Zif bolker_w2

Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References

Estimating environmental heterogeneity fromspatial dynamics data

Ben Bolker, McMaster UniversityDepartments of Mathematics & Statistics and Biology

ZiF

18 June 2012

Ben Bolker ZiF

Spatial estimation


Outline

1 Introduction

2 Endogenous vs exogenous: qualitativeExplanations for spatial patterns

3 Endogenous vs exogenous: quantitativeSpatial synchronyPines

4 Conclusions

Ben Bolker ZiF

Spatial estimation


What do ecologists want?

Explicit questions: explain observed patterns

Implicit questions: explain outcomes of ecological interactions:persistence, coexistence, trait evolution, etc..

Qualitative answers: presence/absence, persistence/extinction,coexistence/exclusion

Quantitative answers: how many? how quickly? scale(s) ofpattern?

Deductive (forward) models: model → outcome

Inductive (inverse) models: data → model

Ben Bolker ZiF

Spatial estimation


Typical examples

Can spatial pattern allow coexistence of similar species?

Is a particular example of coexistence spatially mediated?

Do endogenous or exogenous drivers produce spatialclustering?

What drives spatial clustering in a particular case?

The answer to does process X operate in ecological system Y ismost often Yes.

Ben Bolker ZiF

Spatial estimation


Typical models

Stochastic spatial point processes: continuous time & space,point individuals, innite domains

Spatial (two-point, truncated) correlation functions

Usually assume isotropy and translational invariance

Exogenous heterogeneity expressed as correlation function ofparameters

e.g. spatial logistic:Process Eect Ratecompetition subtract individual at x

∑x∈γ

∑y∈γ a

−(y − x)

death subtract point at x∑

x∈γ m(x)

birth add point at x∫

Ω

∑y∈γ a

+(y − x) dx

Ben Bolker ZiF

Spatial estimation


Explanations for spatial patterns

Outline

1 Introduction



4 Conclusions

Ben Bolker ZiF

Spatial estimation



Templates

assume habitat map reectsunderlying spatial patternmore or less exactly: lowdiusion (but > 0), highgrowth rate

photo: Henry Horn

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Spatial estimation



Nonlinear (deterministic) pattern formation

Turing instabilities: unstablespatial modes of nonlinearsystems

in ecology: tiger bush,spruce waves, predator-preyspirals (Rohani et al., 1997)

requires no noise, just initialperturbation

tiger bush (Wikipedia)

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Spatial estimation

http://en.wikipedia.org/wiki/Tiger_bush



Demographic noise-driven pattern

Correlation equations: Dirac δ function in spatial correlationfunction due to discreteness

Drives pattern in homogeneous, stable systems (e.g.competition): C = 0 is not even an equilibrium for the spatiallogistic equation

Eects could be weak:depend on scale of interaction neighborhood (Bolker, 1999): innumbers of individuals: ≈ 10, 100, 1000 ?eects depend on R = f /µ for equal scale of dispersal &competition, get even pattern if R > 2 (Bolker & Pacala,1999)

Could be stronger with ne-scale heterogeneity (e.g. negativebinomial noise?)

Ben Bolker ZiF

Spatial estimation



General (environmental) noise-driven pattern

Most general case: space(-time) noise ξ(x , t,N(t))

Scale may be > 0 (non-white in space and time)

Amplitude may scale dierently from√N

Space-time correlations:

separable Snyder & Chesson (2004) ?other correlations can be framed as outcomes of dynamicalprocesses

Properties other than 2-point correlation functions?

Ben Bolker ZiF

Spatial estimation



Summary: what do we need?

Lots of interesting questions, but perhaps existing methods aregood enough for ecologists?

Importance of stochastic dynamics at various scales:

Is demographic (endogenous) noise really that important?Perhaps a more traditional separation of scales (individual,patch, site, . . . ) is enough?

How do we estimate the (eects of) heterogeneity?

