Stochastic population growth in spatially heteroge

Authors: Steven N Evans Peter L Ralph Sebastian J Schreiber Arnab Sen

Publish Date: 2012/03/18

Volume: 66, Issue: 3, Pages: 423-476

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Abstract

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 2Δt when the time increment Δt is small The longterm stochastic growth rate lim t to infty t1log Z t for such a population equals mu fracsigma22 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 t1 ldots X tn 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 ti X ti is approximately xi mu i + sum j xj D ji xi 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 ti X ti and X t + Delta tj X tj 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 t1 + cdots + X tn 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 t1log S t equals the spacetime average percapita growth rate sum i mu i mathbbEY inftyi experienced by the population minus half of the spacetime average temporal variation mathbbEsum ijsigma ijY inftyi Y inftyj 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 eg 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

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