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Classical ecological theory predicts that environmental stochasticityincreases extinction risk by reducing the average per-capita growth rate ofpopulations. To understand the interactive effects of environmentalstochasticity, spatial heterogeneity, and dispersal on population growth, westudy the following model for population abundances in $n$ patches: theconditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small theconditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^i\mu_i+\sum_j(x^jD_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $\mu_i$ are the abundance and percapita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is thedispersal rate from the $i$-th to the $j$-th patch, and the conditionalcovariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^ix^j \sigma_{ij}dt$. We show for such a spatially extended population that if$S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$,the vector of patch proportions, converges in law to a random vector $Y_\infty$as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\logS_t$ equals the space-time average per-capita growth rate$\sum_i\mu_i\E[Y_\infty^i]$ experienced by the population minus half of thespace-time average temporal variation $\E[\sum_{i,j}\sigma_{ij}Y_\infty^iY_\infty^j]$ experienced by the population. We derive analytic results for thelaw of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces anoptimal stochastic growth rate for a freely dispersing population, andinvestigate the effect on the stochastic growth rate of constraints ondispersal rates. Our results provide fundamental insights into "ideal free"movement in the face of uncertainty, the persistence of coupled sinkpopulations, the evolution of dispersal rates, and the single large or severalsmall (SLOSS) debate in conservation biology.

Stochastic population growth in spatially heterogeneous environments