Spatiotemporal dynamics of two generic predator–prey models

Abstract

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L p -estimate, uniform in time, for all p≥1, implies L ∞-uniform bounds, given any nonnegative L ∞-initial data. The applicability of the L ∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.