in assimilation. At this point, the resistance to CO2 flow is at its

minimum, the flow rate of this species of leaf is at its maximum, and the photochem-

istry, the light-stimulated capacity of which has been setting the stomatal opening,

yields control of the flow rate to the diffusional resistance. Remember that the limiting

behaviors characterized artificially by the asymptotes of the photosynthetic capacity

curve (but conveniently for our zeroth-order work) indicate that this change of control

occurs abruptly, at the point of asymptote intersection, I� = Is�, whereas it actually

occurs gradually over the whole range of light-induced stomatal opening.

For I� � Is�, we have argued that cc � c∗a and that in response to the capacity of

the photochemical process, the stomates open only enough to admit CO2 at the rate

demanded by that low light. This small stomatal opening makes←→rso very large and

allows us to write, from equation (A10),

lim←→R� →∞

P� = c∗a

←→R�

, (A11)

and establishes

P� = c∗a

←→R�

,←→R� → ∞, (A12)

as the familiar rising (low I�) asymptote of the photosynthetic capacity, P�, when writ-

ten in terms of the independent variable←→R� . The latter will decrease with increasing

I�, until it reaches its minimum value, R�min , for the given species, whereupon I� = Is�,

and the rising asymptote takes its maximum value, P� ≡ Ps . This is illustrated graph-

ically in Figures A3a and A3b, where we indicate that for each species, s,

Ps (s) = c∗a

R�min (s). (A13)

Note from equation (A13) that an increase in ambient CO2 concentration leads to

proportionate increase in Ps for all C3 species.

In summary, at low I�, the CO2 flux is fundamentally controlled by the photo-

chemistry of assimilation, which, to the zeroth order, is a function of the particular

photosynthetic process (C3 in this case) and is not controlled by the leaf geometry.

Therefore the slope of the I� ≤ Is� asymptote represents the potential photochemical

efficiency, εi . It should be independent of the C3 species, except for the species de-

pendence of the reflection and respiration included within the measurements defining

εi . We demonstrate later that this is indeed the case for woody C3 plants using mea-

surements of εi taken from the published photosynthetic capacity curves of C3 trees

and shrubs.



At the other extreme, with the stomates of species, s, already fully open, increasing

light intensity increases the photochemical demand for CO2 beyond that reached at

I� = Is�. With the leaf resistance fixed at its minimum value, R�min , the assimilation

rate for I� � Is� is limited by this diffusional resistance. Thus, for I� > Is�, the

asymptotic approximation for P� is again as given by equation (A13).

To the zeroth order, we can now quantify the departure, �P�, of the Michaelis-

Menten function from its photochemical (low I�) and diffusional (high I�) asymptotes,

as is shown in Figure A3c. For a given species,

�P� = c∗a

←→R�

− c∗a − cc←→R�

= cc←→R�

, I� < Is� (A14)

�P� = c∗a

R�min

− c∗a − cc

R�min

= cc

R�min

, I� ≥ Is�. (A15)

Assimilation modulation by leaf temperature and ambientCO2 concentration

The photochemical processes of carbon assimilation are sensitive to leaf temper-

ature, T�, in such a way that the difference between gross photosynthesis and

respiration yields a net photosynthesis, P�, which maximizes (Ps = Psm) at an

intermediate, species-dependent temperature, T� = Tm , thereby providing a heat-

based, productivity-maximizing species selection mechanism [see Larcher, 1983, Fig-

ure 3.35]. With T� = Tm , Psm varies linearly with the ambient CO2 concentration, c∗a ,

as given by equation (A12). We do not consider these modulations in this zeroth-order

work.

Exponential approximation to the C 3 photosynthetic capacity curve

For analytical convenience and because of the dominance of diffusion in CO2 flux,

we fit the Michaelis-Menten relation for the nonrespiring leaf (i.e., equation (A2))

at optimum temperature, Tm , with its more convenient exponential approximation

[Eagleson, 2002]

P�

Psm≡ Po = 1 − exp

(−I�/

Is�)

, (A16)

as shown in Figure A4a. Note that the rising low-light asymptote of equation (A16)

also has the slope

∣∣∣∣

d Po

d I�

∣∣∣∣

I�=0=

∣∣∣∣

1

Is�exp

(−I�/

Is�)∣∣∣∣

I�=0= 1

Is�, (A17)



FIGURE A4 Proposed local Darwinian selection of woody C3 species. Adapted from Eagleson[2002, Figure 1.1]. Copyright c© 2002 Cambridge University Press.

which intersects the horizontal asymptote, P� = Psm , at I� = Is�, maintaining the

slope, εi , of the rising asymptote of the optimum photosynthetic capacity curve at

εi ≡ Psm

Is�. (A18)

Potential assimilation efficiency of C 3 leaves

Eight paired values of Psm and Is� for individual C3 leaves of differing woody species

are given in Table A1, as taken by Eagleson [2002] from published observed photo-

synthetic capacity curves. The ambient CO2 concentration, c∗a , is assumed to have a

common value in all these observations. The paired values are plotted in Figure A5,

where we see that to the zeroth order, the value of the assimilation efficiency, εi , as



TABLE A1 Parameters of Some C 3 Speciesa

Fitted(1) Parametersb Calculated Parameters

I s� Psm Observed I s�βL t = Ec I εi = Psm/I s�

(MJtotm−2h−1) (gsm−2h−1) Parameter βL tc

(MJparm−2h−1) (gsMJ−1par)

Tropical

Goethalsia 0.44(2) 0.38(2) 2.61(2) 0.57 1.73

Temperate

Creosote bush 2.34(3) 2.08(3) 0.37(3) 0.43 1.78Red oak 0.46(4) 0.28(4) 2.60(5) 0.60 1.22White oak 0.46(4) 0.26(4) 2.60(5) 0.60 1.13Loblolly pine 0.60(4,6) 0.40(4,6) 2.58(5) 0.77 1.33

Boreal

Sitka spruce 0.52(7) 0.55(7) 3.15(8) 0.82 2.12European beach 0.32(9) 0.33(9) 3.19(5) 0.51 2.06Arctic willow 0.38(9) 0.30(9) 3.04(10,11) 0.59 1.58

Average 2.52 0.61 1.62aParenthetical superscript numbers refer to the source listing given subsequently. Columns 2–3 are from published

phytotron observations of individual leaves. Column 4 is from canopy observations by other observers at different

locations; for Sitka spruce, all values reported(12) are averaged. Observed mass of carbon dioxide assimilated (g)

is converted to equivalent solid biomass (gs ) using the accepted(13) conversion factor, υ = gs/g = 0.5. The solar

radiation spectrum is partitioned nominally(14) by MJpar = 0.5MJtot. Sources are as follows: (1) Eagleson [2002];

(2) Allen and Lemon [1976]; (3) Ehleringer [1985]; (4) Kramer andDecker [1944]; (5) Baker [1950]; (6) Kramer and Clark

[1947]; (7) Jarvis et al. [1976]; (8) Landsberg et al. [1973]; (9)Muller [1928], as given by Kramer and Kozlowski [1960,

Figure 3.11]; (10) Cannell et al. [1987]; (11) Lindroth [1993]; (12) Landsberg and Jarvis [1973]; (13) Penning de Vries

et al. [1974]; and (14) Ross [1975].bUsing equation (A16), as shown in Figure A4.cProjected leaf area index (β ≈ κ).

given by equation (A18), has a common value, εi , given by the average of these eight

observations to be, for intercepted total SW radiation,

εi = 1.62 gCO2MJ−1

tot = 1.62 gsMJ−1par = 0.81 gsMJ−1

tot , (A19)

in which grams of CO2 have been converted to grams of equivalent solid biomass

using the generally accepted conversion factor [Penning de Vries et al., 1974]

gs/gCO2= 0.5, and the radiation spectrum is partitioned nominally [Monteith, 1973;

Ross, 1975] by MJpar∼= 0.5MJtot. Note that εi has also been reported as constant for

plants of the same metabolic type by Monteith [1977] and Gosse et al. [1986]. Our

value, as given in equation (A19), is consistent with that (εi = 1.60 gsMJ−1par) found

for the leaves of trees by Linder [1985], slightly larger than that (εi = 1.54 gsMJ−1par)

estimated for forest biomes by Ruimy et al. [1994], but significantly smaller than that

(εi = 2.98 gsMJ−1par) reported for arable crops by Monteith [1977], all for intercepted

radiation. (From remotely sensed εi = 1.5 gsMJ−1par (absorbed) using absorptance co-

efficient [Birkebak and Birkebak, 1964] αT = 0.51.)



FIGURE A5 Potential assimilation efficiency of intercepted light for leaves of woody C3 plants(assumed is that T� = Tm, and c∗

a is identical for each plotted observation). Adapted from Eagleson[2002, Figure 8.10]. Copyright c© 2002 Cambridge University Press.

With εi constant at εi across all woody C3 species, and assuming species to be

“Darwinially” differentiated by their maximum productivity, Psm , Is� becomes the

single, independent parameter defining their potential photosynthetic behavior.

The state of stress

We are assuming herein that the relentless pressure of evolution is toward the state

in which the plant utilizes the local resources dependably available to it in such

a way as to maximize the probability of successful reproduction. We call this the

optimal operating state. Accordingly, when a leaf is at an operating state in which it

is performing suboptimally by reason of one or more inadequate resources (such as

light, heat, water, CO2, or nutrients), we say that organ is stressed. It follows that a

leaf with stomata in a state other than fully open is in a state of stress.

Darwinian operating state of the individual C 3 leaf

Using equation (A17) to fit observations [Ludlow and Jarvis, 1971] of the CO2 transfer

resistance of Sitka spruce needle stomata, Eagleson [2002, Figure 8.8] demonstrated

that the stomata become effectively fully open at I� = Is�. Thus, to the zeroth order,

the leaf of a given species, Is�, is stressed (as defined earlier) only until the SW flux

rises to I� = Is�, whereupon it is unstressed and remains unstressed as I� exceeds

Is�. Eventually, for I� � Is�, water becomes limiting and stress returns. This state of



minimum stress has Darwinian significance when we consider the practical problem

of species selection at a given location where the seasonal average SW flux at the

leaf is I�. Again, working only with the asymptotes of the photosynthetic capacity

curve, as in Figure A4a, we see that species Is� = I� is the maximally productive,

unstressed species. This is clear from Figure A4b, where, with the local average SW

flux having the particular value I� = I ∗� , species 2, with higher productive potential,

will be stressed (i.e., Is�2 > I ∗� ), unless Is�2 = Is�1. The Darwinian operating state of

the C3 leaf is therefore

Is� = I ∗� = I�. (A20)

Should I ∗� be Is�2, instead of Is�1, species 1 would be unstressed but unstable due

to the danger of its displacement by the more productive species 2. Equation (A20)

thus becomes the Darwinian species selection criterion.

