This document discusses explaining spatial patterns in ecology through models. It introduces qualitative explanations for spatial patterns based on endogenous vs. exogenous drivers. Quantitative models are then presented, such as spatial point processes that can be used to model spatial synchrony and clustering. The document outlines different typical models and templates used, including those based on habitat maps, nonlinear pattern formation, and spatial correlation functions.

1. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Estimating environmental heterogeneity from
spatial dynamics data
Ben Bolker, McMaster University
Departments of Mathematics & Statistics and Biology
ZiF
18 June 2012
Ben Bolker ZiF
Spatial estimation

2. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Outline
1 Introduction
2 Endogenous vs exogenous: qualitative
Explanations for spatial patterns
3 Endogenous vs exogenous: quantitative
Spatial synchrony
Pines
4 Conclusions
Ben Bolker ZiF
Spatial estimation

3. 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) of
pattern?
Deductive (forward) models: model → outcome
Inductive (inverse) models: data → model
Ben Bolker ZiF
Spatial estimation

4. 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) of
pattern?
Deductive (forward) models: model → outcome
Inductive (inverse) models: data → model
Ben Bolker ZiF
Spatial estimation

5. 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) of
pattern?
Deductive (forward) models: model → outcome
Inductive (inverse) models: data → model
Ben Bolker ZiF
Spatial estimation

6. 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 spatial
clustering?
What drives spatial clustering in a particular case?
The answer to does process X operate in ecological system Y is
most often Yes.
Ben Bolker ZiF
Spatial estimation

7. 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 spatial
clustering?
What drives spatial clustering in a particular case?
The answer to does process X operate in ecological system Y is
most often Yes.
Ben Bolker ZiF
Spatial estimation

8. 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 spatial
clustering?
What drives spatial clustering in a particular case?
The answer to does process X operate in ecological system Y is
most often Yes.
Ben Bolker ZiF
Spatial estimation

9. 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 of
parameters
e.g. spatial logistic:
Process Eect Rate
−
competition 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

10. 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 of
parameters
e.g. spatial logistic:
Process Eect Rate
−
competition 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

11. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
Outline
1 Introduction
2 Endogenous vs exogenous: qualitative
Explanations for spatial patterns
3 Endogenous vs exogenous: quantitative
Spatial synchrony
Pines
4 Conclusions
Ben Bolker ZiF
Spatial estimation

12. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
Templates
assume habitat map reects
underlying spatial pattern
more or less exactly: low
diusion (but 0), high
growth rate
photo: Henry Horn
Ben Bolker ZiF
Spatial estimation

13. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Explanations for spatial patterns
Nonlinear (deterministic) pattern formation
Turing instabilities: unstable
spatial modes of nonlinear
systems
in ecology: tiger bush,
spruce waves, predator-prey
spirals (Rohani et al., 1997)
requires no noise, just initial
perturbation
tiger bush (Wikipedia)
Ben Bolker ZiF
Spatial estimation

14. 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 correlation
function due to discreteness
Drives pattern in homogeneous, stable systems (e.g.
competition): C =0 is not even an equilibrium for the spatial
logistic equation
Eects could be weak:
depend on scale of interaction neighborhood (Bolker, 1999): in
numbers 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. negative
binomial noise?)
Ben Bolker ZiF
Spatial estimation

15. 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 correlation
function due to discreteness
Drives pattern in homogeneous, stable systems (e.g.
competition): C =0 is not even an equilibrium for the spatial
logistic equation
Eects could be weak:
depend on scale of interaction neighborhood (Bolker, 1999): in
numbers 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. negative
binomial noise?)
Ben Bolker ZiF
Spatial estimation

16. 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 dynamical
processes
Properties other than 2-point correlation functions?
Ben Bolker ZiF
Spatial estimation

17. 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 are
good 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

18. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Outline
1 Introduction
2 Endogenous vs exogenous: qualitative
Explanations for spatial patterns
3 Endogenous vs exogenous: quantitative
Spatial synchrony
Pines
4 Conclusions
Ben Bolker ZiF
Spatial estimation

19. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Spatial synchrony
Eastern spruce budworm,
Choristeroneura fumiferna
non-spatial dynamics: plant
quality, climate, enemies
(Kendall et al., 1999)?
What generates large-scale
spatial synchrony?
Moran eect: large-scale
weather patterns
Dispersal coupling:
movement of larvae
Ben Bolker ZiF
Spatial estimation

