Finite element approximation of a spatially structured metapopulation PDE model

Abstract

We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d≤3 space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model.