Journal of Mathematical Biology

, Volume 66, Issue 3, pp 423–476

Stochastic population growth in spatially heterogeneous environments

  • Steven N. Evans
  • Peter L. Ralph
  • Sebastian J. Schreiber
  • Arnab Sen

DOI: 10.1007/s00285-012-0514-0

Cite this article as:
Evans, S.N., Ralph, P.L., Schreiber, S.J. et al. J. Math. Biol. (2013) 66: 423. doi:10.1007/s00285-012-0514-0


Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita 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 dZtμZtdtσZtdWt, t ≥ 0, where the conditional law of ZttZt given Ztz has mean and variance approximately zμΔt and z2σ2Δt when the time increment Δt is small. The long-term 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 Xtt given Xtx 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 μi is the per capita growth rate in the ith patch and Dij 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 xixjσijΔt for some covariance matrix Σ = (σij). 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 YtXt/St, 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 space-time average per-capita growth rate \({\sum_i \mu_i \mathbb{E}[Y_\infty^i]}\) experienced by the population minus half of the space-time 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 multi-scale 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 density-dependent feedbacks, ideal-free 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.


Stochastic population growth Spatial and temporal heterogeneity Dominant Lyapunov exponent Ideal free movement Evolution of dispersal Single large or several small debate Habitat fragmentation 

Mathematics Subject Classification

92D25 37H15 60H10 

Supplementary material (103 kb)
ESM 1 (ZIP 104 kb)

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Steven N. Evans
    • 1
  • Peter L. Ralph
    • 2
  • Sebastian J. Schreiber
    • 2
  • Arnab Sen
    • 3
  1. 1.Department of Statistics #3860University of CaliforniaBerkeleyUSA
  2. 2.Department of Evolution and EcologyUniversity of CaliforniaDavisUSA
  3. 3.Statistical LaboratoryCentre for Mathematical SciencesCambridgeUK

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