SYMPOSIUM

SPATIAL PROCESSES AND EVOLUTIONARY MODELS:

A CRITICAL REVIEW

by P. DAVID POLLYDepartments of Earth & Atmospheric Sciences, Biology, & Anthropology, Indiana University, 1001 E. 10th Street, Bloomington, IN 47405, USA;

pdpolly@indiana.edu

Typescript received 30 July 2018; accepted in revised form 16 October 2018

Abstract: Evolution is a fundamentally population level

process in which variation, drift and selection produce both

temporal and spatial patterns of change. Statistical model fit-

ting is now commonly used to estimate which kind of evolu-

tionary process best explains patterns of change through time

using models like Brownian motion, stabilizing selection

(Ornstein–Uhlenbeck) and directional selection on traits mea-

sured from stratigraphic sequences or on phylogenetic trees.

But these models assume that the traits possessed by a species

are homogeneous. Spatial processes such as dispersal, gene

flow and geographical range changes can produce patterns of

trait evolution that do not fit the expectations of standard

models, even when evolution at the local-population level is

governed by drift or a typical OU model of selection. The basic

properties of population level processes (variation, drift, selec-

tion and population size) are reviewed and the relationship

between their spatial and temporal dynamics is discussed.

Typical evolutionary models used in palaeontology incorpo-

rate the temporal component of these dynamics, but not the

spatial. Range expansions and contractions introduce rate

variability into drift processes, range expansion under a drift

model can drive directional change in trait evolution, and spa-

tial selection gradients can create spatial variation in traits that

can produce long-term directional trends and punctuation

events depending on the balance between selection strength,

gene flow, extirpation probability and model of speciation.

Using computational modelling that spatial processes can cre-

ate evolutionary outcomes that depart from basic population-

level notions from these standard macroevolutionary models.

Key words: Brownian motion, evolutionary model fitting,

evolutionary rate, metapopulation, trait evolution.

EVOLUT ION is not only a temporal process, but a spatial

one. Just as species change through time, they vary in

space. Their geographical ranges encompass assorted envi-

ronmental conditions and harbour considerable genetic

and phenotypic variation (Fig. 1). But spatial variation is

more than just simple population variation, it is a gradi-

ent of differences between local populations or demes that

arises from metapopulation population processes related

to distance, barriers, gene flow, drift and selection. Spatial

variation interacts with evolutionary processes in ways

that produce patterns of trait change that differ markedly

from the expectations of standard statistical evolutionary

models like Brownian motion, directional selection and

adaptive peak (Ornstein–Uhlenbeck) models. In this

paper, I critically review the ways in which spatial pro-

cesses cause departures from these standard evolutionary

models. My aim is to better inform interpretations drawn

from evolutionary model fitting. No existing statistical

models fully capture the ways in which spatial processes

modify standard evolutionary models, so no remedies are

offered herein; however, the signatures of spatial processes

can be recognized in some cases. This paper aims to

inform the reader what those signatures might look like

in palaeontological data.

Ever since Darwin (1859), variation has been understood

to be a central component of evolution. Evolution in its

quantitative formulation consists of change in a trait’s

mean value as a result of the intergenerational sorting

effects of drift and selection on the heritable variation

within a population (Wright 1929; Fisher 1930; Lande

1976). The Modern Synthesis recognized that rates and

directions of evolution are not simply consequences of

selection, but are constrained by the size of a population,

the amount of variation, and the rate at which variation is

replenished by mutation (Fisher 1930; Haldane 1932;

Dobzhansky 1937; Huxley 1942). Simpson’s (1944) adap-

tive landscapes with their centrifugal (stabilizing), cen-

tripetal (diversifying) and linear (directional) selection

© The Palaeontological Association doi: 10.1111/pala.12410 1

[Palaeontology, 2018, pp. 1–21]

regimes and their horotelic (normal), bradytelic (slow com-

bined with depauperate diversity) and quantum (rapid

adaptive peak jumping) rates were models based on princi-

ples of within-population variation to explain patterns of

evolution in the fossil record. Evolutionary biologists con-

sequently have made important strides in documenting

within-species variation (Darwin 1868; Bateson 1894; Kin-

sey 1929; Mayr 1963; Berry 1977; Yablokov 1977, 1986).

Two very different concepts of within-species variation

exist, however: unstructured variation and spatially struc-

tured variation. Unstructured variation is the classic

model of Fisher (1930) in which a species consists of a

single large, panmictic population with a spatially uni-

form gene pool. In this model, the average values and

range of variation in traits are the same at all geographi-

cal locations. Selection and drift act on this species-wide

population from one generation to the next to change the

average trait value, a change that is again the same for

the entire species. As described below, most of today’s

statistical models of trait evolution are based on this

‘Fisherian’ view.

Spatially structured variation, the alternative, is a con-

cept in which a species is a metapopulation composed of

interacting local populations (or demes), each with their

own trait means and local pool of variation (Levins 1968;

Hanski 1999). In the metapopulation concept local popu-

lations may vary in population size and may occupy dif-

ferent habitats. Drift and habitat-specific selection tend to

cause local populations to differentiate from one another

thus producing spatially structured variation, processes

that are counteracted by dispersal and gene flow that tend

to make the local populations more alike (Levins 1968;

Endler 1977). Geographical connectivity, dispersal ability,

differences in local population fitness, and differences in

interactions with other species govern the dynamics of

metapopulations (Holt 1985; Hanski 1999; Thompson

2005). Waxing and waning of the geographic range of a

metapopulation species due to environmental change,

changes in resource availability, competition, or other

factors, will modify the evolutionary dynamics of the

metapopulation by altering the spatial distribution of its

local populations, population sizes, selection gradients,

decreasin

gly cuspy lower second p

rem

ola

rs

PC 1 skull shape

A

B

F IG . 1 . Geographical variation in fossilizeable traits in two living mammal species. A, variation in the proportion of individuals in

local populations of the red fox, Vulpes vulpes, that possess posterior accessory cusps on the lower second premolar teeth (based on

data from Szuma 2007). B, variation in the shape of the ventral side of the skull of the common shrew, Sorex araneus. Colour coding

represents a spectrum along the first principle component of shape (PC 1), the end members of which are shown as blue and red land-

mark configurations (based on data from Polly & W�ojcik in press).

2 PALAEONTOLOGY

and patterns of connectivity (Brown 1996; Hewitt 1996;

Gaston 2003; Stuart et al. 2004). Population subdivision

thus allows drift and local selection to ‘experiment’ with

new varieties in local populations while maintaining the

overall viability of the species through dispersal and gene

flow. The process of speciation itself also affects evolu-

tionary processes in geographically heterogeneous

metapopulation because daughter species are likely to dif-

fer in their trait means and ranges of variation because

each is composed of only a subset of the total geographi-

cal range of variation in the parent species (consider the

morphological difference if, for example, the British pop-

ulations of shrew in Fig. 1B speciate) whereas daughter

species from a large, unstructured panmictic population

will be essentially identical at the time of speciation

(Mayr 1942, 1963; Bush 1975).

In his ‘shifting balance theory’, Wright (1931, 1969)

argued that without this kind of metapopulation experi-

mentation a species cannot move from one adaptive peak

to another. These factors all make evolution in a spatially

variable metapopulation much more complex than in the

panmictic population concept. The evolutionary out-

comes of panmictic and metapopulation views are so

great that they can be distinguished as ‘Fisherian’ and

‘Wrightian’ respectively (Wade & Goodnight 1998; Wade

2016; Wagner 2017).

The statistical models that are used today to infer evolu-

tionary processes from palaeontological data are based on

fundamentally ‘Fisherian’ principles that are ultimately

derived from Simpson’s adaptive peak metaphor. Regard-

less of whether the models are applied to phylogenetic trees

(Felsenstein 1985, 1988; McPeek 1995; Martins & Hansen

1997; Blomberg et al. 2003; Butler & King 2004; O’Meara

et al. 2006; Harmon et al. 2008; Revell 2008; Revell et al.

