Introduction Analysis Conclusions

Inferring the spatial scale of environmentalvariation in sapling recruitment of Pinus elliotti

(slash pine)

Ben Bolker, Gabriel Herrick, Gordon Fox

McMaster University, University of South Florida

10 August 2011

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Outline

1 IntroductionQuestionsDefinitionsData

2 AnalysisFrameworkSimulation examplesData

3 Conclusions

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Questions

Templates

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Questions

Templates

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Questions

Templates

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Questions

Questions

How do templates modify spatial distributions in general?

Can we work out exact expressions for particular cases?

Can we recover information on templates from messy data?(How?)

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Definitions

Spatial correlation/covariance

cij(|x− y|) ∝ 〈(ni (x)− ni ) · (nj(y)− nj)〉

Positive correlation ⇐⇒ clustering/association

Zero correlation ⇐⇒ random (Poisson)

Negative correlation ⇐⇒ evenness/segregation

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Definitions

Spatial correlations as a function of distance

long−range even

short−rangeuncorrelated

0

1

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1

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Data

Seed data

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empty

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Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Data

Sapling data

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Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Data

Correlation functions

Distance (m)

Aut

ocor

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tion

0.2

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seedling

0 5 10 15 20 25

seed

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Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Framework

Analytical framework

start with a spatial distribution of seeds N(x) . . .

seeds become saplings (recruit) with probability E (x), soS(x) = N(x)E (x) (on average)

describe spatial pattern (autocovariance) of seeds (CNN);establishment process (environment, CEE ); saplings (CSS)

we’ll also need means (N, E ), variances (VarN, VarE )

assume spatial homogeneity/isotropy (!)

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Framework

Analytical framework (2)

assume cross-covariance CEN = 0no correlation between seeds and environment,

e.g. long-distance dispersal

CSS(r) = N2CEE (r) + E 2CNN(r)

(where N=mean seed density, , E=mean establishmentprobability)

or (switch to correlation c)

CSS ∝VarE

E 2cEE +

VarN

N2cNN

. . . a weighted mixture of the two correlation functions

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Framework

Solving for cEE

Therefore,cEE ∝ σ2

ScSS − E 2σ2NcNN

Can we really use this?

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Simulation examples

Gaussian simulations

Distance (m)

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ocor

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tion

0.0

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typeseedlingseedenv

wtrueest

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Simulation examples

Gaussian simulations (2)

Distance (m)

Aut

ocor

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tion

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seedling

0 5 10 15 20 25

seed

0 5 10 15 20 25

env

0 5 10 15 20 25

typeseedlingseedenv

wtrueest

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Simulation examples

Poisson/binomial simulations

Distance (m)

Aut

ocor

rela

tion

0.0

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seedling

0 5 10 15 20 25

seed

0 5 10 15 20 25

env

0 5 10 15 20 25

typeseedlingseedenv

wtrueest

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Simulation examples

Distance (m)

Aut

ocor

rela

tion

0.0

0.2

0.4

0.6

0.8

1.0

seedling

0 5 10 15 20 25

seed

0 5 10 15 20 25

env

0 5 10 15 20 25

typeseedlingseedenv

wtrueest

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Data

Data (!)

Distance (m)

Aut

ocor

rela

tion

−0.2

0.0

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0.6

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1.0

seedling

0 5 10 15 20 25

seed

0 5 10 15 20 25

env

0 5 10 15 20 25

typeseedlingseedenv

west

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Conclusions

we may be able to recover spatial information

still need to characterize / understand / correct for bias

tradeoff between robust pattern analysis and incorporating(sort of) mechanism ?

technically challenging (used gls in R: ask for details!)

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions

Introduction Analysis Conclusions

Acknowledgements

NSERC (Discovery grant, $$$)

SHARCnet (computational resources)

Ben Bolker, Gabriel Herrick, Gordon Fox McMaster University, University of South Florida

Pine distributions