Effect of a protection zone in the diffusive Leslie predator–prey model

Abstract

In this paper, we consider the diffusive Leslie predator–prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω 0 for the prey is introduced. As in earlier work [Y. Du, J. Shi, A diffusive predator–prey model with a protection zone, J. Differential Equations 229 (2006) 63–91; Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Equations 244 (2008) 61–86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω 0 and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω 0 is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator–prey model studied in [Y. Du, J. Shi, A diffusive predator–prey model with a protection zone, J. Differential Equations 229 (2006) 63–91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.