2008, 2012; Harmon et al. 2010; Slater et al. 2012; Slater

2015) or stratigraphic sequences of palaeontological sam-

ples (Bookstein 1987; Gingerich 1993; Roopnarine 2001;

Polly 2004; Hunt 2006, 2007, 2012; Hannisdal 2007) they

produce estimates of rates, directions and modes of evolu-

tion based on equations that treat each species as a single

population whose size remains stochastically constant.

While individual models vary, they are broadly derived

from quantitative genetic equations that are used to predict

the magnitude and direction of trait evolution.

The model of Lande (1976), among the first to evaluate

the contributions of selection and drift to trait evolution

in the fossil record, illustrates the basic principles that

underpin most of today’s statistical models of evolution

(Fig. 2). Lande developed probability models for predict-

ing the patterns of evolutionary change that would be

observed depending on whether selection or drift domi-

nated, including the different expectations for directional

vs stabilizing selection. He drew on Wright’s and Simp-

son’s adaptive landscape concept to model selection,

which Lande implemented as a normally distributed func-

tion (W in Fig. 2A) for the fitness of different trait values

(ln½W �) relative to a selective optimum (h) represented by

the peak of the normal curve (Lande 1976; Felsenstein

1988; Arnold et al. 2001). The contribution of selection

to change in the average value of a trait (DZ) in Lande’s

model is the product of the selection coefficient ß and

the trait’s genetic variance, or bG (Fig. 2A). The selection

coefficient is given by the slope of the fitness curve

(@ ln½W �=@Z), which has a sign that always carries Z

towards the peak where b becomes zero (Fig. 2A). Selec-

tion is stronger the farther the population mean is from

the optimum, so narrow peaks keep the trait close to the

optimum and broad ones allow the trait to vary widely.

G is the trait’s additive genetic variance (heritable vari-

ance), which is the subset phenotypic variance (P) that

comes from offspring resembling their parents. The pro-

portion of P that is genetic is referred to as the trait’s

heritability (h2). Note that P can be estimated directly

from measurements taken from a fossil sample and h2 is

a proportion that ranges from 0 to 1, which allows one

to estimate a range of credible values for G. Drift also

contributes to DZ (Fig. 2A). Drift is a process that arises

from stochastic sampling error in the trait’s mean from

one generation to the next. Specifically, the expectation of

the magnitude of change due to drift is √(G/N) where N

is population size. Drift is equally likely to be in a nega-

tive or a positive direction, which makes it a type of

Brownian motion or random walk process, the long term

outcomes of which are well understood from a statistical

perspective (Felsenstein 1988; Berg 1993).

Most of today’s statistical evolutionary models, especially

those based on Brownian motion (random walk) or adap-

tive peaks (Ornstein–Uhlenbeck or OU processes), are

grounded in the same principles as Lande’s (Felsenstein

1988; Polly 2004; O’Meara 2012; Pennell & Harmon 2013)

and are thus ‘Fisherian’ in concept because they are based

on expectations of how a population with a single mean

will behave in macroevolutionary timescales under drift or

selection. This is illustrated by looking at how a simple

Brownian motion model is fit to a stratigraphic sequence of

samples (Hunt 2006). Brownian motion is a pure random

walk in which the direction of trait change at any step (gen-

eration) is independent of previous changes and has equal

probability of being negative or positive (Felsenstein 1985,

1988; Bookstein 1987). When a Brownian motion process

is repeated over many lineages, it produces a predictable

pattern of outcomes in which the trait values at the ends of

the lineages are, on average, equal to the ancestral value

and their variance equals the variance of the changes at

each step (r2, which is the rate parameter) times the num-

ber of steps tr2 (Fig. 2B). The fit of the changes observed

in real palaeontological data with respect to those expected

from a Brownian motion process is evaluated by fitting a

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 3

normal distribution with a mean (l) of zero (which is

appropriate for Brownian motion because positive and

negative changes are equally likely) and an unknown vari-

ance (i.e. rate r2) that is estimated from the trait data.

Maximum likelihood algorithms like that of Hunt (2006)

yield a likelihood value for all estimates of r2, including the

one that maximizes the likelihood and thus gives the best

estimate for the evolutionary rate given the data and the

Brownian motion model.

Other evolutionary models, such as directional or stabi-

lizing selection (the latter including stasis), can be fit using

a similar likelihood algorithm and an appropriate set of

parameters. Stabilizing selection, for example, is an OU

adaptive peak model with parameters for the location of

the optimum (h) and width of the peak (x) (Fig. 2A, B).Using the Akaike Information Criterion (AIC) to adjust for

the differences in parameters, the relative strength of fit can

be used to select the model that best describes the observed

–4 –2 0 2 4 6 8 10

–4 –2 0 2 4 6 8 10

0

20

40

60

80

100

Mean Trait Value (Z)

Tim

e

–20 0 20 –20 0 20 –20 0 200

20

40

60

80

100

Mean Trait Value (Z)

Tim

e

BrownianMotion

DirectionalSelection

AdaptivePeak (OU)

B C

directionalpeak shift

stabilizingselectionwith drift directional

peak shift

stabilizingselectionwith drift

Z (trait values in population)

Trait value

Z (

mea

n tr

ait v

alue

)

optim

um tr

ait v

alue

(θ

)

fitness = ln[W]

phenotypicvariance (P)

adaptive peak width (ω)(strength of selection)

ΔZ = ßG + E[0,G/N]

ß = ∂ ln[W] / ∂ Z

G = h2 P

W (adaptive peak)

A

F IG . 2 . Statistical evolutionary models. A, characteristics of Lande’s (1976) adaptive peak model for selection and drift; see text for

details. B, simulations of evolution in a stratigraphic lineage, each for 100 steps under Brownian motion, directional, and stabilizing mod-

els of evolution; each model was simulated 500 times (grey lines) with one run highlighted (black line) to illustrate an actual outcome

against the statistical expectation of the model. C, an example of a multiple-peak model of evolution that shifts from stabilizing to direc-

tional selection as lineages move from one peak to another; see Polly (2018b) for an animated version of the multiple-peak model.

4 PALAEONTOLOGY

data (Hunt, 2006). Such procedures are frequently used to

infer the mode of evolution in a lineage or clade (e.g.

whether the evolutionary process had a random, direc-

tional, stabilizing, or static pattern), or to reconstruct

ancestral trait values (Martins & Hansen 1997; Polly 2004;

Revell et al. 2008; Pennell & Harmon 2013; Slater & Har-

mon 2013; Pennell et al. 2014; Hunt & Rabosky 2014).

While statistical models of evolution have many variants,

such as multiple rate and multiple peak models (Fig. 2C;

Butler & King 2004; O’Meara et al. 2006; Hunt 2008; Hunt

et al. 2015) they are all ‘Fisherian’ in that the statistical

outcomes are derived from expectations of how evolution-

ary processes affect the trait mean of a single population.

In this paper, I will present examples of how spatially

explicit ‘Wrightian’ processes can cause long-term patterns

of trait evolution to depart from those expected under

‘Fisherian’ models. For example, pure drift processes com-

bined with range expansions can produce species-level pat-

terns that resemble punctuated equilibrium models, and

stabilizing selection processes in which the optimum varies

geographically can produce patterns that resemble direc-

tional selection. The evolutionary effects of spatial dynamics

will be illustrated using computational modelling of trait

change in geographically explicit settings over geological

time scales. Computational modelling allows control of the

parameters for population size, selection, dispersal proba-

bility, gene flow and probability of extinction in the local

populations of a species (metapopulation) over thousands

or millions of generations, either as a single unbranching

lineage or as a branching phylogenetic tree (Polly 2004;

Polly et al. 2016). We will first look at how drift processes

that are normally associated with Brownian motion can

produce punctuated bursts of evolutionary change driven

solely by changes in the geographical range of the evolving

species. We will then look at how spatially structured selec-

tion along environmental gradients can produce patterns of

trait evolution that mimic directional selection, stasis and

punctuated equilibrium. I will apply Hunt’s (2006, 2007,

2008) likelihood method for evolutionary model selection

to estimate what modes of evolution we might have

inferred if we were ignorant of the role spatial processes

played in generating the simulated data. Finally, I will

review two examples of real data from the fossil record

where spatial processes might logically have played a role

and discuss how ‘Fisherian’ and ‘Wrightian’ explanations of

the patterns might differ.