In northern latitudes, the daylight ambient temperature, T0, may fall such that the

leaf temperature, T�, is less than that, Tm , at which productivity is maximum for the

given species, as we have seen. In such cases, the maximum productivity, and with it

the stressless insolation, Is�, falls as well. It can be assumed that such a circumstance

puts evolutionary pressure on that species to adapt or evolve so as to restore the

optimum condition. Thus, when it is found to occur as the average state, we must

assume it to be a currently limiting condition on the evolutionary timescale.

Using the previously proposed Darwinian selection criterion, Is� = I�, under which

the photochemical capacity of the leaf is tuned to the local average radiational forcing

of the climate, we may refer to the common asymptote, εi , alternatively as the climatic

assimilation potential of C3 leaves, an assumed constant for the C3 photochemical

process.

As a reminder of these conditions for Darwinian selection, we have labeled, in

Figures A3a and A3b, the loci of both maximum efficiency and minimum stress for

all C3 species as well as the maximum productivity for the particular species whose

performance is sketched. To zeroth-order approximation, these three conditions are

all met at the asymptote intersection, which becomes the optimum operating point,

o, given the species shown in Figure A3c. Alternatively, from the Darwinian species

selection perspective, given the local light, I� = Is�, the species sketched will be the

one selected.

The univariate bioclimatic function at leaf scale

We now have the ingredients for a univariate bioclimatic function at leaf scale, in

which the bioclimatic interaction is driven by the single variable, I�, the time-averaged



SW radiation incident on that leaf. Because the individual leaf operates as a single

element of a whole plant, we need now to consider which leaf (or leaf layer) in the

crown we will address. As we will see in Appendix B, the time-averaged SW incident

radiation will fall with depth, ξ , down into the crown (as a function of the “projected

leaf area index,” βLt , an important species parameter) from a maximum of I0 at the

crown surface, ξ = 0, to a lesser value, I (ξ ), at a lower elevation. Consistent with

our zeroth-order approximations we will consider here only that leaf which intercepts

the crown-averaged (designated by the ‘hat’), time-averaged SW flux, Î �, and has the

optimum average temperature T� = Tm .

Equation (A20) has described the optimum photochemical operating state of the

leaf to be I� = Is� (s). Thus setting I� = Is� (s) selects, out of the whole set of species,

that single species which, at the given I�, is maximally productive and unstressed.

We express this maximum value of the (saturating) carbon demand, DC , of the leaf

as

max{

DC (s)| I�

} = εi Is� (s) . (A21)

The saturating carbon supply, SC , expressed as a function of species is, from

equation (A13),

SC (s, u0, ra) = c∗a [rc (s)]

∣∣u0,ra ,ca

R�min (s), (A22)

in which, from Figure A2, u0, ra, and rc (s) represent the carbon-flux resistances of

the atmosphere and canopy above the leaf.

As we will argue in Appendix B, the supply and demand have been (and are being)

individually maximized over evolutionary time through separate modifications of

species leaf and canopy structure, giving, at any time,

max{

DC (s)| I0

} ∼= max {SC (s, u0, ra)} , (A23)

or, using equations (A21) and (A22),

εi Is� (s)︸︷︷︸

Photochemicalcarbon demandby average leaf

∼= max

{

c∗a [rc (s)]

∣∣u0,ra ,ca

R�min (s)

}

︸︷︷︸

Diffusivecarbon supplyto average leaf

, (A24)

in which ca is the CO2 concentration in the free atmosphere above the canopy and

R�min includes only leaf-associated resistances. The species, s, appears directly in Is�

through the shape of the photosynthetic capacity curve; implicitly in c∗a through the

contingence of rc on canopy structure; and implicitly in the minimum leaf resistance,



R�min , through the size, shape, and texture of the leaves, through the size and density

of their stomatal openings, and through the diffusive properties of their mesophyll.

We call equation (A24) the “univariate bioclimatic function” at leaf scale, but

it is of instructional use only because the right-hand side cannot yet be expressed

explicitly in terms of s. Equation (A24) is amplified to the more useful canopy scale

in Appendix B.

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A P P E N D I X B

The Homogenous C3 Canopy

Idealized geometry of the leaf layer

Imagine a tree growing from seed; the emergent shoot produces a single leaf, which

orients itself with respect to the local seasonal solar altitude to maximize its rate of

CO2 assimilation. Because absorption of intercepted photosynthetically active radi-

ation (PAR) is almost complete and virtually independent of incidence angle [Ross,

1981], interception maximization will essentially ensure assimilation maximization,

provided that the leaf angle produces such reflection of longwave energy as will keep

leaf temperature photosynthetically optimal. Idealized, this situation is illustrated on

the right-hand side of Figure B1 for a single opaque leaf of one-sided area, A�, and

angle (with the horizontal), θ�, subjected to direct (i.e., “beam” rather than diffuse)

radiation, R, producing the “full” shadow area, A(1)s (the parenthetical superscript

refers to the number of leaves being considered).

Our shoot may be genetically programmed to produce additional leaves in a “layer”

parallel to the ground. To maximize assimilation (per unit ground area) by the layer,

we now assume that the leaves should arrange themselves [Eagleson, 2002] such that

the upper leaf surfaces are fully illuminated and no photosynthetically active radiation

is “wasted” by passing directly through the layer without interception. Such a highly

idealized layer is illustrated for two adjacent opaque leaves by the whole of Fig-

ure B1, with specular (i.e., nondiffusive) reflection from either leaf surface. The angle

of radiational incidence is α (also with respect to the horizontal); A(2)s is each leaf’s

full-shadow area, Ap is the “projected” (on a horizontal plane) leaf area, and A∗s is the

optimal full-shadow area. We note from the path of the reflected beam in Figure B1

that the assumed conditions for maximum unit area interception and thus (assumed)

111



FIGURE B1 Optically optimal leaf layer geometry. Adapted from Eagleson [2002, Figure 3.20].Copyright c© 2002 Cambridge University Press.

maximum assimilation are satisfied when the geometry of the leaf layer produces

β ≡ cos θ� ≡ Ap

A�

= A∗s

A�

. (B1)

When the leaves of this first layer have formed, continued CO2 assimilation leads

to stem elongation and initiation of a new, higher layer of leaves in a genetically

predetermined structure. These new leaves now receive the radiation that was formerly

incident on the first (now lower) layer. In our idealization of Figure B1, the now lower

layer receives incident radiation reduced in intensity by absorption and backscattering

in the upper layer but (with specular reflection) at the same angle of incidence. The

leaf angles of the lower layer would remain unchanged for optimal assimilation. In

reality, however, some radiation incident on the lower layer will have been transmitted

through translucent upper leaves and diffusively downscattered by them as well. The

lower layer must therefore adjust its leaf orientations for optimum interception of this

altered radiation field. This process proceeds with an expansion of existing layers

and the growth of additional layers over many seasons, which requires continual

environmental adaptation of leaf angles. The leaf angles are a primary factor in

additional radiational extinction through backscattering, and in reality, they vary with

depth into the crown. The growth process will continue adding new leaf area (unless

optimal leaf temperatures can no longer be maintained) until the insolation available


A P P E N D I X B • T H E H O M O G E N O U S C3 C A N O P Y 113

for the bottom leaf layer is just sufficient to satisfy the needs of respiration. (In reality,

of course, the leaves of the individual tree crown vary in structure and performance,

the primary example being the low-assimilation “shade” leaves at the bottom of the

crown as opposed to the high-assimilation “sun” leaves higher up. To the zeroth order

of this work, we omit such differences and consider the photochemistry of all leaves

to be identical.) The tree is then assumed to be assimilating CO2 at the maximum

possible local rate.

Darwinian heat proposition

We now make the Darwinian assumption that to maximize productivity of seed

and hence of reproductive potential, the local species time-and-canopy-average leaf

temperature, ˆT �, is equal to Tm for that species which is selected such that its Tm

equals the local ambient growing season time average surface climatic temperature,

To; that is, the proposition sets

To = ˆT � = Tm . (B2)

Larcher [1995] reports satisfaction of equation (B2) by natural selection, and Eagleson

[2002] verifies this proposition from observations of loblolly pine, American beech,

and Sitka spruce in their natural North American habitats.

Vertical flux of radiation in a closed canopy

Understanding the vertical decay of light is key to the photosynthetic behavior of

the whole crown. However, precise mathematical description of the extinction of

generalized radiation with depth into such a structurally heterogeneous crown is a

daunting task, which is avoided through use of the approximate expedient introduced

by Monsi and Saeki [1953]. Their zeroth-order approximation, which is in the spirit of

this work, treats the mature crown as having a linearly varying extinction coefficient,

as described in this section.

We first follow the development of Monteith [1973] and consider the intercepted

portion, dI, of the vertical flux of shortwave (i.e., SW) radiation, I , where there are

multiple parallel layers of leaves stacked up into an idealized crown, as shown in the

definition sketch (Figure B2). The downward cumulative (one-sided) leaf area per unit

of ground area (i.e., the leaf area index) is L , and for a geometrically homogeneous

crown of leaves having L ≡ Lt at the crown base, the differential leaf area is

d L = Lt dξ, (B3)



FIGURE B2 Homogeneous crown with five leaf layers (closed canopy).

and the SW flux, dI, intercepted at given depth into the crown, ξ , by this differential

leaf area, can be written in terms of the shadow area, As :

A�

d I

d L= −As(ξ )I, (B4)

where the minus sign indicates that I decays with downward increasing L . Integrating

equation (B4) downward into the crown from ξ = 0, using equation (B3) and with

I (ξ = 0) being the ambient insolation, I0, and I (ξ ) being the local leaf insolation,

I� (ξ ), we have

I (ξ )∫

I0

d I�(ξ )

I�(ξ )= −

ξ∫

0

As(ξ )

A�

Lt dξ = −Lt

ξ∫

0

κ(ξ )dξ, (B5)

where

κ(ξ ) ≡ shadow area/foliage area. (B6)

Here enters the underlying principal approximation of Monsi and Saeki [1953]: the

realities of nature ensure that As (and hence κ) is an unknown function of ξ , even with

geometrical homogeneity, because (1) the diffuse component of incident radiation will

bring light from a range of directions simultaneously, thereby invalidating the simple

flux relationship of equation (B4), and (2) even for incident beam radiation, the crown

will likely have variably translucent leaves and diffuse reflections from at least the

rough undersurfaces of the leaves.



