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

This work is devoted to studying the dynamics of a population subject to the combined effects of stochastic environments, competition for resources, and 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 $\lambda$, the stochastic growth rate of the system in the absence of competition, determines the long-term behaviour of the population. The parameter $\lambda$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $\lambda>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $\lambda<0$, the abundances of the patches converge almost surely to $0$ exponentially fast. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Another significant generalization of our work is allowing the environmental noise driving our system to be degenerate. This is more natural from a biological point of view since, for example, the environments of the different patches can be perfectly correlated.