BROWNIAN MOTION IN A CYCLICALENVIRONMENT

Brownian motion is arguably the simplest and most pre-

dictable statistical model for quantitative trait evolution

(Felsenstein 1985, 1988; Bookstein 1987; Roopnarine

2001; Hunt 2006). As described above, Brownian motion

is a pure random walk process in which each step (DZ) isdrawn from a random distribution that is symmetrical

around a mean (l) zero with a variance or step rate (r2),

which produces outcomes that are normally distributed

with a mean equal to the ancestral value and a variance

equal to the tr2 (Fig. 2B; Felsenstein 1988; Hunt 2006;

Revell et al. 2008). Pure genetic drift is a random sam-

pling process where r2 is equal to G/N (Fig. 2A; Lande

1976). Note that pure drift can be distinguished in

palaeontological data from selectional drift, which is a

Brownian motion process involving stochastically fluctu-

ating selection (Kimura 1954; Felsenstein 1988; Revell &

Harmon 2008), by measuring the phenotypic variance P

and combining it with credible estimates of h2 and N to

determine whether the estimate of r2 is substantially

greater than expected (Polly 2004; W�ojcik et al. 2006;

Polly et al. 2013). Note also that the long-term expecta-

tions of drift and selection assume that heritable variance

is replenished. If rate of change exceeds the heritable vari-

ance available in the population, then the rate becomes

limited by the accumulation of new mutations that con-

tribute to genetic variance (Kimura 1965; Turelli & Bar-

ton 1990). In such cases, long-term divergence will be

slower than expected based on P, h2 and N alone, but

divergence under a pure genetic drift process should not

exceed the expectation of Lande’s equation.

Genetic drift in a ‘Wrightian’ spatial context behaves

more like a variable-rate model of evolution than like

typical Brownian motion when it is accompanied by geo-

graphical range changes. It is probably safe to say that

geographical range changes have been the norm for most

species, because of post-speciation expansion from the

founder populations and because of changes in habitat

availability driven by Earth systems processes like sea-level

changes, glacial–interglacial cycles, or long-term climatic

trends (Hewitt 2004; Stigall 2013; Maguire et al. 2015;

Patzkowsky 2017). Range-expansion processes create

‘non-random’ patterns of traits among the local popula-

tions and range contractions create a disconnection

between drift in local populations (which have sizes Ni)

and the behaviour of the trait mean of the total species

metapopulation (which has size Nt = ∑Ni).

Computational model

The impact of geographical range changes on drift is

illustrated here using a computational model in which an

island metapopulation expands and contracts in response

to sea-level cycles (Fig. 3). This model was set on an

island platform that was entirely exposed during low-

stands with three peaks 2 m above platform height that

remained exposed as isolated islands during highstands

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 5

(Fig. 3A). The entire island platform was gridded into

5000 cells (50 9 100), each of which could potentially be

occupied by a local population.

Eustasy was modelled as a sine wave through 2.5 cycles

using the equation �5�Sin(0.005p�x � p) � 4, where x is

time measured in model steps (Fig. 3B). This equa-

tion causes sea-level to start 3 m below platform height

(thus exposing the entire platform at the beginning of

each model run), cresting at 2 m above platform height

at highstands (thus inundating everything except the very

peaks of the three islands) and dropping to 8 m below

platform height during lowstands.

An evolving species was modelled through space and

time using a metapopulation concept. At the beginning of

each run, a single founder population was randomly

placed on the platform with a local population size (N)

of 10, a trait value of 0, a phenotypic variance (P) of

0.002, and a heritability (h2) of 0.5. Recalling that the rate

of genetic drift is h2�P/N, drift in local populations

(demes) was modelled as a random change drawn from a

normal distribution with a mean of zero and a variance

of 0.001. The heritability value was chosen realistic for

morphological traits (Cheverud & Buikstra 1982) but the

other two values were arbitrarily chosen (see discussion

below about consequences of higher or lower rates of

drift). At each model step each local population: (1)

reproduced a new generation in its own cell, undergoing

a drift event at the same time; (2) had a chance of

expanding its range by giving rise to a new local popula-

tion in any or all of the adjacent cells (P = 0.8 for each

cell); and becoming locally extinct (P = 0.2). Each new

population, like its parent, had N = 10, P = 0.2, h2 = 0.5,

and a trait mean equal to the parent’s after a new drift

event. If a population expanded into an already occupied

cell, the two populations were merged through reproduc-

tion by averaging their phenotypes and resetting N = 10.

The averaging of traits in cohabiting populations creates

gene flow since a phenotype in one area can spread via

dispersal and reproduction to another. If the cell was cov-

ered by water the probability of extirpation rose to 1.0.

The geographical range of the species expands when

populations move into empty cells and contracts when

populations are extirpated. Sea-level thus changes the

geographical range and the number of extant local popu-

lations. At lowstands the species tends to grow to approx-

imately 5000 local populations that cover the entire

platform, and at highstands it contracts to as few as 12

populations subdivided between the three isolated islands

(Fig. 3C).

The computational model was run in Mathematica

(Wolfram Research 2018) with the aid of the Phylogenetics

for Mathematica 5.1 package (Polly 2018a) and the Quan-

titative Paleontology for Mathematica 5.0 package (Polly

2016). Code for the model and full outputs of all nine

model runs can be found in Polly (2018b, S2 (data), S8

(raw output)). Models were run on Indiana University’s

Karst high-throughput computer cluster.

Results and discussion

The outcomes of this model, which is a pure drift process

albeit with a spatial component, differ considerably from

the standard expectations of Brownian motion. The trait

mean, when measured as a species-wide average, has an

erratic behaviour, the tempo and direction of which

changes in phases along with the rise and fall of sea-level

(Fig. 3D). The trait values shown in Figure 3D are the

average over all the local populations and thus represent

the average trait value of the species as a whole. Because

it is an emergent property of the underlying metapopula-

tion processes of local drift, gene flow and changes in the

geographical range, the species mean behaves differently

than we would expect under ordinary Brownian motion.

It fluctuates wildly during highstands (i), tends to go

through rapid directional shifts during regressive expan-

sion phases (ii), and has long periods of what looks like

stasis during lowstands (iii). While the variance between

model runs generally increases through time, as one

would expect of a normal Brownian motion process

(Fig. 2B), it does not do so with the regular predictability

of ordinary Brownian motion.

Visually, the patterns produced by the spatial model

appear to be like those produced by a multiple adaptive

peak model involving shifts between stabilizing and direc-

tional selection when sea level rises above or falls below

the platform level (Fig. 2C). But in fact, the patterns are

produced by a multiple-rate random walk. Application of

Hunt’s complex model of trait change (Hunt & Rabosky

2014; Hunt et al. 2015) to the model outputs demon-

strates that this is so. The simple version of Hunt’s mod-

els uses maximum-likelihood and the parameter-corrected

Akaike Information Criterion (AICc) to estimate the rate

and mode of evolution from a temporal series of samples,

such as those modelled here (Hunt 2006, 2007). The

modes used here were unbiased random walks (i.e. ordi-

nary Brownian motion), directional random walks, and

stasis (i.e. stabilizing selection or OU process). Hunt’s

algorithm estimates which evolutionary model best fits

empirical data using the AICc parameter-adjusted maxi-

mum likelihood. In the complex application of the mod-

els, the data sequence is subdivided into segments, each

of which potentially can evolve under a different rate and

mode or have a different adaptive peak optimum (Hunt

et al. 2015). The number of model parameters increases

with the number of segments, so the AICc criterion is

again used to choose between simpler models in which

the sequence is generated by a process with a single rate

6 PALAEONTOLOGY

and mode and more complex models where the process

varies between segments of the lineage.

