Journal of Mathematical Biology
, Volume 66, Issue 3, pp 423476
First online:
Stochastic population growth in spatially heterogeneous environments
 Steven N. EvansAffiliated withDepartment of Statistics #3860, University of California
 , Peter L. RalphAffiliated withDepartment of Evolution and Ecology, University of California
 , Sebastian J. SchreiberAffiliated withDepartment of Evolution and Ecology, University of California Email author
 , Arnab SenAffiliated withStatistical Laboratory, Centre for Mathematical Sciences
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Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average percapita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ _{ t } = μ Z _{ t } dt + σ Z _{ t } dW _{ t }, t ≥ 0, where the conditional law of Z _{ t+Δt }− Z _{ t } given Z _{ t } = z has mean and variance approximately z μΔt and z ^{2} σ ^{2}Δt when the time increment Δt is small. The longterm stochastic growth rate \({\lim_{t \to \infty} t^{1}\log Z_t}\) for such a population equals \({\mu \frac{\sigma^2}{2}}\) . Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model \({{\bf X}_t = (X_t^1, \ldots, X_t^n)}\) , t ≥ 0, for the population abundances in n patches: the conditional law of X _{ t+Δt } given X _{ t } = x is such that the conditional mean of \({X_{t+\Delta t}^i  X_t^i}\) is approximately \({[x^i \mu_i + \sum_j (x^j D_{ji}  x^i D_{ij})] \Delta t}\) where μ _{ i } is the per capita growth rate in the ith patch and D _{ ij } is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of \({X_{t+\Delta t}^i  X_t^i}\) and \({X_{t + \Delta t}^j  X_t^j}\) is approximately x ^{ i } x ^{ j } σ _{ ij }Δt for some covariance matrix Σ = (σ _{ ij }). We show for such a spatially extended population that if \({S_t = X_t^1 + \cdots + X_t^n}\) denotes the total population abundance, then Y _{ t } = X _{ t }/S _{ t }, the vector of patch proportions, converges in law to a random vector Y _{∞} as \({t \to \infty}\) , and the stochastic growth rate \({\lim_{t \to \infty} t^{1}\log S_t}\) equals the spacetime average percapita growth rate \({\sum_i \mu_i \mathbb{E}[Y_\infty^i]}\) experienced by the population minus half of the spacetime average temporal variation \({\mathbb{E}[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]}\) experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multiscale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of densitydependent feedbacks, idealfree dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.
Keywords
Stochastic population growth Spatial and temporal heterogeneity Dominant Lyapunov exponent Ideal free movement Evolution of dispersal Single large or several small debate Habitat fragmentationMathematics Subject Classification
92D25 37H15 60H10 Title
 Stochastic population growth in spatially heterogeneous environments
 Journal

Journal of Mathematical Biology
Volume 66, Issue 3 , pp 423476
 Cover Date
 201302
 DOI
 10.1007/s0028501205140
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Stochastic population growth
 Spatial and temporal heterogeneity
 Dominant Lyapunov exponent
 Ideal free movement
 Evolution of dispersal
 Single large or several small debate
 Habitat fragmentation
 92D25
 37H15
 60H10
 Authors

 Steven N. Evans ^{(1)}
 Peter L. Ralph ^{(2)}
 Sebastian J. Schreiber ^{(2)}
 Arnab Sen ^{(3)}
 Author Affiliations

 1. Department of Statistics #3860, University of California, 367 Evans Hall, Berkeley, CA, 947203860, USA
 2. Department of Evolution and Ecology, University of California, Davis, CA, 956116, USA
 3. Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK