Spatiotemporal dynamics of a diffusive Leslie–Gower prey–predator model with strong Allee effect

Abstract

This paper is concerned with the spatiotemporal dynamics of a diffusive Leslie–Gower prey–predator model with strong Allee effect in the prey population. The necessary and sufficient conditions of Turing instability are explicitly obtained. We demonstrate that at the critical value of the bifurcation parameter d2 a Turing bifurcation occurs (i.e, a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, treating μ as a bifurcation parameter, the Hopf bifurcation and global steady state bifurcation from the positive constant equilibrium solution are investigated. In particular, the local structure of the steady state bifurcation from double eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that strong Allee effect in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without strong Allee effect.