I used Hunt’s strategy to test for differences in rate and

mode associated with three phases of the sea level cycle:

the high-stand phase (i) when populations are isolated on

three small islands; the regressive phase (ii) when sea-level

is retreating across the platform and populations are

expanding; and the low-stand phase (iii) when sea-level is

below the platform edge and the geographic area is stable

(Fig. 2). Fitting nine alternative evolutionary models of

increasing complexity (Table 1) found that the one that

best fits these data is a series of six unbiased random

islands

platform100 500 900

0

–4

–8

Model Step

Sea

Lev

el (

m) i i i

ii ii ii

iii iii

High standphase (i)

Expansionphase (ii)

Brownianmotion phase (iii)

Trai

t val

ue

–0.05

0.15

ii ii ii iii i iiii iii

platform height

0 200 400 600 800 1000

–0.05

0.00

0.05

0.10

0.15

0 200 400 600 800 1000

0

–8Sea

Le

vel (

m)

Trai

t Val

ue (

Z)

Model Step

A

B

C

D

F IG . 3 . Computational model of genetic drift in a changing geographical range. A, evolution of a single terrestrial species was mod-

elled on a shallow marine platform undergoing eustatic cycles that periodically exposes the entire platform and then floods it leaving

only three small islands exposed. B, the species’ geographical range steps through three phases: (i) extreme contraction onto the three

islands during highstand; (ii) expansion across the platform during the regression phase; and (iii) maximum expansion during the low-

stand. C, spatial variation in trait values in local populations through a range expansion sequence: (i) drift causes differentiation to

build up on the three isolated islands during highstand; (ii) expansions from isolated founder populations result in differentiated

patches of trait values during regression phase; (iii) gene flow and drift slowly erase the differentiation during lowstand; see Polly

(2018b) for animated versions of the trait maps. D, nine runs of the model (grey and red lines) showing the mean trait value for the

species through the 1000 steps of each run along with the sea level curve.

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 7

walks (one per sea-level phase in the computational

model), each with a different rate parameter (including

separate rates for the four different expansion phases).

This multiple-rate random walk model not only fit the

run highlighted in Figure 3D (Table 1) but all nine runs

(Polly 2018b, S4). In a few cases an individual segment of

a model run fit a directional or stasis model better than

an unbiased random walk. Of the 81 individual segments

found collectively in the computational model runs (nine

segments (i–iii) in nine runs), only three supported a sta-

sis model and 15 supported a directional model (Polly

2018b, S4). The overall support for a multi-rate random

walk process in these models makes sense, because com-

putation model is driven by pure drift at the local popu-

lation level that produces emergent but related patterns

when measured as change in the species as a whole. Two

spatial processes in particular influence the species-level

pattern.

The first process that makes these spatial models depart

from ordinary random walk is that the overall population

size of the species changes during the course of the runs, as

does the total amount of trait variance. As described above,

genetic drift is a sampling phenomenon in which the mean

trait value of an offspring generation varies from its parent

generation because of randomness in which individuals

reproduce (Wright 1969; Lande 1976). For a single popula-

tion, the rate of drift (h2�P/N) is the evolutionary rate r2

and the Brownian motion process involves repeated resam-

plings from a normal distribution with l = 0 and

r2 = h2�P/N, in other words a single rate random walk

model. However, the total population size of the species as

a whole Nt = ∑Ni, where Ni is size of the ith local popula-

tion (Ni = 10 for all local populations in these models). At

highstands Nt drops to about 120, but at lowstands it may

approach 50 000 when the species covers the entire

platform. Similarly, the trait means of the local populations

vary such that that variance of the metapopulation (i.e. the

species as a whole) is different from the variance within a

given local population and is a function of the rates of drift

in the local populations (which causes them to differenti-

ate, thus increasing the overall variance) vs gene flow

between them (which homogenizes them, thus decreasing

the total variance) both relative to the rate at which the

geographical range expands and contracts. The rate of drift

at the population level r2pop was set at 1.0 9 10�3, which is

about four orders of magnitude higher than the rate for the

species as a whole at highstands (r2i = 3.20 9 10�7 in the

model highlighted in Fig. 3D), five orders of magnitude

higher than during expansion phases (r2iiðaÞ = 1.44 9

10�8, r2iiðbÞ = 3.17 9 10�9, r2

iiðcÞ = 1.68 9 10�8, r2iiðdÞ =

7.04 9 10�8) and six orders of magnitude higher than at

lowstands (r2iii = 4.14 9 10�9). While the rate of drift at

the species level is a complex function of the rates of drift,

gene flow and geographical range area, r2sp will not be con-

stant if Nt and the overall trait variance are not.

The second spatial process that makes trait evolution

depart from ordinary Brownian motion is non-random

resampling of peripheral populations during range expan-

sion. Because range expansion happens by multiplication

of local populations at the edge of the range, the idiosyn-

cratic variants in those populations can be multiplied

across the newly occupied area. This process can produce

the patchworks of spatial differentiation that are seen in

Figure 3C (and even more clearly in the animations

found in Polly (2018b)). The effect is even more pro-

nounced if the range contracts into geographically iso-

lated refugia, like the three islands in the computational

model, where they will differentiate by drift during the

period of separation. Formerly minor variants can quickly

become common during such expansion events, rapidly

TABLE 1 . Results of evolutionary model fitting for the computational model run highlighted in Figure 3D.

Model no. Model type lstep r2step h x Κ l AICc Akaike weight

1 Random walk – Same – – 1 6523.8 �13045.5 0.000

2 Directional Same Same – – 2 6523.8 �13043.6 0.000

3 Stasis – – Same Same 2 4452.9 �8901.8 0.000

4 Random walk – Diff (93) – – 3 7393.5 �13041.5 0.000

5 Directional Diff (93) Diff (93) – – 6 7389.5 �14766.9 0.000

6 Stasis – – Diff (93) Diff (93) 6 2299.7 �4587.4 0.000

7 Random walk – Diff (36) – – 6 12203.0 �24393.9 1.000

8 Directional Diff (96) Diff (96) – – 12 7472.9 �14921.4 0.000

9 Stasis – – Diff (96) Diff (96) 12 5119.6 �10214.9 0.000

Models 1–3 constrain the entire lineage to have the same rate (lstep, r2step) or adaptive peak (h, x) parameters, models 4–6 allowed

independent rate or adaptive peak parameters for each of the three sea-level phases of the computational model (i–iii) and models 7–9allowed independent parameters for each phase, including separate parameters for the four individual expansion phases (ii). K is the

number of parameters in the evolutionary model, l is the log likelihood of the model, AICc is the Akaike adjusted likelihood, and

Akaike Weights is the relative support for each model. Bold indicates the preferred model. See Polly (2018b) for results from all nine

runs of the computational model.

8 PALAEONTOLOGY

changing the trait mean of the species as a whole as seen

by the large, seemingly directional trait changes during

phase ii intervals (Fig. 3D). The effects of this process

have been extensively documented in spatial patterning of

molecular markers, notably in post-glacial recolonization

events of extant terrestrial plant and animal species

(Hewitt 1996, 1999; Avise et al. 1998; Magri 2008; Dap-

porto 2010), but also in marine species (Colson &

Hughes 2007; Maggs et al. 2008). The genetic mechanism

by which a rare allele becomes common as a result of

range expansion, a specialized kind of founder effect

(Ibrahim et al. 1996; Hewitt 2000; Excoffier et al. 2009;

Slatkin & Excoffier 2012), has been referred to as genetic

‘surfing’ because a random gene rides the expanding wave

of the range (Excoffier & Ray 2008).

Both processes are likely to affect the evolution of phe-

notypic traits as measured in the fossil record. While

sampling is often coarse compared to the extant world,

geographical range changes not only logically occurred in

the fossil record but they have been demonstrated in

many cases (Hellberg et al. 2001; Maguire & Stigall 2008;

Blois & Hadly 2009; Stigall 2010; Feurdean et al. 2013).

Refugial expansion and cyclic changes in the geographical

ranges of species in response to Quaternary climate cycles

are especially well documented (Roy et al. 1995; Lyons

2003; Stuart et al. 2004; Stewart 2009). Analyses of geo-

graphical variation in bones and teeth in extant and fossil

species indicates that the ‘surfing’ phenomenon also in

morphological traits in much the same way as it does in

genetic markers (Polly 2001, 2007; Szuma 2007; Polly

et al. 2013). Spatially modified pure drift processes could

therefore drive seemingly non-random trait changes in

palaeontological sequences, at least in principle, a possi-

bility that should be considered among other alternative

models of evolution.

Note that the computational algorithm and the param-

eters used in this paper were specifically chosen to illus-

trate the potential for spatial processes to produce

geographic differentiation and evolutionary rate shifts.

