Mechanistic models for macroecolgy: moving beyond correlation Nicholas J. Gotelli Department of Biology University of Vermont Burlington, VT 05405

Mechanistic models for macroecolgy:

moving beyond correlationNicholas J. Gotelli

Department of BiologyUniversity of VermontBurlington, VT 05405

??What causes geographic variation

in species richness??

Understanding species richness patterns

• Data sources

• A critique of current methods

• Range cohesion and the mid-domain effect

• Mechanistic models for species richness

• Model selection

• Summary

Nicholas Gotelli, University of Vermont

Gary EntsmingerAcquired Intelligence

Rob ColwellUniversity of Connecticut

Gary GravesSmithsonian

Carsten RahbekUniversity of Copenhagen

Thiago Rangel

Federal University of Goiás


• Data sources




• Model selection

• Summary

Data sources

• Gridded map of domain

Avifauna of South America

“There can be no question, I think, that South America is the most peculiar of all the primary regions of the globe as to its ornithology.” P.L. Sclater (1858)

South American Avifauna

• 2891 breeding species

• 2248 species endemic to South America and associated land-bridge islands

Minimum:

18 species

Minimum:

18 species

Maximum:

846 species

Data sources


• Species occurrence records within grid cells

Geographic Ranges For Individual

Species

Myiodoorus cardonai

Phalacrocorax brasilianus

Anas puna

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources


• Species occurrence records within grid cells

• Quantitative measures of potential predictor variables within grid cells (NPP, temperature, habitat diversity)

Climate, Habitat Variables Measured at Grid Cell Scale


• Data sources




• Model selection

• Summary

How are these macroecological data typically analyzed?

Net Primary Productivity (Tonnes/hectare)

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LOESS

Poisson

How are these macroecological data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting

• “Correlation does not equal causation”


• “Correlation does not equal causation”Common to all of macroecology!



• Non-linearity & non-normal, spatially correlated errors



• Non-linearity & non-normal, spatially correlated errorsLOESS, Poisson, Spatial Regression (SAM)




• Choosing among correlated predictor variables




• Choosing among correlated predictor variablesModel selection strategies, stepwise regression, AIC





• Sensitivity to spatial scale, taxonomic resolution, geographic range size





• Sensitivity to spatial scale, taxonomic resolution, geographic range sizeStratify analysis

Conceptual Weakness of Curve-Fitting Paradigm

Predicted Species Richness

(S / grid cell)

Potential Predictor Variables

(tonnes/ha, C°)

Observed Species Richness

(S / grid cell)



(S / grid cell)


(tonnes/ha, C°)


(S / grid cell)

minimizeresiduals



(S / grid cell)


(tonnes/ha, C°)


(S / grid cell)

??MECHANISM

??

minimizeresiduals

ExplicitSimulation

Model

Alternative Strategy:Mechanistic Simulation Models


(S / grid cell)


(tonnes/ha, C°)


(S / grid cell)

ExplicitSimulation

Model



(S / grid cell)


(tonnes/ha, C°)


(S / grid cell)

mechanism

How can we build explicit simulation models for

macroecology?


• Data sources




• Model selection

• Summary

One-dimensional geographic domain


Species geographic ranges randomly placed line segments within domain



Peak of species richness in geographic center of domain



Peak of species richness in geographic center of domain

Species

Number

domain

domain

geographic range

der PfankuchenGuild

Pancakus spp.