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Outline

1 Introduction



4 Conclusions

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Spatial synchrony

Eastern spruce budworm,Choristeroneura fumiferna

non-spatial dynamics: plantquality, climate, enemies(Kendall et al., 1999)?

What generates large-scalespatial synchrony?

Moran eect: large-scaleweather patternsDispersal coupling:movement of larvae

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Correlation equations: via continuous eqns

cf. Lande et al. (1999), Engen et al. 2002:

∂N(x, t)

∂t= F (N(x, t),E (x, t))︸︷︷︸

pop. growth

− mN(x)︸︷︷︸emigration

+ m

∫D(y, x)N(y) dy︸︷︷︸immigration

∂n

∂t≈ −rn(x, t)︸︷︷︸

regulation

+m(D ∗ n − n)︸︷︷︸redistribution

+σ2Ee(x, t)︸︷︷︸noise

2(r + m)c∗ = m(D ∗ c∗) + σ2ECor(e)

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Moth sampling locations

500 1000 1500 2000 2500 3000 3500

0

500

1000

1500

2000

E−W (km)

N−

S (

km)

SBWtemperature

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Spatial estimation


Spatial synchrony

Moth dynamics

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Spatial correlations of moth data

0 500 1000 1500 2000 2500 3000

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

moth densityJune temp.

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Spatial logistic: solution

At equilibrium, the power spectrum of the population densitiesobeys

S =∣∣∣(N∗)∣∣∣2 =

σ2E e

2(r + m(1− D))

where ˜ denotes the Fourier transform. Therefore:

σ2P(p) = σ2E +m

rσ2D

(Lande et al. 1999) where σX represents the standard deviation ofthe autocorrelation function

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Simple expressions for S if we choose

D ∝ e−λ|r | (Laplacian) in 1D

Bessel (K0) in 2D

So that K (λ) = λ/(λ2 + ω2).Then

S = c1K (λe) + c2K (λd√r/(r + m))

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Spectral ratios

Alternatively, calculate the spectral ratio:

R =e

S=

2

σ2E(r + m(1− D)) = c1 − c2D

Re-invert to deconvolve the eects of environmental variability fromthe population pattern . . .

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Reconstructing the dispersal curve

D~

c1

c1−c2

1

~ ~R = c1 − c2 D

We know the limits D(0) = 1, D(∞)→ 1: thus

Dest(ω) =R(∞)− R(ω)

R(∞)− R(0)

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Moth data: spectra

0.000 0.010 0.020

500

10000

Frequency (km−1)

Population

1e+05

2e+06

Environment

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

Moth data: spectral ratios

Frequency (km−1)

Rat

io

0.000 0.005 0.010 0.015

500

400

300

200

100

Ben Bolker ZiF

Spatial estimation


Spatial synchrony

(Putative) moth dispersal curve

0 100 200 300 400 500

0.00

0.05

0.10

0.15

0.20

0.25

Distance (km)

Dis

pers

al

Ben Bolker ZiF

Spatial estimation


Pines

Outline

1 Introduction



4 Conclusions

Ben Bolker ZiF

Spatial estimation


Pines

Pines

Slash pine, Pinus elliottii

Data on seed distribution,seedling distribution, butnot collected on the samequadrats

Sampling scheme highlyirregular; sparse data

Wikipedia

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Spatial estimation

http://en.wikipedia.org/wiki/Pinus_elliottii


Pines

Seed data

A F D

B E G

H J C

01020304050

01020304050

01020304050

0 10 20 30 40 500 10 20 30 40 500 10 20 30 40 50

empty

TRUE

FALSE

count

0

10

20

30

40

Ben Bolker ZiF

Spatial estimation


Pines

Sapling data

D E B J

A F H G

C K

01020304050

01020304050

01020304050

0 1020304050600 102030405060

empty

TRUE

FALSE

count

0

2

4

6

8

10

12

14

Ben Bolker ZiF

Spatial estimation


Pines

Correlation functions

seedling seed

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 0 5 10 15 20 25Distance (m)

Aut

ocor

rela

tion

Ben Bolker ZiF

Spatial estimation


Pines

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 ∝σ2E

E 2cEE +

σ2NN2

cNN

. . . a weighted mixture of the two correlation functions

Ben Bolker ZiF

Spatial estimation


Pines

Solving for cEE

Therefore,cEE ∝ σ2ScSS − E 2σ2NcNN

Can we really use this?

