On population resilience to external perturbations. Research report
2007
Roques , Lionel (INRA (France). UR 0546 Biostatistique et Processus Spatiaux) | Chekroun , Mickaël D. (Ecole Normale Supérieure, Paris cedex 05(France).)
We study a spatially-explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with aspace-dependent nonlinearity of KPP type, and a negative externalforcing term. The domain is either the whole space RN, with periodiccoefficients, or a bounded domain. Analyzing the stationary states, wedefine two main types of solutions: the “significant” solutions, whichalways stay above a certain small threshold value, and the “remnant”solutions, which are always below this value. Using sub- and supersolutionmethods, the characterization of the first eigenvalue and firsteigenfunction of some linear elliptic operators, we obtain existence andnonexistence 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.
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