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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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Outline
1 Introduction
2 Endogenous vs exogenous: qualitativeExplanations for spatial patterns
3 Endogenous vs exogenous: quantitativeSpatial synchronyPines
4 Conclusions
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
Outline
1 Introduction
2 Endogenous vs exogenous: qualitativeExplanations for spatial patterns
3 Endogenous vs exogenous: quantitativeSpatial synchronyPines
4 Conclusions
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
Templates
assume habitat map reectsunderlying spatial patternmore or less exactly: lowdiusion (but > 0), highgrowth rate
photo: Henry Horn
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
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)
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Outline
1 Introduction
2 Endogenous vs exogenous: qualitativeExplanations for spatial patterns
3 Endogenous vs exogenous: quantitativeSpatial synchronyPines
4 Conclusions
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth sampling locations
500 1000 1500 2000 2500 3000 3500
0
500
1000
1500
2000
E−W (km)
N−
S (
km)
SBWtemperature
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth dynamics
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Outline
1 Introduction
2 Endogenous vs exogenous: qualitativeExplanations for spatial patterns
3 Endogenous vs exogenous: quantitativeSpatial synchronyPines
4 Conclusions
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Pines
Slash pine, Pinus elliottii
Data on seed distribution,seedling distribution, butnot collected on the samequadrats
Sampling scheme highlyirregular; sparse data
Wikipedia
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Solving for cEE
Therefore,cEE ∝ σ2ScSS − E 2σ2NcNN
Can we really use this?
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
http://xkcd.com/435/
Ben Bolker ZiF
Spatial estimation
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
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