Stability of stationary distributions in a space-dependent population growth process

Abstract

We consider a spatial population growth process which is described by a reaction-diffusion equation c(x)u _t = (a ²(x)u _x ) _x +f(u), c(x) >0, a(x) > 0, defined on an interval [0, 1] of the spatial variable x. First we study the stability of nonconstant stationary solutions of this equation under Neumann boundary conditions. It is shown that any nonconstant stationary solution (if it exists) is unstable if a _xx⩽0 for all xε[0, 1], and conversely ifa _xx>0 for some xε[0, 1], there exists a stable nonconstant stationary solution. Next we study the stability of stationary solutions under Dirichlet boundary conditions. We consider two types of stationary solutions, i.e., a solution u ₀(x) which satisfies u ₀ x≠0 for all xε[0, 1] (type I) and a solution u ₀(x) which satisfies u _0x = 0 at two or more points in [0, 1] (type II). It is shown that any stationary solution of type I [type II] is stable [unstable] if a _xx ⩾0 [a _xx ⩽0] for all xε[0, 1]. Conversely, there exists an unstable [a stable] stationary solution of type I [type II] if a _xx <0 [a _xx >0] for some xε[0, 1].