Nonlinear reaction-diffusion models for interacting populations

Abstract

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c i ( x, t), 1 ⩽ i ⩽ N (with c i ( x, t) representing the concentration of the ith species at position x in a set Ω at time t ⩾ 0), which exist for all t ⩾ 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.