In a typical zeroth-order approximation, Monsi and Saeki [1953] avoided this

problem by assuming that κ(ξ ) varies linearly with depth, ξ , so that

ξ∫

0

κ(ξ )dξ ∼= ξ

1∫

0

κ(ξ )dξ ≡ ξ κ, (B7)

where κ is the crown-average radiation extinction coefficient, defined at maximum

assimilation rate as

κ = A∗s

A�

. (B8)

With this approximation, we can integrate equation (B5) to obtain the famous

Monsi-Saeki extinction equation:

I�(ξ )

I0= exp(−κLtξ ). (B9)

With the optimum long-term, time-averaged leaf-operating state being I� = Is�, as

has been pointed out in Appendix A (equation (A20)); with the vertical decay of

SW flux, as provided by equation (B9); and with leaf angles varying with elevation,

we recognize that all leaves in a strictly homogeneous multilayered crown cannot

operate optimally at all times. Retaining our homogeneity assumption, we assume

that the crown is optimal in the spatial as well as the temporal average. Accordingly,

we use both hats and overbars in rewriting equation (B1) to define the spatial average

optimum leaf angle over the canopy (as was done for κ), giving

β ≡ cos θ� = A∗s

A�

. (B10)

Note the very important result of these approximations is that under the average

geometrical conditions producing maximum assimilation in our idealized radiation

field,

κ = A∗s

A�

= β, (B11)

as was pointed out by Eagleson [2002], who also showed its limited observational

support, as gathered from the literature [Eagleson, 2002, Figure 3.19]. We will find

equation (B11) to be the key to simplifying the unification of the canopy fluxes of

energy and carbon when considering issues of productivity.

Similarly, I�, the seasonal average leaf SW flux, spatially averaged over the depth

of the crown (due to flux extinction therein) to become Î �, is assumed equal to the

optimal operating state for an individual leaf of the given species (Figure A4); that is,



for a closed canopy (i.e., water and nutrients not limiting),

Î � = Is�. (B12)

With this assumption, equation (B9) yields

Is�

I0=

Î �

I0=

∫ 1

0exp (−κLtξ )dξ = f I (κLt ), (B13)

in which

f I (κLt ) ≡ 1 − e−κLt

κLt. (B14)

Note that at the bottom of the mature crown, where ξ = 1, I�(ξ = 1) ≡ Ik , the

compensation radiation (provided ˆT � = Tm), which is the amount of radiation needed

only to compensate for respiration and thus not provide any net productivity. Then,

from equation (B9),

Ik

I0= exp(−κLt ). (B15)

C 3 species parameters

Since with P� = 0, I� ≡ Ik , and (from Appendix A) Ik ≡ Is�, equation (A1) can be

written, for an isolated bottom leaf at optimum temperature,

Pr

Psm=

Ik

I0

Ik

I0+ Is�

I0

, (B16)

which, together with equations (B13) and (B15), enables us to write

Pr

Psm= f (κLt ). (B17)

Larcher [1983] points out that Pr and Psm are both species constants,which are

fixed here in their separate temperature dependencies by the optimum temperature

associated with Psm . Therefore equation (B17) (together with equation (B11)) tells

us that

κLt = species constant = βLt . (B18)



Bioclimatic function at canopy scale

At leaf scale, we found the functional relation between time averaged local intercepted

radiation, I� = Is�, and species to be (equation (A24))

εi Is�(s)︸︷︷︸

Photochemicalcarbon demandby average leaf

∼= max{

c∗a [rc(s)] |u0, ra, ca

R�min (s)

}

︸︷︷︸


, (B19)

in which εi is assumed to be a constant, whereas Is�, rc, and R�min are all species-

dependent.

At canopy scale, we consider first the carbon demand: at this extended vertical

scale and to the zeroth order, we begin with the time averaged behavior of the canopy

average leaf, which intercepts ¯I �, incident PAR per unit of horizontally projected area,

and time. To get total canopy PAR interception, we multiply ¯I � by the amount of hori-

zontally projected leaf area, βLt , stacked vertically to form a vertically homogeneous

canopy, and to maximize the photosynthetic carbon demand of the stable canopy

(at optimum temperature), we set ¯I � = Is�. This expands the left-hand side of equa-

tion (B19) to

max {carbon demand} ≡ max {DC} = εi Is�(s)βLt = εi¯I �(s)βLt = εi I0

(

1 − e−s).

(B20)

Canopy scale is introduced into the carbon supply side (i.e., right-hand side) of

equation (B19) through the “effective” diffusive resistance to the flux of CO2 from the

free atmosphere to the chloroplasts of each of the βLt , vertically stacked, horizontally

projected leaf surfaces comprising the homogeneous crown. Each of these surfaces

has a different flux path, the average resistance of which is R�c, where

R�c = R� + rc + ra, (B21)

(see Figure A2). The individual leaf resistance, R�, is as defined in Appendix A and

is assumed identical for all leaves in the canopy; rc is the average canopy resistance to

diffusive flow through the leaf layers; and ra is the diffusive resistance of the ambient

atmospheric boundary layer. In this zeroth-order approximation, we consider light

as the sole controlling resource, in which case, for maximum productive potential,

all ground area will be leaf covered, making βLt ≥ 1 (this issue is discussed further

in Appendix C). The βLt separate resistance paths are parallel paths in the canopy,



assumed identical, so that the appropriate total canopy resistance, Rc, is approximated

by

Rc∼= R�c

βLt

, βLt ≥ 1. (B22)

Following equation (A13) and Figure A3b, we seek the minimum of this resistance

for a given species, s ≡ βLt . As a function of species, this minimum is written

Rcmin (s) = R�cmin

(

βLt

)/

βLt , βLt ≥ 1. (B23)

For the lumped canopy, the effective CO2 concentration is now ca , the (constant)

value in the free atmosphere above the canopy, and using equations (B22) and (B23),

the right-hand side of equation (B19) expands to

max {carbon supply} ≡ max {SC} = max

⎧

⎨

⎩

ca

R�cmin

(

βLt

)/

βLt

⎫

⎬

⎭, βLt ≥ 1,

(B24)

which is the maximum (i.e., saturating) carbon supply of a mature, horizontally and

vertically homogeneous canopy as a function of species, s ≡ βLt .

We assume that in the continuing Darwinian search for increased local reproductive

success, both the local canopy photochemical carbon demand and the local canopy

diffusive carbon supply are optimized independently over evolutionary time through

continuing speciation and adaptation. Whenever one or the other (either local supply

or local demand) is even slightly larger at any given moment due to ongoing adaptation

and mutation, there is an excess of capacity for either carbon assimilation or carbon

supply, and hence there is continued pressure for further local species change to utilize

this excess capacity.

Local evolutionary equilibrium: An hypothesis

Eldredge and Gould [1972] and Gould and Eldredge [1977] have argued that this

speciation occurs at the extremes of the local species distribution and is driven by

extremes in the local forcing (climate in our case). Furthermore, they argue that

these extremes of forcing, being temporally sporadic and separated by long periods

of relative calm, produce a speciation process characterized by “long” periods of

equilibrium separated by “short” periods of change. They called this “punctuated

equilibrium.” Here we describe the conditions defining such equilibrium for the

particular case of C3 vegetation.

We now hypothesize that the Darwinian goal has been reached when the canopy

carbon supply and demand are (simultaneously) not only equal, but also at their



respective maximum values, and are therefore in what we shall call here “evolutionary

equilibrium.” Equating equations (B20) and (B24), and remembering that s ≡ βLt ,

the statement of canopy-scale evolutionary equilibrium becomes

εi I0(

1 − e−s) = max

{

ca

R�cmin (s)/

s

}

, s ≥ 1. (B25)

With the ambient atmospheric CO2 concentration assumed everywhere the same, we

carry out the maximization on the right-hand side of equation (B25) by requiring that

d

ds

[

ca

R�cmin (s)/

s

]

= 0, s ≥ 1, (B26)

which is satisfied when

R�cmin (s)

s= constant ≡ c h m−1, s ≥ 1. (B27)

Consistent with our earlier assumption that the species variable, s ≡ βLt , is continu-

ous (rather than discrete), and because equation (B26) must be satisfied for all values

of s ≥ 1(for s < 1, a resource other than light, say, water or nutrients, must limit the

canopy, a condition not considered herein), equation (B27) gives, irrespective of I0,

and for constant ca ,

max {DC} = max {SC} = constant, s ≥ 1. (B28)

Using equations (B20), (B27), and (B28), the energy and carbon flux expressions

of maximum productivity can be equated to express this evolutionary equilibrium

in two instructive forms. First, for physical clarity, we separate the forcing climate

variables from the resulting vegetation variables to obtain the “physical” form:

Carbonand oxygen

︷︸︸︷ca

I0︸︷︷︸

Canopy-topSW flux

=Leaf

︷︸︸︷

εi c︸︷︷︸

C O2

Crown︷︸︸︷[

1 − e−βLt

]

︸︷︷︸

InterceptedSW flux

,

︸︷︷︸

Resources

{

Photo-chemistry

︸︷︷︸

Diffusion

βLt ≥ 1, (B29)

where, for given ca , the species dependence lies solely in the light diffusion term.

Here the physicochemical biological processes have produced, over evolutionary

time, a structural plant form in optimum evolutionary balance with its environmental

resources. (It is therefore not surprising that field studies of mature plant response to

artificially increased ca show little increase in standing biomass over a few years of

observations [Korner et al., 2005].)



Second, for use of evolutionary equilibrium herein as a zeroth-order analytical

connection between climate and species, we rewrite equation (B29) as

I0

[

1 − e−βLt

]

︸︷︷︸

Canopy-interceptedPAR energy

= ε−1i

[ca

c

]

︸︷︷︸

Canopy-assimilatedcarbon

= constant = Ec, βLt ≥ 1, (B30)

the outside terms of which form the desired univariate bioclimatic function, express-

ing, at canopy scale, the local average species (βLt ) as a function of the single local

climate forcing variable ( I0). This evolutionary equilibrium hypothesis gives an im-

portant new theoretical result, which we will evaluate from observational data by two

independent methods in Appendix C.

Finally, when considering the expected effects of climate change, it may be helpful

to isolate the potential assimilation efficiency, εi , in the revealing form

εi ≡ assimilated carbon

intercepted PAR=

Climaticpotential

︷︸︸︷ca

I0·

Canopy efficiency︷︸︸︷(

βLt

)/

R�cmin

1 − e−βLt. (B31)

In this form, εi displays its role as the dimensional similarity parameter governing C3

plant growth.

