Analysis of Stochastic/Nonlinear Models

Roques, L., and M. D. Chekroun. 2007. “On Population resilience to external perturbations.” SIAM Journal on Applied Mathematics 68 (1): 133—153. Publisher's Version Abstract
We study a spatially explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with a spatially dependent nonlinearity of Kolmogorov–Petrovsky–Piskunov type, and a negative external forcing term $-\delta$. Using sub- and supersolution methods and the characterization of the first eigenvalue 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 two critical values $\delta^*$ and $\delta_2$ such that, if $\delta$ is smaller than $\delta^*$, the population density converges to a “significant" state, which is everywhere above a certain small threshold, whereas if $\delta$ is larger than $\delta_2$, the population density converges to a “remnant" state, everywhere below this small threshold. Our results are shown to be useful for studying the relationships between environmental fragmentation and maximum sustainable yield from populations. We present numerical results in the case of stochastic environments.
Mickaël D. Chekroun, Lionel J. Roques. 2006. “Models of population dynamics under the influence of external perturbations: mathematical results.” Comptes Rendus Mathématique 750 (5): 291-382. Publisher's Version Abstract
Abstract In this note, we describe the stationary equilibria and the asymptotic behaviour of an heterogeneous logistic reaction-diffusion equation under the influence of autonomous or time-periodic forcing terms. We show that the study of the asymptotic behaviour in the time-periodic forcing case can be reduced to the autonomous one, the last one being described in function of the size' of the external perturbation. Our results can be interpreted in terms of maximal sustainable yields from populations. We briefly discuss this last aspect through a numerical computation. To cite this article: M.D. Chekroun, L.J. Roques, C. R. Acad. Sci. Paris, Ser. I 343 (2006). Résumé Cette Note a pour objet lʼétude des états stationnaires et du comportement asymptotique dʼéquations de réaction-diffusion avec coefficients hétérogènes en espace, auxquelles nous ajoutons un terme de perturbation stationnaire ou périodique en temps. Nos résultats peuvent sʼinterpreter en termes de prélèvement maximal supportable par une population. Nous soulignons cet aspect à lʼaide dʼun calcul numérique. Pour citer cet article : M.D. Chekroun, L.J. Roques, C. R. Acad. Sci. Paris, Ser. I 343 (2006).