Mechanistic models for macroecolgy: moving beyond correlation

Mechanistic models for macroecolgy: moving beyond correlationNicholas J. GotelliDepartment of BiologyUniversity of VermontBurlington, VT 05405

??What causes geographic variation in species richness??

Understanding species richness patternsData sourcesA critique of current methodsRange cohesion and the mid-domain effectMechanistic models for species richnessModel selectionSummary

Nicholas Gotelli, University of VermontGary Entsminger Acquired IntelligenceRob Colwell University of ConnecticutGary Graves SmithsonianCarsten Rahbek University of Copenhagen Thiago RangelFederal University of Gois

Data sourcesGridded map of domain

Avifauna of South AmericaThere 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 Avifauna2891 breeding species

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

Minimum:18 species

Minimum:18 speciesMaximum: 846 species

Data sourcesGridded map of domainSpecies occurrence records within grid cells

Geographic Ranges For Individual SpeciesMyiodoorus cardonaiPhalacrocorax brasilianusAnas puna

Geographic Ranges Species Richness

Data sourcesGridded map of domainSpecies occurrence records within grid cellsQuantitative measures of potential predictor variables within grid cells (NPP, temperature, habitat diversity)

Climate, Habitat Variables Measured at Grid Cell Scale

How are these macroecological data typically analyzed?

How are these macroecological data typically analyzed?

Curve-fitting!

Criticisms of Curve-FittingCorrelation does not equal causation

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors LOESS, Poisson, Spatial Regression (SAM)Choosing among correlated predictor variables

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors LOESS, Poisson, Spatial Regression (SAM)Choosing among correlated predictor variables Model selection strategies, stepwise regression, AIC

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors LOESS, Poisson, Spatial Regression (SAM)Choosing among correlated predictor variables Model selection strategies, stepwise regression, AICSensitivity to spatial scale, taxonomic resolution, geographic range size

Criticisms of Curve-FittingCorrelation does not equal causation Common to all of macroecology!Non-linearity & non-normal, spatially correlated errors LOESS, Poisson, Spatial Regression (SAM)Choosing among correlated predictor variables Model selection strategies, stepwise regression, AICSensitivity to spatial scale, taxonomic resolution, geographic range size Stratify analysis

Conceptual Weakness of Curve-Fitting ParadigmPotential Predictor Variables (tonnes/ha, C)

Conceptual Weakness of Curve-Fitting ParadigmPotential Predictor Variables (tonnes/ha, C)minimize residuals

Conceptual Weakness of Curve-Fitting ParadigmPotential Predictor Variables (tonnes/ha, C)?? MECHANISM ??minimize residuals

Alternative Strategy:Mechanistic Simulation ModelsExplicit Simulation ModelPotential Predictor Variables (tonnes/ha, C)

Alternative Strategy:Mechanistic Simulation ModelsExplicit Simulation ModelPotential Predictor Variables (tonnes/ha, C)mechanism

How can we build explicit simulation models for macroecology?

One-dimensional geographic domain

One-dimensional geographic domainSpecies geographic ranges randomly placed line segments within domain

One-dimensional geographic domainSpecies geographic ranges randomly placed line segments within domainPeak of species richness in geographic center of domain

One-dimensional geographic domainSpecies geographic ranges randomly placed line segments within domainPeak of species richness in geographic center of domainSpeciesNumber

domain

domaingeographic range

der PfankuchenGuildPancakus spp.

Reduced species richness at margins of the domain

Mid-domainpeak of species richnessin the center of the domain

2-dimensional MDE ModelRandom point of origination within continent (speciation)Random spread of geographic range into contiguous unoccupied cellsSpreading dye model (Jetz & Rahbek 2001) predicts peak richness in center of continent (r2 = 0.17)

Assumptions of MDE modelsPlacement of ranges within domain is random with respect to environmental gradientsControversial, but logical for a null model for climatic effects

Assumptions of MDE modelsPlacement of ranges within domain is random with respect to environmental gradientsControversial, but logical for a null model for climatic effectsGeographic ranges are cohesive within the domainRarely discussed, but important as the basis for a mechanistic model of species richness

Range CohesionRange Scatter

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

Causes of Range CohesionExtrinsic Causes

Causes of Range CohesionExtrinsic CausesCoarse Spatial ScaleSpatial Autocorrelation in Environments

Causes of Range CohesionExtrinsic CausesCoarse Spatial ScaleSpatial Autocorrelation in EnvironmentsIntrinsic Causes

Causes of Range CohesionExtrinsic CausesCoarse Spatial ScaleSpatial Autocorrelation in EnvironmentsIntrinsic CausesLimited DispersalPhilopatry & Site FidelityMetapopulation & Source/Sink StructureFine-scale Genetic Structure & Local AdaptationSpatially Mediated Species Interactions

Strict Range CohesionStepping 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 EnvironmentHeterogeneous EnvironmentAlmost all MDE models have assumed a homogeneous environment: grid cells are equiprobable

EnforcedRelaxedHomogeneousHeterogeneousRANGE COHESIONENVIRONMENT

EnforcedRelaxedHomogeneousHeterogeneousRANGE COHESIONENVIRONMENTClassic MDEStatistical Null (slope = 0)

EnforcedRelaxedHomogeneousHeterogeneousRANGE COHESIONENVIRONMENTClassic MDEStatistical Null (slope = 0)Range Scatter ModelsRange Cohesion Models

EnforcedRelaxedHomogeneousHeterogeneousRANGE COHESIONENVIRONMENTClassic MDEStatistical Null (slope = 0)Range Scatter ModelsRange Cohesion ModelsRange 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.

Alternative Strategy:Mechanistic Simulation ModelsExplicit Simulation ModelPotential Predictor Variables (tonnes/ha, C)mechanism

Modeling StrategyEstablish simple algorithms that describe P(occupancy) based on environmental variables

Modeling StrategyEstablish simple algorithms that describe P(occupancy) based on environmental variablesSimulate origin and placement of each species geographic range in heterogeneous landscape (with or without range cohesion)

Modeling StrategyEstablish simple algorithms that describe P(occupancy) based on environmental variablesSimulate 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.)

What determines P(cell occurrence)?Simple environmental modelsP(occurrence) Measured Environmental Variable (NPP, Temperature, etc.)Formal analytical models

What determines P(cell occurrence)?Simple environmental modelsP(occurrence) Measured Environmental Variable (NPP, Temperature, etc.)Formal analytical modelsSpecies-Energy Model (Currie et al. 2004)Temperature Kinetics (Brown et al. 2004)

What determines P(cell occurrence)?Simple environmental modelsP(occurrence) Measured Environmental Variable (NPP, Temperature, etc.)Formal analytical modelsSpecies-Energy Model (Currie et al. 2004)P(occurrence) (NPP)(Grid-cell Area)Temperature Kinetics (Brown et al. 2004)P(occurrence) e-E/kT

Model-Selection in Curve-Fitting AnalysesSimple tests against the null hypothesis that b=0No consideration of what expected slope should be with a specific mechanismLeast-square and AIC criteria to try and select a subset of variables that best account for variation in S

H0: b = 0

Model Selection with Mechanistic Simulation ModelsModels make quantitative predictions of expected species richnessTest slope of observed richness versus predicted richnessHypothesis of an acceptable fit H1: b = 1.0Rank acceptable models according to slope, intercept, and r2AIC criteria not appropriate

Predicted SObserved STheoretical b = 1.0Observed b

SummaryCurve-fitting framework does not incorporate explicit mechanisms

SummaryCurve-fitting framework does not incorporate explicit mechanismsUse mechanistic simulations to define the placement of geographic ranges in a gridded domain

SummaryCurve-fitting framework does not incorporate explicit mechanismsUse mechanistic simulations to define the placement of geographic ranges in a gridded domainSpecify rules for P(occurrence)= f(environmental variables)

SummaryCurve-fitting framework does not incorporate explicit mechanismsUse mechanistic simulations to define the placement of geographic ranges in a gridded domainSpecify rules for P(occurrence)= f(environmental variables)Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & RejoindersEach 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.

Criticisms & RejoindersEach 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)?

Criticisms & RejoindersEach 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)?

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

Criticisms & Rejoinders1:1 scaling of environmental variables with P(occurrence) is unrealistic and arbitrary.

Criticisms & Rejoinders1:1 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.

Criticisms & Rejoinders1:1 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))

Criticisms & Rejoinders1:1 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))

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

Criticisms & RejoindersMany environmental variables, but especially NPP, show non-linear relationships with peaks in richness at intermediate levels. This is not captured by linear models.

Criticisms & RejoindersMany 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

Criticisms & RejoindersUsing slopes comparisons will not successfully distinguish between models with intercorrelated predictor variables.

Criticisms & RejoindersUsing 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.

Criticisms & RejoindersThe model is not truly mechanistic because it does not model the sizes of the geographic ranges, only their placement.

Criticisms & RejoindersThe 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.

Criticisms & RejoindersThe range cohesion and range scatter models dont 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?

Criticisms & RejoindersThe range cohesion and range scatter models dont 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

To Be ContinuedCarsten Rahbek. Perception of Species Richness Patterns: The Role of Range Sizes

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Mechanistic models for macroecolgy: moving beyond correlation