Qualitative results for solutions of the steady fisher-KPP equation

Abstract

In this letter, the homogeneous Dirichlet problem involving the N-dimensional Fisher-KPP equation, a reaction-diffusion model which arises in study of population genetics, is investigated for a class of nonlinear polynomial growth laws. Existence and uniqueness conditions for positive (i.e., physically realistic), steady-state solutions on finite domains, or habitats, are noted and stability questions are addressed. Of particular interest are habitats that can be modeled as open balls. For these cases, two relatively recent and powerful theorems from nonlinear analysis are employed to ascertain important qualitative information. Specifically, these solutions are shown to be strictly decreasing and radially symmetric, as well as achieving a stationary maximum at the habitat's center. In addition, the function spaces containing these solutions are determined. Last, the effects of the solution parameters are investigated numerically for the physically relevant cases of N = 2 and 3, the temporal evolution of a particular solution is illustrated, and connections to nuclear reactor science, as well as other fields, are noted.