Stochastic population growth in spatially heterogeneous environments: the density-dependent case

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 [Formula: see text]. 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 [Formula: see text], the population abundances converge polynomially fast to a unique invariant probability measure on [Formula: see text], while when [Formula: see text], 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, [Formula: see text], 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.

Keywords: Density-dependence; Dispersion; Ergodicity; Habitat fragmentation; Lotka–Volterra model; Lyapunov exponent; Spatial and temporal heterogeneity; Stochastic environment; Stochastic population growth.

Fig. 1

Consider (3.2) when $\mathit{\alpha }=\mathit{\beta }$ and the Brownian motions $B_1$ and $B_2$ are assumed to have correlation $\mathit{\rho }$. The graphs show the stochastic growth rate r as a function of the dispersal rate $\mathit{\alpha }$ for different values of the correlation. Note that if $\mathit{\rho }=0$ we get the setting when the Brownian motions of the two patches are independent while when $\mathit{\rho }=1$ we get that one Brownian motion drives the dynamics of both patches. The parameters are $\mathit{\alpha }=\mathit{\beta },a_1=3,a_2=4,{\mathit{\sigma }}_1^2={\mathit{\sigma }}_2^2=7$