Reduced species richnessat margins of the domain

Mid-domainpeak of species richnessin the center of the domain

2-dimensional MDE Model

• Random point of originationwithin continent (speciation)

• Random spread of geographicrange into contiguousunoccupied cells

• Spreading dye model (Jetz & Rahbek 2001) predicts peak richness incenter of continent (r2 = 0.17)

Assumptions of MDE models

• Placement of ranges within domain is random with respect to environmental gradients– Controversial, but logical for a null model for

climatic effects

Assumptions of MDE models

• Placement of ranges within domain is random with respect to environmental gradients– Controversial, but logical for a null model for

climatic effects

• Geographic ranges are cohesive within the domain– Rarely discussed, but important as the basis

for a mechanistic model of species richness

Range Cohesion Range Scatter

At the 1º x 1º scale, > 95% of species of South American birds have contiguous

geographic ranges

Causes of Range Cohesion

• Extrinsic Causes


• Extrinsic Causes– Coarse Spatial Scale– Spatial Autocorrelation in Environments



• Intrinsic Causes



• Intrinsic Causes– Limited Dispersal– Philopatry & Site Fidelity– Metapopulation & Source/Sink Structure– Fine-scale Genetic Structure & Local Adaptation– Spatially Mediated Species Interactions

Strict Range Cohesion Stepping Stone

* The mid-domain effect does not require strict range cohesion. A mid-domain peak in species richness will also arise from stepping stone models with limited dispersal and from neutral model dynamics (Rangel & Diniz-Filho 2005)

Homogenous Environment

HeterogeneousEnvironment

Almost all MDE models have assumed a homogeneous environment: grid cells are equiprobable

Enforced

Relaxed

Homogeneous Heterogeneous

RANGECOHESION

ENVIRONMENT

Enforced

Relaxed


RANGECOHESION

ENVIRONMENT

Classic MDE

Statistical Null(slope = 0)

Enforced

Relaxed


RANGECOHESION

ENVIRONMENT

Classic MDE


Enforced

Relaxed


RANGECOHESION

ENVIRONMENT

Classic MDE


Range ScatterModels

Range Cohesion Models

Enforced

Relaxed


RANGECOHESION

ENVIRONMENT

Classic MDE


Range ScatterModels

Range Cohesion Models

Range Cohesion Models are a hybrid that describes a stochastic MDE model in a more realistic heterogeneous environment.

Range Scatter Models also incorporate environmental heterogeneity, but do not place any constraints on species geographic ranges.

ExplicitSimulation

Model



(S / grid cell)


(tonnes/ha, C°)


(S / grid cell)

mechanism


• Data sources




• Model selection

• Summary

Modeling Strategy

• Establish simple algorithms that describe P(occupancy) based on environmental variables

Modeling Strategy


• Simulate origin and placement of each species geographic range in heterogeneous landscape (with or without range cohesion)

Modeling Strategy


• Simulate origin and placement of each species geographic range in heterogeneous landscape (with or without range cohesion)

• Repeat simulation to estimate predicted species richness per grid cell

Geographic Ranges

Species Richness

What determines P(cell occurrence)?

• Simple environmental modelsP(occurrence) Measured Environmental

Variable (NPP, Temperature, etc.)




• Formal analytical models




• Formal analytical models– Species-Energy Model (Currie et al. 2004)– Temperature Kinetics (Brown et al. 2004)




• Formal analytical models– Species-Energy Model (Currie et al. 2004)

P(occurrence) (NPP)(Grid-cell Area)– Temperature Kinetics (Brown et al. 2004)

P(occurrence) e-E/kT


• Data sources




• Model selection

• Summary

Model-Selection in Curve-Fitting Analyses

• Simple tests against the null hypothesis that b=0

• No consideration of what expected slope should be with a specific mechanism

• Least-square and AIC criteria to try and select a subset of variables that best account for variation in S


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H0: b = 0

Model Selection with Mechanistic Simulation Models

• Models make quantitative predictions of expected species richness

• Test slope of observed richness versus predicted richness

• Hypothesis of an acceptable fit H1: b = 1.0

• Rank acceptable models according to slope, intercept, and r2

• AIC criteria not appropriate

Predicted S

Observed S

Theoretical b = 1.0

Observed b


• Data sources




• Model selection

• Summary

Summary

• Curve-fitting framework does not incorporate explicit mechanisms

Summary


• Use mechanistic simulations to define the placement of geographic ranges in a gridded domain

Summary



• Specify rules for P(occurrence)= f(environmental variables)

Summary



• Specify rules for P(occurrence)= f(environmental variables)

• Test model fit against expected slope = 1.0

Criticisms & Rejoinders


• “Each species has a unique and distinctive response to different environmental variables. Species ranges should be modeled independently, not with a single function for all species.”