For a wide-ranging evaluation of how the balance

between drift, migration, local extinction and geographi-

cal ranges affects differentiation within a species see Slat-

kin (1977), Wade & McCauley (1988), Lande (1992),

Ibrahim et al. (1996) and Whitlock (1999).

SELECTION IN SPATIALLYHETEROGENEOUS ENVIRONMENTS

Selection and speciation in a spatially heterogeneous envi-

ronment can also cause notable departures from standard

evolutionary models, even in unchanging environments.

Characteristics that affect the performance of an organism

with respect to its environment are known as functional

traits. The morphology of teeth (King et al. 2005), the

strength of crocodilian skulls (Rayfield 2011) or the shape

of angiosperm leaves (Nicotra et al. 2011) are examples

of traits that affect the performance and therefore the rel-

ative fitness of an organism in different environments

(Arnold 1983). Directional and stabilizing selection (OU)

models of evolution are often used to test for this kind of

trait-environment relationship, but they operate in a

‘Fisherian’ manner that assumes that there is a single

optimal value for the entire species (or more precisely,

that DZ for the species is responding as though it were a

single population with a single G value experiencing the

same selection gradient ß, and thus responding to the

same adaptive peak W).

Even though directional selection causes change in a

trait and stabilizing selection prevents it from changing,

the two modes of evolution are both produced by stan-

dard adaptive peak (OU) models like Lande’s (Fig. 2)

and thus deserve comment. An OU model produces

directional change when traits are far away from the opti-

mum but results in stabilizing selection or stasis when

traits are near the optimum (Butler & King 2004).

Palaeontologists in particular are concerned with the dif-

ference between directional selection and stasis, in part

because of interest in testing Eldredge & Gould’s (1972)

punctuated equilibrium model of evolution, and therefore

tend to use two separate statistical models for the two

modes in order to distinguish between them. Hunt (2006,

2007, 2008), for example, represented directional selection

with a generalized random walk model (GRW) in which

the selectional events are drawn from a normal distribu-

tion with a non-zero mean (lstep) that produces a direc-

tional bias because the selection events are, on average, in

the same positive or negative direction. In contrast,

Hunt’s unbiased random walk model (URW) describes

Brownian motion because its events are described by a

normal distribution with a mean of zero, which repre-

sents an equal chance of moving in a positive or negative

direction. Note that Hunt’s directional model describes a

directional trend that has no reference to an optimum

and could, in principle, continue infinitely, but it ade-

quately describes the expectation for a directional trend

through a stratigraphical sequence of finite duration. To

represent stasis or stabilizing selection, Hunt (2006, 2007,

2008) used a simplified OU model that followed Sheets &

Mitchell’s (2001) ‘white noise’ model with only two

parameters for a trait optimum h and adaptive peak

width x (cf. Fig. 2) to which the DZ of a real data set is

fit, thus capturing only the stabilizing part of the process

without any directional component. Hunt’s models are

used below because they have the power to distinguish

directional trends from static ones, but readers should

note their differences from other OU implementations

(Hansen 1997; Butler & King 2004) that have directional

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 9

components when the trait is far from the optimum, sta-

bilizing components when it is near the optimum, and

Brownian motion components when it is at the

optimum.

A species evolving in a spatial mosaic of different selec-

tive environments does not respond to selection in the

‘Fisherian’ manner just described. If the range of the spe-

cies extends across a gradient of environments, its con-

stituent local populations will experience functional

selection toward different local optima that will tend to

produce geographical variation in traits (Fig. 1). Selective

differentiation among local populations is partially coun-

terbalanced by gene flow from other parts of the species

range. If speciation occurs by vicariance in a geographi-

cally differentiated species, the offspring species will be

different from one another. This effect is most extreme in

speciation by peripheral isolation, which is considered to

be the most common mechanism (Mayr 1942; Coyne &

Orr 2004) because the peripheral founder population is

likely to have a trait value that is a substantial outlier

from the mean of the parent populations, thus producing

a punctuated jump between parent and offspring

(Eldredge & Gould 1972). Expansion and contraction of

the species’ range adds to the ‘Wrightian’ effects because

it alters the number of local populations exposed to a

particular local selective optimum and thus changes the

evolutionary trajectory of the trait’s mean at the species

level. The evolutionary outcomes of a spatially distributed

species thus depend on the balance between local selec-

tion, gene flow, range changes and speciation (Endler

1977; Slatkin 1977; Wade & McCauley 1988; Lande 1992;

Frey 1993).

Computational model

The impact of spatial selection gradients and speciation

are illustrated using runs from the computational model

described by Polly et al. (2016). The model was set in a

mid-latitude island continent with a north–south trending

mountain ranges, latitudinal and altitudinal temperature

gradients, a rain shadow effect east of the mountains, and

four different vegetation biomes (Fig. 4A). Like with the

genetic drift model above, the model space was gridded

into approximately 25 9 40 cells (with irregular conti-

nental margins, the exact number of cells was 822).

A univariate trait (tooth crown height) was simulated in

a clade (mammalian herbivores) whose fitness depends on

the match between tooth type and local conditions: high

crown teeth are favoured in arid, gritty, grassy environ-

ments (trait value = 3) and low crown teeth in moist

forested environments (trait value = 1). The optimal value

of the trait in each cell is a function the local combination

of climatic and environmental conditions (Fig. 4B).

As with the genetic drift computational model above,

each species consisted of a set of local populations that

experienced drift (N = 100, P = 0.05, and h2 = 0.5, except

where otherwise stated), had a chance of dispersing into

adjacent cells (P = 0.5), experienced gene flow when two

or more conspecific local populations occupied the same

grid cell, and had a chance of becoming locally extinct

(function of the distance between the local optimum and

the trait value in the local population). Unlike the genetic

drift model, the local populations were also under selection

using Lande’s adaptive peak model (described above)

where the optimum was a function of the local conditions

(Fig. 4B) and the adaptive peak width (strength of selec-

tion) was set at 0.5 (except as otherwise noted). This model

also included four speciation events in which each extant

species gave rise to a new species arbitrarily every 100 mod-

els steps to yield a total of 30 species (two species in steps

1–100, four in steps 101–200, eight in steps 201–300, and16 in steps 301–400) except where lineages may have

become prematurely extinct. Speciation occurred by

peripheral isolation in which the population that was most

distant from the geographical centre of the species’ range

became the founder of the new species.

Runs lasted 400 steps, with 100 steps between specia-

tion events (including one at the first step). Note that

with 822 grid cells and up to 16 contemporaneous spe-

cies, there can be more than 13 000 evolutionary events

per step at the local population level late in each run.

The computational model runs discussed here

(Fig. 4C–F) are a subset of experiments in which key

parameters (strength of selection, phenotypic variance,

extirpation probability and ancestral starting location)

were systematically varied (see Polly et al. 2016 for a full

description). The runs were selected because they illus-

trate the range of effects that spatial processes can have

on trait evolution.

This computational model is fully described in Polly

et al. (2016) and its Mathematica code and outputs are

archived with that paper. The runs were performed on

the Karst high-throughput computing cluster at Indiana

University.

Results and discussion

The outcomes of this computational model, which used

simple adaptive peak processes at the level of local

populations, produces a wide variety of patterns in trait

evolution at the species level. The outcomes of four

runs are shown in Figure 4C–F, chosen because they

illustrate the range of patterns that can arise from the

spatial interactions of selection, gene flow, extirpation

and speciation. Each run was part of a larger

computational experiment in which most parameters

10 PALAEONTOLOGY

were held constant at a mid-level value and one was

systematically varied from low to high values (Polly

et al. 2016).

Weak selection. The run in Figure 4C was from an experi-

ment in which the intensity of selection (i.e. the width of

the adaptive peak) was varied. This run had a weak selec-

tion (i.e. an adaptive peak that was 40 times wider than

the phenotypic variance) which lessened the effects of

selection relative to gene flow and drift and lessened the

probability of extirpation. The lineages appeared to

undergo a form of parallelism in which they all were stea-

dily selected toward higher trait values. Speciation consis-

tently produced small punctuation toward lower trait

values in this run. This pattern emerged because each

species was able to expand its range across the entire con-

tinent (i.e. into all available environments) due to the

weakening of the extirpation effect (see Polly et al. 2016

for spatially explicit maps for all the runs discussed here).

