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.
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