As the ambient CO2 concentration, ca , rises with time, leaf temperature may be

expected to rise also, and biomes that were operating optimally will experience a

decrease in R�cmin [Larcher, 1983, Figure 3.35]. Secondarily, the SW flux may be

reduced by increased scattering and absorption, which will likely induce an increase

in βLt (equation (B30)), and the aggregate expected ecological response is an increase

of plant biomass and perhaps a change to more leafy species.

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A P P E N D I X C

Evaluation of the Evolutionary Equilibrium Hypothesis

The equilibrium hypothesis at leaf scale

In equation (A24), we developed the bioclimatic function of that particular stressless,

stable, and maximally productive leaf in a canopy of species, s, to be

Is� (s)︸︷︷︸

PAR interceptedby average leaf

≈ ε−1i max

{

c∗a [rc (s)]

∣∣u0,ra ,ca

R�min (s)

}

︸︷︷︸


. (C1)

Evaluation of equation (C1) is difficult, however, due to the unknown functions of

species (s) and position within the canopy (ξ ), and consequently, this leaf-scale

analysis serves only an instructional function.

The equilibrium hypothesis at local canopy scale

In equations (B25)–(B28), after assuming the canopy average leaf to be stressless,

and maximally productive, we further assumed the whole plant to be in a state

of evolutionary equilibrium, in which the maxima of carbon supply and of carbon

demand are equal at plant scale. We write this “evolutionary equilibrium hypothesis”

for the lumped, local, spatially homogeneous canopy as

I0

[

1 − e−βLt

]

︸︷︷︸

InterceptedPAR energy

= Ec = ε−1i

[

ca

R�cmin

/

βLt

]

︸︷︷︸

Assimilatedcarbon

. (C2)

121



We now obtain independent estimates of the hypothesized constant, Ec, by separate

evaluations of the left-hand side (EcI ) and right-hand side (EcC ) of equation (C2).

On the left-hand side, in columns 2 and 3 of Table A1, values of Is� and Psm

(respectively) are given for a variety of C3 species, as estimated by Eagleson [2002]

by graphically fitting published photosynthetic capacity curves for single leaves. In

column 4 of the same table are values of the species parameter, βLt , again collected

by Eagleson [2002] from the literature, as obtained from full crown or canopy ob-

servations. It is important to note that in some cases, the paired estimates of Is� and

βLt , although from the same species, came from different observers and/or different

stands of the species. Resulting values of intercepted PAR for these species using the

left-hand side of equation (C2) are given in column 5 of Table A1, where we see that

they scatter about a sampled species average, 〈· · ·〉, of

〈Is�βLt 〉 ≡⟨

I0

[

1 − e−βLt

]⟩

= 〈EcI 〉 = 0.61MJparm−2p h−1, (C3)

in which EcI signifies Ec as determined from intercepted PAR and in which (as a

reminder) m2p signifies square meters of projected (i.e., horizontal) canopied area.

For subsequent analysis, we are interested in the variability of Ec across species.

Although eight observations is a very small sample, we calculate the coefficient of

species variation of EcI to be

CV (EcI ) = σ (EcI )

〈EcI 〉 = 0.20. (C4)

We now separately estimate the numerical value of each side of equation (C2). The

right-hand side of equation (C2) is evaluated for sun leaves of four tree species using

observations from the literature, as summarized in Table C1.

Assimilated carbon Turning first to the numerator of the right-hand side of equa-

tion (C2), we have, from Gates [1980],

ca∼= 12.5 mmCO2 m−3 (millimoles of CO2 per cubic meter);

from stoichiometry,

1 mmCO2 ≡ 0.044 gCO2;

and from Penning de Vries et al. [1974], the conversion of assimilated mass of carbon

dioxide to mass of solid matter, gs , is approximated by

gs/

gCO2∼= 0.50.

Thus

ca∼= (12.5) (0.044) (0.5) ∼= 0.275 gs m−3. (C5)


A P P E N D I X C • T H E E V O L U T I O N A R Y E Q U I L I B R I U M H Y P O T H E S I S 123

TABLE C1 Carbon Supply Evidence for Evolutionary Equilibrium in Certain Tree Speciesa

ra (1) r c (2) r�a (3) r smin(3) rm(3) R�cmin

(4) R�cmin

βL t

ca/ε(7)i

R�cmin/βL t= EcC

Species (s m−1) (s m−1) (s m−1) (s m−1) (s m−1) (s m−1) βL t (s m−1) (MJparm−2h−1)

Norway maple(Acerplatanoides)

6.25 6.25 79.7 1350.0 803.3 2245.5 1.73(5) 1298.0 0.47

European whitebirch (Betulaverrucosa)

6.25 6.25 78.6 209.3 610.0 910.4 1.51(6) 602.9 1.02

European aspen(Populustremula)

6.25 6.25 59.3 391.3 750.0 1213.1 1.38(6) 879.1 0.70

Chestnut oak(Quercusrobur)

6.25 6.25 85.5 1980.0 1010.0 3088.0 1.50(5) 2058.7 0.30

Average 1864 1.53 1210 0.62aSuperscripted parenthetical numbers refer to the following: (1) equation (C9); (2) equation (C11); (3)Holmgren et al.

[1965]; (4) equations (C7) and (C8); (5) Baker [1950]; (6) Rauner [1976]. The experimental apparatus and its use is

described by Bjorkman and Holmgren [1963].

From Figure (A7) and equation (A19),

εi = 1.62 gs MJ−1par,

where the radiation is intercepted light, giving

ca/εi = 0.275/1.62 = 0.17 MJpar m−3. (C6)

Leaf andcanopy resistances For the denominator of the right-hand side of equation

(C2), we first turn to equation (B21), from which

R�cmin = R�min + rc + ra. (C7)

We use equations (A7) to obtain

R�min ≡ r�a + rsmin + rm . (C8)

For trees with height, h ≥ 2m, Earth’s atmospheric boundary layer resistance, ra , has

the estimator [Eagleson, 2002]

ra∼=

(

k2u0

)−1, (C9)

in which k = von Karman’s constant (dimensionless) = 0.40 and u0 = atmospheric

“free stream” velocity (meters per second). At the common average wind speed,

u0 = 1 ms−1, equation (C9) gives

ra∼= 6.25 s m−1. (C10)



FIGURE C1 Linearity of leaf resistance at canopy scale.

Broad-leaved species have a single stomated surface per leaf, for which Eagleson

[2002] has found rc to satisfy

rc/

ra∼= 1. (C11)

Holmgren et al. [1965] give repeated observed values of the remaining leaf resis-

tances, r�a , rsmin , and rm , for sun leaves of four broad-leaved tree species, the averages

of which are reproduced in Table C1. The desired total resistance, R�cmin , is given

(from equation (C8)) in column 6, and crown-average values of the projected leaf

area, βLt , not measured by Holmgren et al. [1965], are taken from the literature (as

referenced) for the same tree species.

From equation (C2), we see that the zeroth-order condition for constancy of Ec

across all leaves of all species requires R�cmin to be linear in the species variable, βLt .

We explore this linearity in Figure C1 using the observations combined in column

6 of Table C1. In this figure, the solid line represents a linear least squares fit to

the plotted observations of column 6 of Table C1, constrained to pass through the

origin, R�cmin = 0, βLt = 0 [Benjamin and Cornell, 1970], whereas the dashed line

represents the unconstrained linear least squares fit of these same four data points.

Note that this dashed line projects to R�cmin = 0, very close to the minimum full ground

cover condition, βLt = 1, required for canopy utilization of maximum available light

and thus for maximum production per unit ground area.


A P P E N D I X C • T H E E V O L U T I O N A R Y E Q U I L I B R I U M H Y P O T H E S I S 125

Returning now to evaluation of the right-hand side of equation (C2) and using

Table C1, the average observed EcC (where EcC means Ec as determined by carbon

flux) is, for this small sample,

〈EcC〉 ≡ ca/εi

〈R�cmin (s)/

βLt〉= 0.62 MJpar m−2

p h−1, (C12)

with a CV = 0.87. Note that the species observed for estimating EcC are different than

those observed for estimating EcI (Table A1), yet their average values of Ec are essen-

tially identical; that is, 〈EcI 〉 = 0.61 MJpar m−2p h−1 and 〈EcC〉 = 0.62 MJpar m−2

p h−1.

Summary

The two independent estimators of E are remarkably close in value. Over the species

range of these observations, 0.37 ≤ βLt ≤ 3.19, E is given by its average:

〈E〉 = 0.62 MJpar m−2p h−1 = 172 Wtot m−2

p , (C13)

with

σE = 0.17 MJparm−2p h−1, (C14)

making

CVE = 0.27. (C15)

For current purposes, we take the species-controlling bioclimatic function from the

left-hand side of equation (C2) as

〈 I0〉⎡

⎣1 − e−⟨

βLt

⟩

1

⎤

⎦ ≡ E1 = 172 Wtot m−2p . (C16)

We note that Enquist and Niklas [2001, p. 655], in extending allometry theory to

closed plant communities, including those of mixed species, found that “the intrinsic

capacity to produce biomass on an annual basis will vary little across communities.”

We find here that biomass productivity is essentially uniform across primary canopies

of a range of communities. Written in the useful form of equation (1.3), that is,

s = g (c) , (C17)

the bioclimatic function of equation (C16) becomes, finally, the univariate “state

equation”⟨

βLt

⟩

1= �n

[1

1 − E1/〈 I0〉]

, 〈 I0〉 > E1 = 172 Wtot m−2p . (C18)

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Notation

A area of vegetation community in which species are counted

(m2).

A� one sided leaf area (cm2).

A� spatial average leaf area (cm2).

An projection of leaf area onto a plane perpendicular to the inci-

dent beam radiation (cm2).

Ap leaf area as projected on a horizontal plane (cm2).

As leaf shadow area (cm2).

A(1)s full shadow area of a single leaf (cm2).

A(2)s each leaf’s full shadow area in a two-leaf system (cm2).

A∗s optimum full shadow area of leaf (see Figure B1) (cm2).

A∗s spatial average optimum full shadow area (cm2).

a coefficient.

C3 class of vegetation utilizing the Calvin-Benson photochemi-

cal pathway.

C4 class of vegetation utilizing the Hatch-Slack photochemical

pathway.

CAM class of vegetation utilizing the Crassulacean acid metabolism

photochemical pathway.

CO2 carbon dioxide.

c local constant in the “species-area” relationship (m−2z).

c local (i.e., pixel) randomly time-variable climate.

c ≡ E(c) time average of local climate.

c∗a = ca ambient concentration of CO2 (gCO2

m−3).