Furthermore, gene flow dominated local selection making

the trait values fairly homogeneous over all local popula-

tions. Each species was selected toward a trait value of

1.08, which is the net mean of all the local selective

optima on the continent (i.e. the average of the values

shown in Fig. 4B). Punctuation events occurred because

the populations on the periphery of each species range

received less input from gene flow than those at the cen-

tre, therefore the peripheral founders of each new species

were different from the mean of their parent species. Spe-

ciation always produced a consistently lower trait value

because each species was spread across the entire conti-

nent so that the peripheral populations was consistently

located in local environments that selected for low trait

values (Fig. 3B).

0

100

200

300

400

0 1 2 3 0 1 2 3

0 1 2 3 0 1 2 3

0

100

200

300

400

Trait Value Trait Value

Tim

e (model step)

Tim

e (model step)

Longitude

Latit

ude

Elevation(m)

4000

0

–30–20

–10

020

40

60

Trait optim

um0

1

2

3

A B

C D

E F

speciation 2

speciation 1

speciation 3

speciation 4

F IG . 4 . Computational model of

evolution with speciation on spa-

tially complex selection gradients.

A, models were run on a virtual

island continent with latitudinal and

elevational temperature gradients

and a rain shadow effect. B, the

environmental variation creates a

spatial pattern of selection in which

the optimum value of a trait varies

from place to place. C, results of a

model run with weak selection; all

model runs involve four speciation

events that produce 16 tip taxa by

the end of the run; the mean trait

value for each species is shown for

all branches in grey with one com-

plete tip-to-ancestor path high-

lighted. D, results of a model run

with strong probability of extirpa-

tion. E, results of a model run with

high trait variance. F, results of a

model run in which the ancestor

population inhabited an extreme

environment.

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 11

To show how this pattern might be interpreted using

standard evolutionary model fitting, Hunt’s (2006)

method was applied to a path through the tree from tip

to ancestor (highlighted in Fig. 4C). This path is analo-

gous to the kind of real palaeontological data that might

be assembled for a stratigraphic sequence of fossils when

the record is sparse enough for speciation events to go

unnoticed. The data shown here best support a Brownian

motion model (URW) with reasonable alternative support

for a directional model (GRW) (Table 2). Even though

the highlighted path has a strongly directional trend, the

punctuation event at step 200 is inconsistent enough with

the GRW model to tip the support toward Brownian

motion. The punctuation events can be ignored by sepa-

rately analysing its 30 individual branches (16 tip

branches plus 14 internal ones). These paths represent the

ideal palaeontological data in which only a single species

lineage is included. All 30 branches support a directional

model (GRW; Table 2). The data generated by this run

thus appear to support Brownian motion or directional

models of evolution, even though they are produced by

what are fundamentally OU processes. In real life such

processes could play out over any number of temporal

and spatial scales, some of which would be resolvable in

the fossil record and some of which would not.

Strong extirpation. The run in Figure 4D shows what

happens when the probability of local extirpation is high.

All of the branches were selected toward a low trait value

because the species were unable to expand out of their

ancestral environments (which happened to favour low

trait values in this run). This outcome was produced by

the same factors as in the weak selection run, except that

the species was exposed to a narrower range of local con-

ditions meaning that less change was needed to reach the

net optimum. Selection intensity was also stronger in this

run which helped each species to converge on the net

optimum faster. The highlighted lineage (Fig. 4D) sup-

ported a Brownian motion model (URW) because the

punctuation events were strongly inconsistent with direc-

tional or stasis models and branch-by-branch analysis

favoured directional evolutionary models (GRW) for 29

out of 30 branches (Table 2).

High phenotypic variance. The run in Figure 4E had high

phenotypic variance P in the local populations, which

produces a high rate of genetic drift and a strong

response to selection equations (as explained above).

With all other parameters set to mid-level values, the

result was that all species were able to expand their ranges

to the entire continent (as happened in the weak selection

run) and had strongly differentiated local populations (in

contrast to the weak selection run) because of the com-

bined effects of high rates of drift and strong selection

toward the local optimum. The geographical differentia-

tion made the punctuation events larger than in the pre-

vious two runs because the founder populations were

more differentiated. The run converged on the same net

optimum trait value as in the weak selection run (1.08),

but here each lineage was selected back to the net opti-

mum more quickly in about 100 model steps. The combi-

nation of large punctuations at speciation and quick

return to the net optimum produced a much different

pattern of evolution than in the weak selection run. The

highlighted lineage in this run also supported a Brownian

motion model of evolution (URW), largely because of the

punctuation events were inconsistent with the two alter-

native models, and branch-by-branch analysis supported

directional selection (GRW) in 27 out of 28 species

(Table 2).

Note that parameters used in this run have connec-

tions, albeit opposite ones, to Sheldon’s (1996) plus c�achange hypothesis of evolution. Sheldon argued that

wildly fluctuating environments produce generalists whose

traits do not track the changes but instead hover about a

mean optimum. This implies that these generalists

respond slowly to selective changes and that they are able

to flourish in a wide variety of environments. In terms of

the parameters used in this computational model, one

way of getting a slow response to selection is by having a

small phenotypic variance (or low heritability) and one

way of surviving in a wide range of environments is to

have a low probability of extirpation. See Polly et al.

(2016) for a companion simulation that was parameter-

ized with low phenotypic variance and is thus similar to

the expectation of the plus c�a change hypothesis.

Extreme ancestral environment. The most unique out-

come of the four runs is one in which all parameters

were set to mid-level values and the ancestral population

was placed in an ‘extreme’ environment where the opti-

mal trait value was much higher than the starting point

of other runs (Fig. 4F). ‘Extreme’ here means ‘uncom-

mon’ because the cells favouring high trait values are far

less numerous than those favouring low values (Fig. 4B).

As the run unfolded, the species repeatedly colonized

regions that favoured lower trait values so the overall

pattern of trait evolution appeared to be directional.

However, the pattern of punctuation events and within-

branch anagenetic changes were much more idiosyn-

cratic than in the other runs. In this run there was a

nearly equal balance between the strength of local selec-

tion, the rate of drift, the rate of gene flow, and the

probability of extirpation which made chance events

more important for determining the final range size of

each species, and thus the number of local environments

they occupied. Some species became specialists for a

narrow range of environments and some because

12 PALAEONTOLOGY

generalists over a wide range, resulting in a wider range

of traits at the tips of the tree by step 400 than in the

other runs. The trends in this run were strong enough

that the highlighted lineage (Fig. 4F) favoured a direc-

tional evolutionary model (GRW) despite the punctua-

tion event (Table 2). The branch-by-branch analysis

supported directional evolution for 28 out of 29 species

but note that the direction in this run was variable in

contrast to the uniformly positive directions of the other

runs. The individual branches in this run also had a

stronger random component due to greater influence of

random processes (drift and extirpation) compared to

the smoothly deterministic branches seen in the other

runs.

General principles. The departures of these computa-

tional model runs from the patterns of trait evolution

expected from standard evolutionary models can be

seen by comparing Figures 2B and 4C–F. Technically

speaking, all of the runs in Figure 4C–F are governed

by OU processes with a small component of Brownian

motion drift. The departure from the standard evolu-

tionary models is due to the spatial selection gradient

and the differences between the runs is due to the bal-

ance between the strength of selection, the rate of gene

flow, and the probability of extirpation. Generally

speaking, a spatial gradient of selection will produce

geographical variation within a species. The species-level

trait mean is therefore an emergent property of the

trait means in the local populations, unlike a

‘Fisherian’ species in which the mean is a property of a

single pool of variation. In a ‘Wrightian’ context, trait

evolution at the species-level therefore depends on

whether a species has a range that extends into more

than one selective environment (weaker selection and

less probable extinction make this more likely), whether

it is geographically differentiated (stronger local selec-

tion and slower gene flow make this more likely) and

how rapidly it colonizes new selective environments

(higher probability of extirpation and steeper environ-

mental gradients make this a slower process). Species

whose ranges encompass many environments and have

high rates of gene flow tend to evolve toward a mean

trait value that converges on the average of all the local

optima in its range (but note that the fact that it con-

verges specifically on the arithmetic mean is dependent

on how gene flow was implemented in this computa-

tional model). If such species reach the net optimum

slowly, as would happen if selection is weak or gene

flow is slow, then trait evolution will take on a direc-

tional trend from its ancestral state towards the net

value (Fig. 4C). However, if selection is strong and

gene flow and dispersal happen quickly, then the spe-

cies will quickly reach the net optimum, closer to the

expectation of a standard OU model (Fig. 4E). Only

when the drift component is more substantial than

selection or when selection, gene flow and extirpation

are balanced enough that chance events determine the

outcome will a ‘Wrightian’ system on a selection gradi-

ent be truly consistent with Brownian motion (but see

TABLE 2 . Results of evolutionary model fitting for the computational model runs shown in Figure 4C–F.

