Nonlinear population dynamics in a bounded habitat

Abstract

A key issue in ecology is whether a population will survive long term or go extinct. This is the question we address in this paper for a population in a bounded habitat. We will restrict our study to the case of a single species in a one-dimensional habitat of length L. The evolution of the population density distribution ρ(x, t), where x is the position and t the time, is governed by elementary processes such as growth and dispersal, which, in standard models, are typically described by a constant per capita growth rate and normal diffusion, respectively. However, feedbacks in the regulatory mechanisms and external factors can produce density-dependent rates. Therefore, we consider a generalization of the standard evolution equation, which, after dimensional scaling and assuming large carrying capacity, becomes ∂tρ=∂x(ρν−1∂xρ)+ρμ, where μ,ν∈R. This equation is complemented by absorbing boundaries, mimicking adverse conditions outside the habitat. For this nonlinear problem, we obtain, analytically, exact expressions of the critical habitat size Lc for population survival, as a function of the exponents and initial conditions. We find that depending on the values of the exponents (ν, μ), population survival can occur for either L ≥ Lc, L ≤ Lc or for any L. This generalizes the usual statement that Lc represents the minimum habitat size. In addition, nonlinearities introduce dependence on the initial conditions, affecting Lc.