127



cc leaf average concentration of CO2 in the chloroplasts

(gCO2m−3).

ci i th climatic forcing variable.

ci temporal mean of the i th climatic forcing variable.

cm ≡ c mass density of the diffusant CO2 (gCO2m−3).

DC saturating carbon demand (gCO2m−2 h−1).

E ≡ E1 SW flux intercepted by the primary canopy = 172 Wtot m−2.

Ec canopy-intercepted PAR flux at “evolutionary equilibrium”

(W m−2) and its equivalent, canopy-assimilated carbon flux

at “evolutionary equilibrium” (gs m−2 h−1).

EcC optimum canopy-assimilated carbon flux at evolutionary

equilibrium (see Figure B3) (W m−2).

EcI optimum SW flux (PAR) intercepted by canopy at evolution-

ary equilibrium (W m−2).

E1 solar energy intercepted by primary canopy of light-limited

forest (W m−2).

e base (2.718. . . ) of natural logarithm.

F(n) probability mass of normal distribution at n standard devia-

tions from mean (see Figure 3.6).

G (κ, λh) two-parameter Gamma function (dimensionless).

gCO2 grams of assimilated CO2.

gs grams of solid plant matter.

h depth of canopy (m).

h j random amount of SW radiant energy intercepted by a

species-supporting cloud event, i0 ≤ I0 (W h m−2).

ha hectare, a measure of land area (1 ha = 104 m2).

I SW flux (quantity of SW solar energy intercepted by given

surface in given time) (W m−2).

Ik I� (ξ = 1) (W m−2).

I� photosynthetically active SW flux intercepted per unit time

by leaf (Wpar m−2).

I� seasonal average SW flux intercepted by leaf (W m−2).

I� canopy average SW flux intercepted by the leaf (W m−2).Î � crown average of the seasonal average SW flux intercepted

by leaf (W m−2).¯I � time average of the canopy average SW flux intercepted by

the leaf (W m−2).

I ∗� particular value of seasonal average intercepted SW flux (Fig-

ure A4b) (W m−2).


N O T A T I O N 129

I0 local pixel growing-seasonal-average canopy-top SW flux in

a given year (W m−2).

I0k pixel annual SW flux in year k (W m−2).

I −0 SW flux at � = �+ (see Figure 3.3) (W m−2).

I0 or I0 long-term time average of local pixel growing-seasonal

canopy-top SW flux (W m−2).

I +0 average SW flux at � = �+ (Figure 3.16a) (W m−2).

I 00 local maximum of the seasonal average SW flux, located at

�00 (Figure 3.16b) (W m−2).

I L0 mean annual SW flux at �0

L (Figure 3.16b) (W m−2).

I R0 mean annual SW flux at �0

R (Figure 3.16b) (W m−2).

Is� species parameter measuring the leaf effectiveness in utilizing

SW flux (Figure A1) (Wpar m−2).

Is�1, Is�2 leaf effectiveness of species 1 and 2 in utilizing SW flux

(Figure A4b) (Wpar m−2).

Is�(βLt ) SW flux intercepted by crown-average leaf of the local average

species (Wtot m−2).

I� SW flux at the top of the atmosphere during daylight in the

growing season (W m−2).

I� time average (June–September, inclusive) of I� at each lati-

tude (W m−2).

i0 instantaneous pixel-average SW flux at canopy top (W m−2).

j counting variable.

KCO2 spatially averaged (along the flow path) value of the CO2 mass

diffusivity (m2 s−1).

Km ≡ KCO2 mass diffusivity of CO2 (m2 s−1).

k von Karman’s constant =0.40 (dimensionless).

k number of years of record (dimensionless).

kgs kilograms solid.

L or LAI leaf area index, one-sided leaf area per unit of ground area

(dimensionless).

Lt leaf area index of a crown.

M = m2p/m2 canopy density (dimensionless).

mh mean of SW flux intercepted by a species-supporting cloud

event (W h m−2).

m2p horizontal projection of canopied area (m2).

m R(ν) mean of R(ν) (W h m−2).

m Rτ≡ E[Rτ ] mean of Rτ and R(τ ) (W h m−2).

mtb mean time between cloud events (hours).



mtc mean duration of cloud event (hours).

mν mean number of species-supporting events in mτ .

mτ daylight length of local average annual growing season

(h yr−1).

N.A. North America.

N.H. Northern Hemisphere.

nI number of standard deviations away from the mean annual

insolation, I0 (W2 m−4).

n I latitudinal average number of standard deviations away from

the mean annual, I0.

ni number of standard deviations of ν above the local mean at

which new species can no longer be identified.

ns number of standard deviations, σs , away from the mean

species.

P� net rate of leaf photosynthesis (gCO2 m−2 h−1).

Pr rate of nighttime respiratory release of CO2 (i.e., “dark” res-

piration) (gCO2 m−2 h−1).

Ps light-saturated rate of photosynthesis at given temperature

(Figure A1) (gCO2 m−2 h−1).

Psm maximum light-saturated rate of photosynthesis (i.e., at opti-

mum temperature, Tm) (gCO2 m−2 h−1).

Pt total rate of assimilation of CO2 (i.e., the photosynthetic ca-

pacity) (gCO2 m−2projected leaf area h−1).

P0 relative leaf productivity or efficiency P�/Psm (dimension-

less).

PAR photosynthetically active radiation (Wpar m−2 ≈ 1/2Wtot m−2

= 1/2W m−2).

p�|τ (ν) discrete probability that exactly ν species-supporting events

will occur.

Qm ≡ QCO2 CO2 mass flux density (gm−2 s−1).

R intensity of beam radiation (W m−2).

R generic range (deg latitude).

R(ν) total SW radiant energy intercepted by a random number, ν,

of species-supporting cloud events, i0 ≤ I0 (W h m−2).

R(τ ) total SW flux intercepted in season of length, τ (W h m−2).

Rτ total SW flux intercepted in season of length, τ (W h m−2).

Rc total canopy CO2 flux path resistance (s m−1).

Rcmin minimum total canopy CO2 flux path resistance (s m−1).

R� CO2 flux path resistance of individual leaf in crown (s m−1).


N O T A T I O N 131

←→R� variable, total (i.e., mesophyll + diffusive) resistance of single

leaf (s m−1).

R�c average resistance of the separate leaf CO2 flux paths in a

homogeneous crown (s m−1).

R�min minimum value of the individual leaf variable resistance

(s m−1).

R�cmin minimum average resistance of the separate leaf CO2 flux

paths in a homogeneous crown (s m−1).

Rs range of modal species at � = �◦ (deg).

Rs|�◦(�) ≡ range in degrees latitude of species, s ≡ βLt , at a local site

having � = �◦.

Rs|�◦(�) ≡ range in degrees latitude of the mean species, s ≡ βLt , at

a local site of latitude �◦.Rs (�◦) ≡ Rs|�◦(�) ≡ Rs range in degrees latitude of the modal

species, s, at a local site of latitude �◦.

Rs|�◦(�) ≡ Rs (�◦) ≡ Rs range in degrees latitude of the modal

species, s, at a local site of latitude �◦.

R p

s|�0 (� = 0) range in degrees latitude of the modal species, s, at � =0, projected from the range gradient at high latitudes in an

imaginary world having low-moisture atmosphere.

Rs|�◦(�) mean of the ranges in degrees of all species, βLt , at a local

site at �◦.

R〈s〉|�◦(�) ≡ range in degrees of the zonal average local mean species,

〈s〉, at �◦.

Rss|�◦(�) sample (superscript s) zonal average range of species, s, in

zone �0 (deg latitude).

R�◦ = �◦ − �− range of mean species at � = �◦ (deg).

R s|�00

range (identically zero) of the most frequent (i.e., modal)

species occurring at a maximum of I0(�) (deg).

R s|�0L

that part of the range of the least frequent (i.e., largest) species

occurring at a maximum of I0(�) that is found on the rising

(i.e., left-hand) branch of I0(�) (deg).

R s|�0R

that part of the range of the least frequent (i.e., largest) species

occurring at a maximum of I0(�) that is found on the falling

(i.e., right-hand) branch of I0(�) (deg).

RN (�0) Northern Hemisphere portion of range that straddles the equa-

tor (deg).

RNs|�0 northern portion of the range of the most frequent (i.e., modal)

species occurring at a minimum of I0(�) (deg).



r amount of SW flux intercepted by � cloud events (W m−2).

ra atmospheric boundary layer resistance (s m−1).

rb� = ra + r�a boundary layer resistance (s m−1).

rc canopy (i.e., interleaf) resistance (s m−1).←→rd diffusive resistance (s m−1).

ri intercellular stomatal resistance (s m−1).

r�a leaf boundary layer resistance (s m−1).

rm leaf mesophyll resistance (s m−1).←→rs = ←→

rso + ri∼= ←→

rso variable stomatal resistance (s m−1).

rsmin minimum stomatal resistance (s m−1).←→rso variable stomatal opening resistance (s m−1).

SC saturating carbon supply to leaf (gCO2 m−2 h−1).

SC saturating carbon supply to a mature homogeneous canopy

(gCO2 m−2 h−1).

SE standard error of estimate.

S.H. Southern Hemisphere.

SW shortwave radiative flux = Wtot m−2 or W m−2.

s = βLt = g(c), numerical representation of optimally sup-

ported vegetation species (dimensionless).

s ≡ E(s) = βLt = g(c), local community-average species (dimen-

sionless).

s�0 local average species at latitude �0.

s local modal (i.e., most frequent) species (dimensionless).

T0 daylight ambient temperature (◦C).

T0 growing season time-average daylight ambient surface tem-

perature (◦C).

T� leaf temperature (◦C).

T� optimum average temperature of leaf-intercepting crown-

averaged SW flux, Î � (◦C).ˆT � local species time (growing season) and canopy-average leaf

temperature (◦C).

Tm leaf temperature at which light-saturated rate of net photosyn-

thesis is maximized (◦C), and (Figure 3.17 only) mean daily

maximum atmospheric temperature at surface.

tb duration of rectangular pulses of SW flux for which i0 > I0

(hours).

tc duration of rectangular pulses of SW flux for which i0 ≤ I0

(hours).

u0 free stream wind speed (m s−1).


N O T A T I O N 133

Wpar m−2 photosynthetically active radiative flux ∼= 1/2Wtot m−2.

Wtot m−2 ≡ W m−2 SW radiative flux.

x position along the flux path (m).

z vertical distance from base of canopy (m).

z locally constant exponent of the area in the “species-area”

relationship (dimensionless).

max{DC} maximum carbon demand of a mature homogeneous canopy

(gCO2 m−2 h−1).

max{SC} maximum (i.e., saturating) carbon supply of a mature homo-

geneous canopy, (gCO2 m−2 h−1).

