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Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References            ...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesOutline     ...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat do ecol...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat do ecol...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat do ecol...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical exam...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical exam...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical exam...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical mode...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical mode...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial sync...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesOutline...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesPines  ...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSeed da...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSapling...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesCorrela...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesAnalyti...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSolving...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesResults...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSimulat...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesCaveats/assu...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesGoals       ...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesAcknowledgem...
Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesMeta-issues ...
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  1. 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 2012Ben Bolker ZiFSpatial estimation
  2. 2. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesOutline 1 Introduction 2 Endogenous vs exogenous: qualitative Explanations for spatial patterns 3 Endogenous vs exogenous: quantitative Spatial synchrony Pines 4 ConclusionsBen Bolker ZiFSpatial estimation
  3. 3. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat 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 → modelBen Bolker ZiFSpatial estimation
  4. 4. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat 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 → modelBen Bolker ZiFSpatial estimation
  5. 5. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesWhat 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 → modelBen Bolker ZiFSpatial estimation
  6. 6. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical 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 ZiFSpatial estimation
  7. 7. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical 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 ZiFSpatial estimation
  8. 8. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical 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 ZiFSpatial estimation
  9. 9. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical 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 ) dxBen Bolker ZiFSpatial estimation
  10. 10. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesTypical 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 ) dxBen Bolker ZiFSpatial estimation
  11. 11. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsOutline 1 Introduction 2 Endogenous vs exogenous: qualitative Explanations for spatial patterns 3 Endogenous vs exogenous: quantitative Spatial synchrony Pines 4 ConclusionsBen Bolker ZiFSpatial estimation
  12. 12. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsTemplates assume habitat map reects underlying spatial pattern more or less exactly: low diusion (but 0), high growth rate photo: Henry HornBen Bolker ZiFSpatial estimation
  13. 13. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsNonlinear (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 ZiFSpatial estimation
  14. 14. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsDemographic 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 ZiFSpatial estimation
  15. 15. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsDemographic 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 ZiFSpatial estimation
  16. 16. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsGeneral (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 ZiFSpatial estimation
  17. 17. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesExplanations for spatial patternsSummary: 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 ZiFSpatial estimation
  18. 18. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyOutline 1 Introduction 2 Endogenous vs exogenous: qualitative Explanations for spatial patterns 3 Endogenous vs exogenous: quantitative Spatial synchrony Pines 4 ConclusionsBen Bolker ZiFSpatial estimation
  19. 19. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronySpatial 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 larvaeBen Bolker ZiFSpatial estimation
  20. 20. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyCorrelation 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 ZiFSpatial estimation
  21. 21. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyMoth 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 ZiFSpatial estimation
  22. 22. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyMoth 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 3000Ben Bolker ZiFSpatial estimation
  23. 23. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronySpatial 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 ZiFSpatial estimation
  24. 24. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronySpatial 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 functionBen Bolker ZiFSpatial estimation
  25. 25. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial 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 ZiFSpatial estimation
  26. 26. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronySpectral 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 ZiFSpatial estimation
  27. 27. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyReconstructing 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 ZiFSpatial estimation
  28. 28. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyMoth data: spectra Environment Population 10000 2e+06 500 1e+05 0.000 0.010 0.020 Frequency (km−1)Ben Bolker ZiFSpatial estimation
  29. 29. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial synchronyMoth data: spectral ratios 100 200 Ratio 300 400 500 0.000 0.005 0.010 0.015 −1 Frequency (km )Ben Bolker ZiFSpatial estimation
  30. 30. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesSpatial 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 ZiFSpatial estimation
  31. 31. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesOutline 1 Introduction 2 Endogenous vs exogenous: qualitative Explanations for spatial patterns 3 Endogenous vs exogenous: quantitative Spatial synchrony Pines 4 ConclusionsBen Bolker ZiFSpatial estimation
  32. 32. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesPines Slash pine, Pinus elliottii Data on seed distribution, seedling distribution, but not collected on the same quadrats Sampling scheme highly irregular; sparse data WikipediaBen Bolker ZiFSpatial estimation
  33. 33. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSeed 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 50Ben Bolker ZiFSpatial estimation
  34. 34. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSapling 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 0Ben Bolker ZiFSpatial estimation
  35. 35. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesCorrelation 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 ZiFSpatial estimation
  36. 36. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesAnalytical 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 functionsBen Bolker ZiFSpatial estimation
  37. 37. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSolving for cEE Therefore, c ¯ EE ∝ σS cSS − E σN cNN 2 2 2 Can we really use this?Ben Bolker ZiFSpatial estimation
  38. 38. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesResults: 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 ZiFSpatial estimation
  39. 39. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesPinesSimulation 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 ZiFSpatial estimation
  40. 40. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesCaveats/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 ZiFSpatial estimation
  41. 41. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesGoals 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 samplesBen Bolker ZiFSpatial estimation
  42. 42. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesAcknowledgements People Ottar Bjørnstad, Sandy Liebhold, Aaron Berk, Gordon Fox, Stephen Cornell, Mollie Brooks, Emilio Bruna Funding NSF, NSERC (Discovery Grant)Ben Bolker ZiFSpatial estimation
  43. 43. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions ReferencesMeta-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 ZiFSpatial estimation
  44. 44. Introduction Endogenous vs exogenous: qualitative Endogenous vs exogenous: quantitative Conclusions References http://xkcd.com/435/Ben Bolker ZiFSpatial estimation
  45. 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 ZiFSpatial estimation

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