20. 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 )
= F (N ( x, t ), E (x, t )) − mN ( x)
∂t
pop. growth emigration
+m D( y, x)N (y) d y
immigration
∂n
≈ −rn(x, t ) + m(D ∗ n − n) + σE e (x, t )
2
∂t
regulation redistribution noise
∗ ∗ 2
2(r + m)c = m (D ∗ c ) + σE Cor(e )
Ben Bolker ZiF
Spatial estimation

21. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth sampling locations
2000
q
q q
q q q q q
q q q q q q q q q
1500 q q q q q q q q qq q
q q q q q q
q q q q q q q q qq q q q
q qqq q
q q q q
N−S (km)
q q q q q q q q q q q q q q q qqq q q q
q q q q
q q q q q q q q qqq q q q q q qq q qq q qqq q
q q q q q
q
q q q q q qq q
q q qq
q q q q q q qqq q q q qqq qq q q qq q q q q q
qq
q q q
1000 q
q qq q q q q q
q q qq
q q q q q q q q q q q q qq qq q q q q qq q q q
qq
q
qq q q q q
q
q q q q q q q q
q
q
q q q
q
qq
q q q q q q q q qq q q q q q q qqq q q q q q q qqqq q qq
q q q q qq q
q q q
q q qq
q q q q q
q q q q q q q q q q q q qqq qq qqq q q q q q q
q q q qq q qq qq qq
q q
q q q
qq q
q q
q q q q q qq qqq
q
q q q q qqq qq qq q q q q q q q
q q qq q q
q q q q q qqqqq qq
q qq q
qq q q
500 q
qq q q q
q
qq q qq
qqqq
q
qq qq
q
q qq
qq
qq q q
q
q SBW
0 q temperature
500 1000 1500 2000 2500 3000 3500
E−W (km)
Ben Bolker ZiF
Spatial estimation

22. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth dynamics
q
qq
qq 1945 q qq
1500 qqqq q qqqq
qqq qq q qqqqq qqqqq z
qq q qqqqqqqqqq qqqqqq q 0.0
qqqqq qq qqqqqqqqqqqq qqq q q 0.2
qqqqqqq qqqqqqqqqqqqqqqq qq
1000 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q 0.4
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q 0.6
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q 0.8
qqqqq qqqqqqqqqqqqqqqqqq q 1.0
qqqqq qqqqqqqqqqqqqqqq
500 qq qqqqqqqq qq
qqqqq
qqq
qqq
0 1000 2000 3000
Ben Bolker ZiF
Spatial estimation

23. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Spatial correlations of moth data
1.0
moth density
0.8 June temp.
0.6
Correlation
0.4
0.2
0.0
−0.2
0 500 1000 1500 2000 2500 3000
Distance (km)
Ben Bolker ZiF
Spatial estimation

24. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Spatial logistic: solution
At equilibrium, the power spectrum of the population densities
obeys
2
˜ ˜ ∗
2 σE e
˜
S = N =
2(r ˜
+ m(1 − D ))
where ˜ denotes the Fourier transform. Therefore:
2 2
m 2
σP (p ) = σE + σD
r
(Lande et al. 1999) where σX represents the standard deviation of
the autocorrelation function
Ben Bolker ZiF
Spatial estimation

25. 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 + ω 2 ).
Then
˜
S ˜ ˜
= c1 K (λe ) + c2 K (λd r /(r + m))
Ben Bolker ZiF
Spatial estimation

26. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Spectral ratios
Alternatively, calculate the spectral ratio:
˜ ˜
e 2
˜ ˜
R = = (r + m(1 − D )) = c1 − c2 D
˜
S
2
σE
Re-invert to deconvolve the eects of environmental variability from
the population pattern . . .
Ben Bolker ZiF
Spatial estimation

27. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Reconstructing the dispersal curve
1
~
D
c1
~ ~
R = c1 − c2 D
c1−c2
˜
We know the limits D (0) ˜
= 1, D (∞) → 1: thus
˜
R (∞) ˜
− R (ω)
˜
Dest (ω) =
˜ ˜
R (∞) − R (0)
Ben Bolker ZiF
Spatial estimation

28. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth data: spectra
Environment
Population
10000 2e+06
500
1e+05
0.000 0.010 0.020
Frequency (km−1)
Ben Bolker ZiF
Spatial estimation

29. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
Moth data: spectral ratios
100
200
Ratio
300
400
500
0.000 0.005 0.010 0.015
−1
Frequency (km )
Ben Bolker ZiF
Spatial estimation

30. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Spatial synchrony
(Putative) moth dispersal curve
0.25
0.20
Dispersal
0.15
0.10
0.05
0.00
0 100 200 300 400 500
Distance (km)
Ben Bolker ZiF
Spatial estimation

31. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Outline
1 Introduction
2 Endogenous vs exogenous: qualitative
Explanations for spatial patterns
3 Endogenous vs exogenous: quantitative
Spatial synchrony
Pines
4 Conclusions
Ben Bolker ZiF
Spatial estimation

32. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Pines
Slash pine, Pinus elliottii
Data on seed distribution,
seedling distribution, but
not collected on the same
quadrats
Sampling scheme highly
irregular; sparse data
Wikipedia
Ben Bolker ZiF
Spatial estimation

33. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Seed data
A F D
50
40
30
20
10 empty
0 TRUE
B E G
FALSE
50
40 count
30 0
20
10
10
0 20
H J C 30
50 40
40
30
20
10
0
0 10 20 30 40 50
0 10 20 30 40 50
0 10 20 30 40 50
Ben Bolker ZiF
Spatial estimation

34. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Sapling data
D E B J
50
40
30 empty
20
TRUE
10
FALSE
0
A F H G
count
50
0
40
30 2
20 4
10 6
0
8
C K
10
50
12
40
30 14
20
10
0
0 102030405060 102030405060
0
Ben Bolker ZiF
Spatial estimation

35. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Correlation functions
seedling seed
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
0 5 10 15 20 25 0 5 10 15 20 25
Distance (m)
Ben Bolker ZiF
Spatial estimation

36. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Analytical framework (2)
assume cross-covariance C EN = 0
no correlation between seeds and environment,
e.g. long-distance dispersal
C ¯
SS (r ) = N
2
C ¯
EE (r ) + E
2
NN (r )
C
¯ ¯
(where N =mean seed density, , E =mean establishment
probability)
or (switch to correlation c )
2
EE + ¯N cNN
σ2E σ
C SS ∝ ¯ 2
c
2
E N
. . . a weighted mixture of the two correlation functions
Ben Bolker ZiF
Spatial estimation

37. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Solving for cEE
Therefore,
c ¯
EE ∝ σS cSS − E σN cNN
2 2 2
Can we really use this?
Ben Bolker ZiF
Spatial estimation

38. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Results: observed/inferred correlation equations
seedling seed env
1.0
w
Autocorrelation
est
0.5
type
seedling
seed
env
0.0
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
Distance (m)
Ben Bolker ZiF
Spatial estimation

39. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Pines
Simulation results
seedling seed env
1.0
0.8
type
Autocorrelation
seedling
0.6
seed
env
0.4
w
true
0.2 est
0.0
−0.2
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
Distance (m)
Ben Bolker ZiF
Spatial estimation

40. 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 is
dicult (!)
Advantages vs. direct estimation (e.g. via MCMC) ?
Ben Bolker ZiF
Spatial estimation

41. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Goals
Simple, light-weight (!!), non-parametric (??) approach to
spatial estimation
leverage unreasonable eectiveness of linearization
(Gurney Nisbet, 1998)
Use all available information:
snapshots, before/after, time/series
non-matching spatial samples
Ben Bolker ZiF
Spatial estimation

42. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Acknowledgements
People Ottar Bjørnstad, Sandy Liebhold, Aaron Berk, Gordon
Fox, Stephen Cornell, Mollie Brooks, Emilio Bruna
Funding NSF, NSERC (Discovery Grant)
Ben Bolker ZiF
Spatial estimation

43. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
Meta-issues
Why do interdisciplinary work? Public vs. private explanations:
for fun
to nd interesting math problems
to solve biological problems (basic or applied)
career enhancement (publications, grants, publicity . . . )
Interactions of science with mathematics
Physics vs. biology vs. social sciences
Quantitative or qualitative dierences?
Which dierences matter? Scale, noisiness, data
quality/quantity, culture (lumpers vs. splitters), . . . ?
Are dierent strategies required?
Ben Bolker ZiF
Spatial estimation

44. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References
http://xkcd.com/435/
Ben Bolker ZiF
Spatial estimation

45. 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. ISBN
0195104439.
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. ISSN
0169-5347. 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. URL
http://www.jstor.org/stable/10.1086/424969.
Ben Bolker ZiF
Spatial estimation