On Population Resilience to External Perturbations

We study a spatially-explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with a space-dependent nonlinearity of KPP type, and a negative external forcing term. The domain is either the whole space R , with periodic coefficients, or a bounded domain. Analyzing the stationary states, we define two main types of solutions: the “significant” solutions, which always stay above a certain small threshold value, and the “remnant” solutions, which are always below this value. Using suband supersolution methods, the characterization of the first eigenvalue and first eigenfunction of some linear elliptic operators, we obtain existence and nonexistence results, as well as results on the number of stationary solutions. We also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude. In particular, we define critical thresholds on the forcing term below which the population density converges to a significant state, while it converges to a remnant state whenever the forcing term lies above the highest threshold. These bounds were shown to be useful in studying the influence of environmental fragmentation on the long-time behavior of the population density, in terms of the forcing term amplitude. We also present numerical results in the case of stochastic environments.