� (κ) single-parameter gamma function (dimensionless).

angle between beam radiation and a normal to the leaf surface

(deg).

�c minimum total canopy resistance (h m−1).

α angle between beam radiation and the horizontal (i.e., angle

of incidence) (deg).

αc fraction of season experiencing i0 ≤ I0 (dimensionless).

αT absorption coefficient, (dimensionless).

β cosine of the leaf angle, θ� (deg).

β spatial average cosine of the leaf angle (deg).

δ solar declination (deg).

E dimensionless ecodynamic similarity parameter for latitudi-

nal dispersion of C3 plants.

εi = Psm/Is� potential utilization efficiency of intercepted light (also

called potential photochemical efficiency) (gCO2 W−1 h−1 or

gs MJ−1tot ).

εi average across all woody C3 species of potential utilization

efficiency of intercepted light (gCO2 MJ−1tot or gs MJ−1

par).

η ≡ m−1h (W−1 h−1 m2).

� number of cloud events that together can intercept r species–

supporting energy in time, t = τ.

θ� leaf angle with the horizontal (deg).

κ leaf shadow area/leaf foliage area ≡ radiation extinction co-

efficient (dimensionless).

κ shape parameter of the gamma distribution of local SW flux

event arrivals (hj ) (dimensionless).

κ crown average of the radiation extinction coefficient at max-

imum assimilation rate (equals A∗s / A�) at which it equals β

(dimensionless).



κ geographical average of the gamma shape parameter from

observed series of rainstorm arrivals.

κ (2) shadow area/foliage area in two-leaf system (dimensionless).

κν = νκ (see equation 4.12).

λ scale parameter of the gamma distribution of the hj (W−1tot

h−1 m2).

λν = λ.

ν number of species-supporting local SW flux events, i0 ≤ I0,

in the growing season.

ν = gs/g = 0.5.

νmax maximum number of discrete local SW flux disturbances in

the growing season, mτ .

ξ dimensionless depth downward into crown (ξ =1 at base of

canopy) (m).

ρa mass density of air (mass of air per unit volume of air).

σ( ) standard deviation of ( ).

σE standard deviation of SW flux intercepted by primary canopy

(Wtot m−2).

σI0 standard deviation over time of the local (pixel) seasonal SW

flux (W m−2).

σ I0standard deviation (over the zonal pixels) of the time average

of the annual pixel SW flux (W m−2).⟨

σI0

⟩

zonal average of the standard deviation (over time) of the

average (for each year) pixel SW flux (W m−2).⟨

σI0

⟩

meridional average of the zonal averages of the standard de-

viation (over time) of the average (for each year) pixel SW

flux (W m−2).

σR(ν) standard deviation of R (ν) = (νκ)1/2/λ (W h m−2).

σh standard deviation of h = κ1/2/λ (W h m−2).

σs(�) standard deviation of local species in latitude units.

σs(βLt ) = σs(s) standard deviation of local species in species units.

σs(�−) ≡ σs|�−(�) standard deviation of species at latitude �− in latitude units.

σ s|� standard deviation of s at latitude �.

σs|�−(�) ≡ σs(�−) standard deviation of species at latitude �− in latitude units.

σs|�+(�) ≡ σs(�+) standard deviation of species at latitude �+ in latitude units.

σμ standard deviation of the resistance of sun leaves (s m−1).

σν standard deviation of the number of local SW flux events in

mτ .

σ〈I0〉 standard deviation (over time) of the zonal average of the

average (for each year) pixel SW flux (W m−2).


N O T A T I O N 135

σ 2( ) variance of ( ).

σ 2Rs

s|�◦ (�) variance of the sample range of species, s, in zone �0.

τ daylight-hour length of the local growing season (hours).

ψ carbon concentration in air (mass of carbon per unit mass of

air, dimensionless).

ω seasonal rate of cloud event arrivals (h−1).

� latitude (deg).

�◦ latitude of modal species whose range is being sought (Figures

1.4, 3.1, and 3.5) (deg).

�− latitude where largest species is the modal species at �◦ (Fig-

ures 1.4 and 3.1) (deg).

�+ latitude where the smallest species is the modal species at �◦

(Figures 1.4, 3.1, and 3.5) (deg).

�0L latitude south of SW flux maximum at which mean annual

SW flux is the minimum annual SW flux at the latitude, �00,

of the SW maximum (Figure 3.16b) (deg).

�0R latitude north of SW flux maximum at which mean annual

SW flux is the minimum annual SW flux at the latitude, �00,

of the SW maximum (Figure 3.16b) (deg).

�00 latitude of SW flux maximum (deg).

βLt species-defining total leaf area of plant per unit of horizontal

area (equals κLt ), i.e., “projected” leaf area index; corre-

sponds to the climate in that pixel during a particular growing

season (dimensionless).

βLt pixel average species (i.e., average of the local species distri-

bution) (dimensionless).

βL+t largest stable species at �0 (Figure 3.3).

ωτ ≡ mν mean number of zonal arrivals of SW flux events per season.

ωτ ≡ σ 2ν variance of number of zonal arrivals of SW flux events per

season.

(· · ·) increment of (· · ·).COVz[ . . . ] zonal covariance of [. . . ].

CV(· · ·) coefficient of variation of (· · ·).CV I0

(�) coefficient of zonal variation of temporal mean seasonal pixel

canopy-top SW flux at latitude �.

E (· · ·) expected value of (· · ·).max{· · ·} maximum of {· · ·}.

VAR(· · ·) variance of (· · ·).f (· · ·) functional notation.



f I (· · ·) the relationship of I as a function of (· · ·).f I0

(�) the relationship of I0 as a function of �.

fR(ν)(r ) pdf of R (ν).

fR(τ )(r ) pdf of R(τ ).

G (κ, λh) gamma probability distribution of the energy intercepted by

a species-supporting cloud event (dimensionless).

G R(ν)(λr ) dimensionless pdf of R(ν).

g (· · ·) a function of (· · ·).g(c) bioclimatic function.

gσI (· · ·) the relationship of σI0 as a function of (· · ·).h(�) = s one-to-one functional relationship between s and �.

hn (· · ·) the relationship of n as a function of (· · ·).�n (· · ·) natural logarithm of (· · ·).max�s potential number of local species (assumed ≤ maximum num-

ber of discrete local SW flux disturbances, νmax, in the grow-

ing season).⟨

d I0d�

⟩

zonal average of the latitudinal (i.e., meridional) gradients of

the time average of the annual pixel SW flux (W m−2 deg−1).d〈 I0〉

d�latitudinal (i.e., meridional) gradient of the zonal aver-

age of the time-averaged annual pixel canopy-top SW flux

(W m−2 deg−1).

pdf probability density function.

P� departure of the net rate of photosynthesis from its asymptotes

(Figure A3c) (gCO2 m−2 h−1).

(· · ·) increment of (· · ·).�s number of local species.

|· · ·| absolute value of · · ·.〈· · ·〉 zonal average of · · ·.〈· · ·〉 ≡ 〈· · ·〉 time average of the zonal average of · · ·, equal to the

zonal average of the time average of · · ·.· · · average of . . . over a stated area.

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Glossary

Abundance: Number of organisms of a given species per unit of area.

Allometry: Defines the relation between size and shape of objects and classes thereof.

Assimilation: Uptake of CO2 by plant for photosynthesis.

Bioclimatic function: Mathematical relationship defining plant species, s, resulting

from specific climatic forcing, c, i.e., s = g(c).

Biome: “A major type of natural vegetation that occurs wherever a particular set of

climatic and edaphic conditions prevail but that may have different taxa in different

regions” [Brown and Gibson, 1983, p. 558].

C3: The pentose phosphate pathway (i.e., Calvin-Benson cycle) for CO2 assimilation:

saturates at high light intensity; used by most plants, including most agricultural

crops and trees (both hardwoods and conifers).

C4: The dicarboxylic acid (i.e., Hatch-Slack) pathway for CO2 assimilation; utilizes

even the most intense solar radiation; tropical grasses and agricultural plants such

as millet, sorghum, and maize.

CAM: The Crassulacean acid metabolism pathway for CO2 assimilation; minimizes

water loss, opening stomata to take up CO2 only during the cooler nighttime;

succulent plants such as cacti.

Canopy density: Fraction of land surface covered by horizontal projection of crowns.

Chloroplast: A small, chlorophyll-containing mass in a plant cell.

Climate: Temporal and spatial variation of the pixel SW flux during the growing

season.

Combinatorics: The branch of mathematics studying the enumeration, combination,

and permutation of sets of elements and the mathematical relations that characterize

their properties.

137



Community: A local land surface area of homogeneous properties and unspecified

size.

Compensation radiation: That light intensity at which photosynthesis fixes an

amount of CO2 just equal to that given up in respiration.

Compensation ratio: Compensation SW flux divided by canopy-top SW flux.

Cytoplasm: The protoplasm of a leaf cell (excluding the nucleus).

Diversity: The number of local species per unit area.

Dormancy: A state of temporary inactivity.

Ecosystem: A biological community plus the physical environment that it occupies.

Edaphic: Related to or caused by particular soil conditions.

Envelope: Line connecting the maximum values of the dependent variable at each

value of the independent variable.

Evenness: An ecosystem descriptor of the degree of equality in the number of each

species present (i.e., “high” or “low” evenness).

Evolutionary equilibrium: Proposed optimal state of plant growth at which its

demand for and supply of CO2 are at once equal and at their maximum values.

Gedankenexperiment (thought experiment): “A device of the imagination used

to investigate the nature of things” [Brown, 2007] (see http://plato.stanford.edu/

entries/thought-experiment/).

Germinate: To begin to grow or develop; to sprout forth.

Hectare: 10,000 m2.

Insolation (incoming solar radiation): The rate at which direct solar radiation is

received at the canopy top.

Mean: Numerical average of all observations.

Meridional: Along a line of constant longitude.

Mesophyll: Plant tissue forming the interior parts of a leaf.

Metacommunity: The biogeographic unit in which most member species spend their

entire evolutionary lifetimes (the Amazon basin or the arctic tundra, for example).

Michaelis-Menten equation: Equation (A1), also called the “photosynthetic capacity

curve.”

Modal: Element of a distribution having the largest frequency of occurrence.

Moment: (Mathematical) property such as mean and variance of a distributed

variable.

Monocultural: Single species.

NDVI: Normalized Difference Vegetation Index.

