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 modelled as the
solutions of a system of nonlinear stochastic differential equations living on
[0,∞)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,∞)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
(2014 J. Math. Biol.) 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. 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.
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.Comment: 43 pages, 1 figure, edited according to the suggestion of the
referees, to appear in Journal of Mathematical Biolog