Abstract
This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on [0,infty )^n. We prove that r, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if r>0, the population abundances converge polynomially fast to a unique invariant probability measure on (0,infty )^n, while when r<0, the population abundances of the patches converge almost surely to 0 exponentially fast. This generalizes and extends the results of Evans et al. (J Math Biol 66(3):423–476, 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting. As an example we fully analyze the two-patch case, n=2, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.
Highlights
For numerous models of population dynamics it is natural to assume that time is continuous
There have been a few papers dedicated to the study of stochastic differential equation models of interacting, unstructured populations in stochastic environments
Examples of structured populations can be found by looking at a population in which individuals can live in one of n patches
Summary
66(3):423–476, 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Continuous-space discrete-time population models that disperse and experience uncorrelated, environmental stochasticity have been studied by Hardin et al (1988a, b, 1990) They show that the leading Lyapunov exponent r of the linearization of the system around the extinction state almost determines the persistence and extinction of the population. We propose a density-dependent model of stochastic population growth that captures the interactions between dispersal and environmental heterogeneity and complements the work of Evans et al (2013). Few theoretical studies have considered the combined effects of spatio-temporal heterogeneities, dispersal, and density-dependence for discretely structured populations with continuous-time dynamics As seen in both the continuous (Evans et al 2013) and the discrete (Palmqvist and Lundberg 1998) settings, the extinction risk of a population is greatly affected by the spatio-temporal correlation between the environment in the different patches.
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