Neutral Theory: Assumes that all individuals of every species in a nutritionally

defined community obey exactly the same ecological rules [Hubbell, 2001]; how-

ever, these ecological rules are unspecified, thus requiring model calibration using

field observations of the vegetation. (We use this dual capitalization herein when

intending this strict Hubbell definition.)


G L O S S A R Y 139

neutral theory or “neutral theory”: Used with respect to the theory developed

herein, yielding latitudinal distributions of both local species richness and the

range of the local modal species under a defined (but approximate) bioclimatic

forcing.

NPP: Net primary productivity.

PAR: Photosynthetically active solar radiation, normally taken to be one-half total

SW radiative flux.

pdf: Probability density function.

Photoperiod: The interval in a 24-hour period during which a plant is exposed to

light.

Photosynthetically active SW flux, Wpar m−2: That fraction of the total shortwave

radiation, Wtot m−2 or just W m−2, that is involved in the photosynthetic process.

Pixel: Earth surface area covered by a single remotely sensed snapshot from an

orbiting satellite (77,312 km2 in this work).

Potential photochemical efficiency: εi = Psm/Is�; also called “climatic assimilation

potential” or “potential assimilation efficiency.”

Projected area: Total area times the cosine of the angle of the area with the horizontal.

Range: The continuous interval of latitude over which the mean or modal species at

another latitude is found.

Rapoport’s rule: The name given by Stevens [1989] to the observed correlation

between latitude and north-south range for a variety of taxa.

Realizable species: A species that is unstressed by the average growing season

insolation to which it is exposed.

Respiration: The process by which plants take up O2 and release CO2.

Richness: The maximum number of separate species that can be supported locally.

RUE: Radiation utilization efficiency (i.e., PAR conversion efficiency).

Scaling: The property of obeying a power law (adj.); proportioning (v.).

Shade leaves: Leaves with single-layered palisade cells resulting in low unit area

assimilation rates.

Species: Plant having a distinctive projected leaf area.

Species abundance: Number of individuals of the species per unit area (usually the

hectare).

Species diversity: Number of different species in a particular area each weighted by

its abundance.

Species evenness: Relative abundance with which each species is represented in an

area.

Species richness: Number of different species in a particular area of unlimiting

size.

Species supporting: Condition compatible with stability of particular species.

Specular: Nondiffusive.



Stomata: The small orifices in the epidermis of leaves.

Sun leaves: Leaves with multilayered palisade cells resulting in high unit area assim-

ilation rates.

SW flux: The rate at which the total shortwave solar energy arrives at a surface

per unit time and per unit area, Wtot m−2 or just W m−2. (Although the solar

spectral division is somewhat arbitrary, the PAR component is commonly taken as

approximately 1/2 Wtot m−2.)

Taxa: Plural of “taxon”; a grouping of like organisms in a systematic biological (in

this case) classification system.

Thought experiment: See Gedankenexperiment.

Total SW solar radiative flux: Approximately twice PAR.

Understory: Any canopy beneath the primary (i.e., topmost) canopy.

Vascular plants: Those plants having special tissues for conducting water, minerals,

and photosynthetic products through the plant from soil to leaf (e.g., ferns, flowering

plants, trees); nonvascular plants have no roots, stems, or leaves (e.g., mosses, green

algae, liverwort).

Visible radiation: The photosynthetically active half of the total SW flux.

Zeroth order: “First approximation”; as used here, it implies use of the simplest

mathematical formulations such as single independent variables and locally linear

functions.

Zonal: Concerning all land surface pixels at a constant latitude (actually over pixel

width of 5◦ meridionally).

P1: JZZ Trim: 7in × 10in Top: 42.5pt Gutter: 78ptAGUB001-BIB AGU001/Eagleson December 15, 2009 17:35

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Additional Reading

These additional references were pointed out to the author by an anonymous reviewer after themanuscript had gone to bed.

Gaston, K. J. (2003), The Structure and Dynamics of Geographic Ranges, 266 pp., OxfordUniv. Press, New York.

Hawkins, B. A., et al. (2003), Energy, water, and broad-scale patterns of species richness,Ecology, 84, 3105–3117.

Kraft, H., and W. Jetz (2007), Global patterns and determinants of vascular plant diversity,Proc. Natl. Acad. Sci. U. S. A., 104, 5925–5930.

Weiser, M. D., et al. (2007), Latitudinal patterns of range size and species richness of NewWorld woody plants, Global Ecol. Biogeogr., 16, 679–688.

Willig, M. R., D. M. Kaufman, and R. D. Stevens (2003), Latitudinal gradients of biodiversity:Pattern, process, scale and synthesis, Annu. Rev. Ecol. Evol. System., 34, 273–309.



Woodward, F. I. (1987), Climate and Plant Distribution, 174 pp., Cambridge Univ. Press, NewYork.

World Resources Institute/International Institute for Environmental Development (WRI/IIED)(1988), World Resources 1988–1989, Basic Books, New York.

Wright, D. H. (1983), Species-energy theory: An extension of species-area theory, Oikos, 41,496–506.

Additional Reading

These additional references were pointed out to the author by an anonymous reviewer after themanuscript had gone to bed.

Gaston, K. J. (2003), The Structure and Dynamics of Geographic Ranges, 266 pp., OxfordUniv. Press, New York.

Hawkins, B. A., et al. (2003), Energy, water, and broad-scale patterns of species richness,Ecology, 84, 3105–3117.

Kraft, H., and W. Jetz (2007), Global patterns and determinants of vascular plant diversity,Proc. Natl. Acad. Sci. U. S. A., 104, 5925–5930.

Weiser, M. D., et al. (2007), Latitudinal patterns of range size and species richness of NewWorld woody plants, Global Ecol. Biogeogr., 16, 679–688.

Willig, M. R., D. M. Kaufman, and R. D. Stevens (2003), Latitudinal gradients of biodiversity:Pattern, process, scale and synthesis, Annu. Rev. Ecol. Evol. System., 34, 273–309.

Author Index

Allen, L.H., 106

Anderson, P.W, 7, 15

Bailey, R.G., 19,20,67

Baker, ES., 106, 123

Benjamin, IR, xvii, 9, 10, 11,28,36,40,

42,68,74,75,76,78,124

Berliand, I.G., 50

Bird, RB., 64

Birkebak, R, 106

Bjorkman, IE, 123

Bonan, G.B., 8

Brockman, C.E, xiv, 4, 13,29,31,37,39-40,

42,43,45,46,49,52,54,57,60,89-91

Brown,IH., 1,7,17,83,141

Brown, IR., 142

Cannell, M.G.R, 106

Caro, R, 80

Carpenter, S.R, xv, 5, 69

CerIing, I.E., xv, 6

Clark, WS., 106

Condit, R., 3, 31

Connor, E.F., 71

Cornell, c.A., xvii, 9, 10, 11,28,36,40,42,

68, 74, 75, 76, 78, 124

Cox, D.R., 75

Currie, DJ., 4, 69

Davis, S.D., 70, 71, 72, 73, 79, 81, 82, 84, 93

Decker, IP., 106

deLaubenfe1s, 58, 59

Eagleson, P.S., xv, 9, 14,34,35,66,67, 73,

80,81,87,92,97, 104, 105, 106, 107,

111, 112, 113, 115, 122, 123, 124

Ehleringer, I, xv, 6, 106

Eldredge, N., xv, 118

EI-Hemry, 1.1., 80

Enquist, B.1., 4, 70, 79, 125

Entekhabi, D., 24, 28,40,41,43,44,45,46,

50,55,56,57,78

Fischer, A.G., 4, 69

Gates, D.M., 97, 101, 122

Gentry, A.H., 4, 13, 14,70, 71, 79, 91, 93

Gibson, A.c., 137

Goldenfeld, N., 70, 71 .

Gosse, G.C., 106

Gould, S.I, xv, 118

Harte, I, 1,7

Hawk, K.L., 80, 81

Heal, O.W, 67

Holmgren, P., 123, 124

Hom, H.S., 7, 17,97,98

149

150 RANGE AND RICHNESS OF VASCULAR LAND PLANTS

Hubbell, S.P., 5, 12,63, 84, 92, 140

Huston, M.A., 5, 6, 10, 13,69, 70, 71, 72,

83,91

Jar vis, P.G., 106, 107

Keely, IE., 6

Kodric-Brown, A., 83

Komer, c.R., 119

Kozlowski, T.T., 106

Kraft, N.J.B., 12, 14,63,93

Kramer, P.I, 106

Kussell, E., 72

Landsberg, J.J., 106

Larcher, W, 14,66,67,69,92,98, 104, 113,

116, 120

Laszlo, I., 21, 73

Legg, B., 101

Leibler, S., 72

Lemon, E.R., 106

Lewis, P.A.W, 75

Li,M.,6

Linder, S., 106

Lindroth, A., 106

Lomolino, M.V., 71

Lovelock, IE., 95

Ludlow, M.M., 107

MacArthur, R.H., 7

Marshall, c.R., 69

Martin, H.G., 70, 71

McCoy, E.D., 71

Miller, K.R., xv, 13,70, 71, 79, 82, 83, 91,

93

Monsi, M., 113-115

Monteith, IL., 97, 101, 106, 113

Mosteller, E, 42

Miiller, D., 106

Niklas, I, 4, 70, 80, 125

Oort, A.H., 49, 50, 51

Paquin, v., 4, 69

Peixoto, IP., 49, 50, 51

Penning de Vries, EWT., 106, 122

Pickett, S. T.A., xiv, 72

Pinker, R.T., 21, 73

Rapoport, E., xv, 3, 92

Rauner, IL., 123

Reid, Wv., xv, 13, 70, 71, 79, 82, 83, 91, 93

Rey-Benayas, 1M., 5, 69

Rinaldo, A., 70

Rodriguez-lturbe, I., 70

Ross, I, 106, 111

Rosswall, T., 67

Roy, I, 4

Ruimy, A.B., 106

Saeki, T., 113-115

Scheiner, S.M., 5, 69

Schluter, D., 69

Stevens, G.C., xv, 3, 4, 7,14,31,69,83,84,

93, 141

Strahler, A.N., 48, 49, 52

Strokina, L.A., 50

Svenning, I-C., 3, 31

Thorn, A.S., 101

Tilman, D., 7,14

Todorov ic, P., 73

Trewartha, G.T., 19,21

Wallace, A.R., 4, 69

Weir, I, 69

West, G.B., 1,7,17,70

W hite, P.S., xiv, 72, 97

Williamson, M., 71

Wilson, E.o., xv, 5, 7, 69, 83, 85

Woodward, EI., 3

Wright, D.H., 4

P1: JZP Trim: 7in × 10in Top: 42.5pt Gutter: 78ptAGUB001-SIN AGU001/Eagleson December 17, 2009 0:44