Ben Bolker ZiF

Spatial estimation


Pines

Results: observed/inferred correlation equations

seedling seed env

0.0

0.5

1.0

0 5 10 15 20 250 5 10 15 20 250 5 10 15 20 25Distance (m)

Aut

ocor

rela

tion w

est

typeseedlingseedenv

Ben Bolker ZiF

Spatial estimation


Pines

Simulation results

seedling seed env

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 250 5 10 15 20 250 5 10 15 20 25Distance (m)

Aut

ocor

rela

tion

typeseedlingseedenv

wtrueest

Ben Bolker ZiF

Spatial estimation


Caveats/assumptions

Assumes linearization/moment truncation

Assumes isotropy/homogeneity

Estimating spectra of small, irregular, noisy data sets isdicult (!)

Advantages vs. direct estimation (e.g. via MCMC) ?

Ben Bolker ZiF

Spatial estimation


Goals

Simple, light-weight (!!), non-parametric (??) approach tospatial estimation

leverage unreasonable eectiveness of linearization(Gurney & Nisbet, 1998)

Use all available information:

snapshots, before/after, time/seriesnon-matching spatial samples

Ben Bolker ZiF

Spatial estimation


Acknowledgements

People Ottar Bjørnstad, Sandy Liebhold, Aaron Berk, GordonFox, Stephen Cornell, Mollie Brooks, Emilio Bruna

Funding NSF, NSERC (Discovery Grant)

Ben Bolker ZiF

Spatial estimation


Meta-issues

Why do interdisciplinary work? Public vs. private explanations:

for funto nd interesting math problemsto solve biological problems (basic or applied)career enhancement (publications, grants, publicity . . . )

Interactions of science with mathematics

Physics vs. biology vs. social sciencesQuantitative or qualitative dierences?Which dierences matter? Scale, noisiness, dataquality/quantity, culture (lumpers vs. splitters), . . . ?Are dierent strategies required?

Ben Bolker ZiF

Spatial estimation


http://xkcd.com/435/

Ben Bolker ZiF

Spatial estimation

http://xkcd.com/435/


Bolker, BM, 1999. 61:849874.

Bolker, BM & Pacala, SW, 1999. American Naturalist, 153:575602.

Gurney, WSC & Nisbet, R, 1998. Ecological Dynamics. Oxford University Press, Oxford, UK. ISBN0195104439.

Kendall, B, Briggs, C, Murdoch, W, et al., 1999. Ecology, 80:17891805.

Lande, R, Engen, S, & Sæther, B, 1999. The American Naturalist, 154(3):271281.doi:10.1086/303240. URL http://dx.doi.org/10.1086/303240.

Rohani, P, Lewis, TJ, Grunbaum, D, et al., 1997. Trends in Ecology & Evolution, 12(2):7074. ISSN0169-5347. doi:10.1016/S0169-5347(96)20103-X. URL http://www.sciencedirect.com/science/article/B6VJ1-3X2B51F-19/2/752d6d91ad9282ab7ec3707fdf4e5c7e.

Snyder, RE & Chesson, P, 2004. The American Naturalist, 164(5):633650. URLhttp://www.jstor.org/stable/10.1086/424969.

Ben Bolker ZiF

Spatial estimation

http://dx.doi.org/10.1086/303240

http://www.sciencedirect.com/science/article/B6VJ1-3X2B51F-19/2/752d6d91ad9282ab7ec3707fdf4e5c7e

http://www.sciencedirect.com/science/article/B6VJ1-3X2B51F-19/2/752d6d91ad9282ab7ec3707fdf4e5c7e

http://www.jstor.org/stable/10.1086/424969

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