If this is true, why are there widespread repeatable patterns of species richness (e.g., latitude, elevation, area, productivity)?



If this is true, why are there widespread repeatable patterns of species richness (e.g., latitude, elevation, area, productivity)?

Often not enough data to model each species individually. We need a simple framework for analysing entire floras and faunas at a biogeographic scale.


• “1:1 scaling of environmental variables with P(occurrence) is unrealistic and arbitrary.”



Perhaps, but this is a parsimonious mechanistic model that relates environmental variables to geographic range placement.




Linearity in P(occurrence) is not unreasonable over the empirical ranges of environmental variables measured in South America. (Linearity of P(occurrence) ≠ Linearity of (Species Richness))


• “1:1 scaling of environmental variables with P(occurrence) is unrealistic and arbitrary”


Linearity in P(occurrence) is not unreasonable over the empirical ranges of environmental variables measured in South America. (Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to begin somewhere!


• “Many environmental variables, but especially NPP, show non-linear relationships with peaks in richness at intermediate levels. This is not captured by linear models.”


• “Many environmental variables, but especially NPP, show non-linear relationships with peaks in richness at intermediate levels. This is not captured by linear models.”

At least at this spatial scale, no evidence for a diversity hump of avian species richness when plotted with NPP or other variables


• “Using slopes comparisons will not successfully distinguish between models with intercorrelated predictor variables.”


• “Using slopes comparisons will not successfully distinguish between models with intercorrelated predictor variables.”

Not a problem for these analyses. From an initial set of ~ 100 candidate models (10 variables x 2 algorithms x 5 range size quartiles), we reduced the set down to only 4 or 5 possible contenders.


• “The model is not truly mechanistic because it does not model the sizes of the geographic ranges, only their placement.”



True! Our model takes range sizes as a given and then uses algorithms to place them in a heterogeneous domain. A more realistic model would describe the processes of speciation, dispersal, and extinction of an evolving fauna.




But how can the parameters of such a model (e.g. speciation and dispersal rates) ever be measured in the real world? Same problems have plagued most empirical evaluations of the neutral model.




But how can the parameters of such a model (e.g. speciation and dispersal rates) ever be measured in the real world? Same problems have plagued most empirical evaluations of the neutral model.

Our models are designed to analyze the data that macroecologists typically have: gridded maps of environmental variables and species geographic ranges.


• “The range cohesion and range scatter models don’t’ seem like they would give predictions that are any different from just a regression with the underlying variables themselves. What is the added value of these simulation models?”


• “The range cohesion and range scatter models don’t’ seem like they would give predictions that are any different from just a regression with the underlying variables themselves. What is the added value of these simulation models?”

The predictions are not the same. For species with large geographic ranges, the range cohesion models always fit the data better than the range scatter models, regardless of which environmental variable is considered.

Key Differences

Curve-Fitting Mechanistic Models

Unit of Study Species Richness Underlying geographic ranges

Predicted values Minimization of residuals (data dependent)

Algorithms for origin and spread of geographic ranges (data independent)

Model Selection Criteria Smallest number of variables that reduce residual sum of squares

Quantitative fit to model predictions

Statistical Tests H0: (b = 0) tests for any effect that is larger than 0

H0: (b = 1.0) tests for quantitative match between observed and predicted S

To Be Continued…

Carsten Rahbek. Perception of Species Richness Patterns:

The Role of Range Sizes

Documents

Mechanistic models for macroecolgy: moving beyond correlation Nicholas J. Gotelli Department of Biology University of Vermont Burlington, VT 05405