Subject Index

Acacia craspedocarpa, 66annual land surface, 23–24arctic willow, 106arrival process, 92assimilated carbon, 122assimilation modulation, 104atmospheric temperature, latitudinal

distribution of, 66autumnal equinox, 49

bioclimatic basis, for local communitystructure, 7–8

bioclimatic control, high-latitude shift in,65–68

bioclimatic dispersion, 63–65southward latitudinal, 64

bioclimatic function, 33, 88–89at canopy scale, 117–118one-to-one, 45for primary canopies, 62species-controlling, 125Taylor expansion of, 34Taylor series approximation to,

9univariate, 108–110zeroth-order, 11

biodiversity, 5biological transformation, of local

distributions, 36

biomeslatitudinal boundaries of, 20of North America, 19

C3 leafassimilation modulation, 104biochemical structure of, 98Darwinian operating state, 107–108idealized geometry of, 111–113optically optimal geometry, 112photosynthetic capacity of, 97–99,

104–105potential assimilation efficiency of,

105–107structure of, 102

C3 species, 6, 8, 9canopy, 111–121cloud-supporting events, 74, 78–79common saturating property of, 99distribution, 32, 75–77germination, 79–80in growing season, 75–77intercepted in growing season, 77–78modal local, 38parameters of, 106, 116restriction, 84stress-constrained local distribution of,

37supporting radiation, 75–77

151



C4 pathway, 6CAM pathway, 6Canadian Climate Program, 67canopy resistances, 123–125

interleaf, 100canopy scale, 117–118

evolutionary equilibrium, 121–122carbon, 122

supply evidence, 123chloroplasts, 99–104climate, 87climate variability, 69climate zonal homogeneity, 28climatic assimilation potential, see potential

assimilation efficiencyclimatic disturbance, 79–80climatic forcing

analytical summary for, 43–45theoretical estimation of range with,

36–39closed canopy, 113–116CO2, 8

ambient concentrations, 97, 104assimilation rates, 97flux resistance, 100intercellular concentration, 102

cold pulse, 92common saturating property, 99complete distribution, 40continuous distribution, 72–73convective variability, see spatial variabilitycovariance, 41creosote bush, 106

Darwinian heat proposition, 113Davis curve, 72density functions, 76

dimensionless, 77probability, 88

derived distributions, 9dimensionless density function, 77dimensionless Schmidt number, 64discontinuous local modal species,

62discontinuous range, 62discrete distribution, 72–73

dispersion of species, 63–65distribution

C3 species, 33, 37, 75–77complete, 40continuous, 72–73derived, 9discrete, 72–73gamma, 76–77, 81geographic, 5latitudinal, 48, 52, 58, 62, 66local, 36, 37local species, 34–35, 37, 42–43, 72–73one-sided, 8Poisson, 75single-sided, 40stress-constrained local, 37

dry-atmosphere latitudes, 52

equinoctial average growing season, 89European beach, 106evolutionary equilibrium, 11, 88

at canopy scale, 121–122carbon supply evidence, 123at leaf scale, 121local, 118–120

Fick’s law, 101flux-gradient relationship, 101free atmosphere, 99–104free stream velocity, 123functional types, 6

gamma distribution, 76–77shape factor, 81

gamma function shape factor, 80Gedankenexperiment, 49, 50geographic distribution, 3GISS, see NASA-Goddard Institute for

Space Studiesglobal pixel SW flux, 50global zonal average, 21

of annual land surface, 23–24of daytime average SW flux, 26of latitudinal gradient, 27of meridional gradients, 26of observed pixel climate, 24, 28


S U B J E C T I N D E X 153

seasonal canopy-top, 23of standard deviation, 23–24, 25

global zonal standard deviation, 25, 26gradient estimation

latitudinal distribution of local species,61

of range, 52–55, 60–62growing seasons, 10

C3 species in, 75–77C3 species-supporting radiation

intercepted in, 77–78estimated, 21, 24local climate, 19

heat, 65–68heating-cooling cycles, 72high-latitude shift, 65–68historical summary, 1–3homogeneity scaling, 71homogeneous C3 canopy, 111–121homogeneous crown, 114

idealized geometry, 111–113idealized local time series, 74instability, 87interannual variability, 9interleaf canopy resistance, 100International Satellite Cloud Climatology

Project (ISCCP), 21, 40land-only pixels in zonal band from,

22intertropical convergence, 84ISCCP, see International Satellite Cloud

Climatology Projectisotropic atmosphere, SW flux for, 49–51

land-only pixels, 21number of, 24in zonal band, 22

latitudinal boundaries, of biomes, 20latitudinal distribution

of atmospheric temperature, 66of mean latitudinal range of local

species, 31–68latitudinal envelopes, of observed plant

richness, 70

latitudinal gradient, 3–4, 69global zonal average of, 27

latitudinal rangeestimation of, 56of local modal species, 47, 48, 52, 56

latitudinal variationof gamma distribution shape factor, 81of local species richness, 83

leaf resistances, 123–125leaf scale, 108–110, 121least squares, 48light, 65–68light saturated systems, 103loblolly pine, 106local area, 70–71local canopy scale, 121–122local climate, 19–29

growing seasons, 19major biomes, 19variations in, 36

local community structure, bioclimatic basisfor, 7–8

local distributionsbiological transformation of, 36stress-constrained, 37

local evolutionary equilibrium, 118–120local modal species

C3, 38discontinuous, 62latitudinal range of, 47, 48, 52, 56range of, 39–41, 51–52

local richness, maximum envelope of, 70local species

bioclimatic dispersion of, 63–65continuous distribution of, 72–73discrete distribution of, 72–73frequency of distribution of, 37latitudinal variation of, 83mean latitudinal range of, 31–68mean of ranges of, 39–41modal, 39–41, 47, 48, 52, 56, 62observed, 42–43piecewise latitudinal linearization of, 57probability mass of distribution of

observed, 42–43range of, 32–36



local species (cont.)richness of, 69–84standard deviations, 38

local variability, see point variabilitylow-latitude smoothing, 62–63low-moisture atmosphere, 49–50lumped canopy, 118

major simplifications, 15mass diffusivity, 101mass flux density, 101mass transfer, 99–104maxima of SW flux, 51–52mean latitudinal range

definition, 31–32of local species, 31–68

mean point values, 53meridional gradients, global zonal average

of, 26Michaelis-Menten equation, 97–99, 104Millennium Ecosystem Assessment, 5minima of SW flux, 51–52modal local C3 species, constrained range of,

38modal species, range of, 51–52, see also

local modal speciesmodeling philosophy, 5–7moist forests, 28Monsi-Saeki extinction equation, 115multicultural symbiosis, 15multiple forcing variables, 68

NASA-Goddard Institute for Space Studies(GISS), 21, 40, 43, 56

land-only pixels in zonal band from, 22normalized fluctuation around zonal

mean, 44satellite data set, 45

neutral theory, 12, 91nighttime respiration, 98nitrogen, 12–13normalized fluctuation, around zonal mean,

44, 55

observed local species, probability mass ofdistribution of, 42–43

observed pixel annual shortwavefluctuations, 55

observed richness, 82–84off-mode species, 89Ohm’s law, 101one-sided distribution, 8optimally supported species, 9–10

parameter estimation, 80–81percentage mass, 42photosynthetic behaviors, 6

of C3 leaf, 97–99, 104–105piecewise latitudinal linearization, 46

of components of local species range,57

Pinus cembra, 66pixel climate, see local climatepoint rainfall, 73point variability, 65point-by-point estimation, 45–49, 91

first method, 46of latitudinal distribution of local species

range, 58of range, 45–49, 55–60second method, 46–48

Poisson distribution, 75Poisson probability mass function, 81potential assimilation efficiency, 35

of C3 leaf, 105–107potential richness, 82–84power laws, 70primary canopies, bioclimatic function,

62principal assumptions, 15principal findings, 15probability density function, 88probability mass, of observed local species,

42–43

range, 9–13bioclimatic dispersion and, 63–65discontinuous, 62extension of forecasts, 68gradient estimation of, 52–55, 60–62idealized, of mean local species, 12latitudinal, 47, 48, 52, 56


S U B J E C T I N D E X 155

of local mean species, 32–36, 39–41, 58,61

of local modal species, 39–41, 51–52low-latitude smoothing of, 62–63observation, 45–49point-by-point estimation of, 45–49,

55–60richness and, 84theoretical estimation of, 36–39of vascular land plants, 91

Rapoport’s rule, 3, 5rectangular pulse, 73red oak, 106reductionism, 85resistances

canopy, 123–125CO2 flux, 100interleaf canopy, 100leaf, 123–125

Reynolds number, 65richness, 14

gradient, 5local, 70local species, 69–84observed plant, 70potential v. observed, 82–84range and, 84of vascular land plants, 91zonal, 14

saturation mechanisms, 35scaling, 70

homogeneity, 71short dashes, 48shortwave radiative flux, 8, 16single-sided distribution, 40Sitka spruce, 106, 107solar radiation, 20–27southward latitudinal bioclimatic dispersion,

64spatial variability, 64species-area relationship, 70–71species-controlling bioclimatic function, 125species-supporting cloud events, 74stationary Poisson stochastic process, SW

flux as, 73–75

stoichiometry, 122stomatal control circuit, 102–103stress, 107stress-constrained local distribution, 37stressing, 73SW flux, 88, 102

astronomical, 78climatic forcing by, 36–39, 43–45disturbance pairs, 81estimation of, 59global pixel, 50instantaneous, 92in isotropic atmosphere, 49–51local maximum in, 59local minimum in, 58maxima, 51–52minima, 51–52seasonal average, 115–116seasonal canopy-top, 74as stationary Poisson stochastic process,

73–75

Taylor expansion, 37bioclimatic function, 34

Taylor series approximation, 36bioclimatic function of, 9

top-of-the-atmosphere SW flux, 27trees, 3–4Trewartha, 19tropical forest dynamics, 63tropical preeminence, 5

univariate bioclimatic function, 108–110univariate state equation, 125

variabilitypoint, 65spatial, 65

vascular land plants, 91vertical flux of radiation, 113–116von Karman’s constant, 123

warm pulse, 92water, 12–13wavelike oscillations, 45white oak, 106



zeroth-order, 7, 8bioclimatic function, 11estimate of local species distribution,

34–35zonal average species, 32zonal bands, 31

zonal climate, 41zonal homogeneity, 27–29

climate, 28zonal mean, normalized fluctuation around,

44, 55zonal richness, 14

Documents

Range and Richness of Vascular Land Plants: The Role of Variable Light