Weak selection Strong extirpation High variance Ancestral environment

Best model URW URW URW GRW

N 404 404 404 404

lstep 0.001 0.000 0.002 �0.003

r2step 0.001 0.0001 0.002 0.001

h 0.605 0.293 0.9 1.41

x 0.019 0.015 0.082 0.262

AICc URW �1848 �2169.4 �1280.1 �1596.1

AICc GRW �1846.6 �2167.5 �1278.5 �1597.8

AICc stasis �447.78 �549.28 137.36 607.81

wt URW 0.67 0.73 0.69 0.29

wt GRW 0.33 0.28 0.31 0.71

wt stasis 0.00 0.00 0.00 0.00

N branches URW 0 1 1 1

N branches GRW 30 29 27 28

N branches stasis 0 0 0 0

For each of the four tip-to-ancestor lineages highlighted in Figure 4, the best estimates of the rate (r2step), direction (lstep), and selec-

tion parameters (h, x) are reported along with the AICc values and associated Akaike weights that determine which evolutionary

model is best supported by the data. The last three lines of each table report the number of branches (out of max 30 except where lin-

eages became extinct) that support each evolution model of evolution when the test is repeated branch by branch. Abbreviations:

URW, unbiased random walk; GRW, generalized random walk (directional selection); stasis, stabilizing selection (OU process).

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 13

the genetic drift section above for the behaviour of a

‘Wrightian’ system under Brownian motion).

The magnitude of punctuated events in this system is

controlled by how much geographical variation is present

in the parent species. In homogeneous species (ones with

high gene flow or weak local selection) peripheral popula-

tions are like the parent, which produces little trait change

at speciation (Fig. 4C, D). But in geographically differenti-

ated species (strong local selection, rapid response to selec-

tion, or weak gene flow) peripheral populations are likely

to be uniquely different and larger punctuation events are

likely to occur (Fig. 4E). If punctuated speciation events

are inadvertently included in an evolutionary model fitting

analysis, which might happen if cladogenetic events are

not recognized in a stratigraphic time series, the excur-

sions they produce may result in poor fits with directional

or stabilizing models and thus cause the data to support a

Brownian motion process (Table 2).

INTERPRETING REAL DATA

The potential implications of spatial processes for evolu-

tionary model fitting in palaeontology can be illustrated

by examining alternate interpretations of two real-world

data sequences. Figure 5 shows the change in area of the

lower first molar as the natural log of length 9 width for

two lineages of mammalian carnivoramorphans, Viver-

ravus laytoni–V. acutus and Didymictis proteus–D. pro-tenus (Polly 2018b, S5).

These data are from the late Paleocene and early

Eocene sequences of the Bighorn and Clarks Fork basins

in Wyoming and pass through the Paleocene–Eocenethermal maximum (PETM), which was a rise in global

temperature of 5–8°C that lasted less than 20 kyr (Zachos

et al. 2003, 2006). Profound biogeographical changes were

associated with the PETM, including immigration of

many new taxa into Europe and North America from

some unknown, probably Asian source, as well as specia-

tion events and evolutionary changes in body size and

other traits (McInerney & Wing 2011; Hooker & Collin-

son 2012). Locally in the Bighorn and Clarks Fork basins

there was a pronounced change in vegetation from

broad-leaved tropical forests before and after the PETM

to a more open vegetation characteristic of dry, tropical

savannahs or shrublands (McInerney & Wing 2011).

Many mammalian taxa in the Bighorn Basin sequence

had much smaller body size during the Wa0 subage of

the Wasatchian North American Land Mammal Age that

corresponds with the PETM (Gingerich 2006). The

response to the PETM involved a complex mixture of

geographical range changes, evolutionary change, and

community reorganization (Clyde & Gingerich 1998;

Rankin et al. 2015).

Both of these carnivoran lineages have unknown geo-

graphical range sizes and putatively contain a cryptic spe-

ciation event. Paleocene and Eocene Viverravus and

Didymictis are primarily known from the Bighorn and

nearby basins in western North America, but isolated

occurrences are found elsewhere, including the late Pale-

ocene of Mississippi (Beard & Dawson 2009) and post-

PETM western Europe (Hooker 2010; Sol�e et al. 2016).

Whether these extralimital occurrences belong to different

species or are indicative of widespread but largely undoc-

umented ranges of the Bighorn Basin species is poorly

understood. Because good sampling of these taxa is geo-

graphically limited, we are forced to treat them in a ‘Fish-

erian’ way by presuming that the traits measured in

Wyoming are characteristic of the average for the entire

species. Polly (1997) interpreted each of these lineages as

Cf1

Cf2

Cf3

Wa0

Wa1

Wa2

Wa3

Wa4

Wa5

Wa6

Cla

rkfo

rkia

nW

asat

chia

n

Pal

eoce

neE

ocen

e

PETMCladogenic event? Cladogenic event?

stratigraphic level (m)

lower first molar area (log mm2)

Viverravus Didymictis

3.6 3.8 4.0 4.2 4.4 4.6 4.8

1400

1600

1800

2000

2200

2400

2.0 2.2 2.4 2.6 2.8 3.0

F IG . 5 . Change in the natural log

of molar area (length 9 width) in

two lineages of Palaeogene carni-

vores (Mammalia, Carnivora) from

the Bighorn Basin, Wyoming. A,

Viverravus laytoni–V. acutus lineage.B, Didymictis proteus–D. protenuslineage. Grey points are individual

specimens and black points joined

by broken line are the species mean

for each stratigraphic level.

14 PALAEONTOLOGY

representing two species (ancestor and descendant) sepa-

rated by a cladogenetic event early in the sequence that

gave rise to a second descendant species that is not shown

here. However, previous and subsequent authors have

interpreted these taxa as being divided into either more

or fewer species, with and without cladogenic events (cf.

Gingerich & Winkler 1985; Secord et al. 2006; Secord

2008). The uncertainty whether data represent a single

evolving lineage or a more complex set of closely related

taxa is typical of most stratigraphic sequences of palaeon-

tological data (Benton & Pearson 2001). These data there-

fore make interesting examples for exploring ‘Fisherian’

and ‘Wrightian’ interpretations because either kind of

process would be credible.

The molars of both lineages show a general trend

toward becoming larger with seemingly erratic excursions

around the time of the PETM, which is also about the

time of the putative speciation events (Fig. 5). The data

best support a Brownian motion model of evolution for

Viverravus and an OU model for Didymictis (Table 3)

but support for an alternative directional model is high

in Viverravus (0.34) and support for an alternative Brow-

nian motion model is low but credible for Didymictis

(0.10). A purely ‘Fisherian’ face-value interpretation of

these data might therefore be that, with Brownian motion

in one taxon and stasis in the other, the PETM had little

effect on the evolution of either lineage. But several fea-

tures of these data sequences are similar to ones illus-

trated above. Both lineages seem to have a pronounced

excursion in trait values at what might have been specia-

tion events, which are followed by more gradual direc-

tional trends toward larger molar size. Given both the

environmental perturbations and the range changes that

have been documented at the PETM, these data are, in

principle, consistent with punctuation events associated

with speciation in two geographically widespread and dif-

ferentiated species, followed by a directional trend driven

by range expansion, either under a pure Brownian

motion process like the ‘surfing’ effects seen in the com-

putational model of drift (Fig. 3) or under an adaptive

peak selection process accompanied by gene flow as seen

in the computational model of environmental gradients

(Fig. 4).

CONCLUSIONS

Spatial dynamics can produce unexpected departures

from standard evolutionary statistical models, even when

the evolutionary processes at the level of local populations

are fully consistent with the same formulations of drift

and adaptive peak selection that form the basis of the

TABLE 3 . Results of evolutionary model fitting two mam-

malian data sets from the Palaeogene of the Bighorn Basin,

Wyoming.

Viverravus Didymictis

Best model URW Stasis

N 8 8

lstep 0.001 0.001

r2step 0.0001 0.0023

h 2.241 4.212

x 0.071 0.087

AICc URW �1.036 14.254

AICc GRW 0.246 18.264

AICc stasis 8.357 9.772

wt URW 0.65 0.10

wt GRW 0.34 0.01

wt stasis 0.01 0.89

Abbreviations: URW, unbiased random walk; GRW, generalized

random walk (directional selection); stasis, stabilizing selection

(OU process).

TABLE 4 . Summary of species level patterns that emerge from spatial processes acting on local populations.

Species level pattern Spatial processes

Evolutionary rate changes • geographic range changes affect drift rate by changing total N of the metapopulation

• geographic differentiation in traits can cause short term punctuation events under

some models of speciation

• species level evolutionary rate is an emergent property of local level processesBrownian motion patterns • punctuation events can tip model fitting support toward Brownian motion even

under directional or stabilizing selection

• complex spatial OU processes can tip support toward Brownian motion if they

introduce excursions from the species’ net optimumDirectional trends • range expansion can cause directional trends under a drift model through ‘surfing’.

• range expansion plus gene flow can cause directional trends under a selection-based

model by exposing more local populations to OU selection in new environmentsStabilizing selection or stasis • sharp drops in rate due to increased N from an expanded geographic range can cause

appearance of stasis

POLLY : SPAT IAL PROCESSES & EVOLUTIONARY MODELS 15

statistical models. Discrepancies arise because the expecta-

tions of virtually all evolutionary models make the ‘Fishe-

rian’ assumption that drift and selection are acting

uniformly on a species-wide trait distribution. But

‘Wrightian’ metapopulation processes involving local

populations, drift, selection gradients, gene flow and

range changes can produce trait patterns at the species

level that are unexpected from the perspective of most

evolutionary statistical models. The examples discussed in

this paper are summarized in Table 4. Ignoring spatial

processes may lead to misleading or incorrect interpreta-

tions when palaeontological data are assessed using stan-

dard evolutionary model fitting.

A ‘Wrightian’ system of trait evolution on a spatial

selectional gradient is likely to have affected many clades

that are preserved in the palaeontological record. Most of

the objects preserved in the fossil record (teeth, shells,

bones, leaves) play some sort of functional role in the life

of the organism and are thus likely to be subject to selec-

tion. Whether the selection is weak or strong, directional

or stabilizing, balanced by gene flow, or distributed uni-

formly across the range of a species is context dependent,

but that it occurred can probably assumed for the major-

ity of palaeontological traits. It is also likely that most

species in the fossil record had geographical ranges that

encompassed at least some variety of environments, if for

no other reasons than commonness in the fossil record is

associated with geographical range and abundance (Jern-

vall & Fortelius 2004), because geographical range is posi-

tively correlated with temporal duration (Jablonski 1987),

and because geographical variation in traits is common in

extant species with large ranges (Fig. 1). The ‘Wrightian’

processes discussed in this paper are therefore likely to

influence species-level trait evolution as preserved in the

fossil record and may cause departures from standard

models that require some independent means of detec-

tion.

Pennell et al. (2015) demonstrated that standard statis-

tical evolutionary models are often inadequate to describe

real data. They assessed the adequacy of Brownian

motion, single-peak OU, and early burst models to

explain the evolution of more than 300 functional traits

in angiosperms. Their analysis was based on phylogenies

and traits in tip taxa rather than stratigraphic sequences,

but their first two models are essentially the same as the

ones considered here (the early burst model is not mean-

ingful for a stratigraphic sequence). Only 133 of 337 traits

were fully consistent with their best supported model,

meaning that nearly two-thirds of the data sets were not

adequately explained by any of these standard models.

One reason for the inadequacy of standard models may

well be that spatial dynamics produce evolutionary pat-

terns that do not fit ‘Fisherian’ expectations, even when

the processes of drift and selection at the local

population-level are effectively the same as standard

Brownian motion and OU models. Using an approach

that compared data to models in both absolute and rela-

tive terms, Voje et al. (2018) found that more than one-

third of the fossil data sets they examined were not ade-

quately explained by the statistical models they evaluated.

Unfortunately, few tools are available to help us distin-

guish between the ‘Fisherian’ and ‘Wrightian’ interpreta-

tions of comparative data. Already, a new generation of

evolutionary statistical models based on more complex

scenarios of trait evolution is emerging, although none of

them explicitly incorporate the spatial effects discussed in

this paper. Most of these are implemented only for phylo-

genetic trees rather than palaeontological sequences. They

include variable rate models (Mooers & Schluter 1999;

O’Meara et al. 2006; Eastman et al. 2011; Uyeda et al.

2011; Smaers et al. 2016; Landis & Schraiber 2017) and

multiple peak OU models (Hansen et al. 2008). Hunt and

colleagues have done the most to elaborate model fitting

techniques for stratigraphic sequences by allowing for

hypotheses involving changes in tempo and mode (Hunt

& Rabosky 2014; Hunt et al. 2015; see Table 1 for

example).

Even though spatially explicit statistical models for trait

evolution are not readily available, many palaeontologists

have studied spatial processes in the fossil record. The mech-

anisms of punctuated equilibrium have frequently been

analysed in a spatially explicit context, including speciation

by peripheral isolation (Eldredge & Gould 1972), mecha-

nisms for stasis (Lieberman & Dudgeon 1996) and related

mechanisms for turnover in clades and communities (Vrba

1993; Lieberman et al. 2007). Spatial gradients of selection

(Hoffmeister & Kowalewski 2001), phylogeographical turn-

over within species (Polly 2001), extinction (Foote et al.

2007; Jablonski 2008) and metacommunity spatial dynamics

(Lyons 2003; Liow & Stenseth 2007; Blois & Hadly 2009; Sti-

gall 2013) have been studied in the fossil record, as have

contributions of spatial heterogeneity to speciation, trait dis-

tributions and biodiversity (Badgley & Finarelli 2013; Lyons

& Smith 2013). The role of spatial processes in trait evolu-

tion is an active area that invites new methodological

advances in modelling of trait evolution.

Acknowledgements. Thank you to Gene Hunt for verifying my

Mathematica code for evolutionary model fitting against his

original R code and for offering insightful comments about

model fitting. Thanks also to Paul Smith and Mark Sutton for

organizing the 2017 Palaeontological Association Symposium

where this work was originally presented and to Sally Thomas

and Andrew Smith for their patience and editorial help with this

manuscript. I have benefitted from discussions about spatial

processes with many insightful people over the years, including

Catherine Badgley, Tony Barnosky, Jessica Blois, Andrea Cardini,

Pavel Borodin, Jussi Eronen, Mikael Fortelius, Marianne Fred,

Islam G€und€uz, Jeremy Hooker, David Jablonski, Christine Janis,

16 PALAEONTOLOGY

Jacques Hausser, Michelle Lawing, Adrian Lister, Richard

Nichols, Svetlana Pavlova, Andrei Polyakov, Danielle Schreve,

Jeremy Searle, Nils Stenseth, John Stewart, Chris Stringer, Elwira

Szuma, Marvalee Wake, and Jan W�ojcik, as well as Graham Sla-

ter and two anonymous reviewers (none of whom are to be

blamed for any shortcomings in this paper). This work was sup-

ported by US National Foundation Grant EAR-1338298. The use

of the Karst high-throughput computing cluster was by the Indi-

ana University Pervasive Technology Institute and the Indiana

METACyt Initiative, both of which received partial support from

the Lilly Endowment, Inc.

DATA ARCHIVING STATEMENT

Data for this study are available in the Dryad Digital Repository:

https://doi.org/10.5061/dryad.99pj691

Editor. Andrew Smith

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