Spatiotemporal patterns in a diffusive predator–prey system of Leslie–Gower type with Smith growth for prey

Abstract This paper concerns a two-species predator-prey system with Smith-type growth for the prey, Leslie-Gower-type growth for the predator, a functional response of Beddington-DeAngelis type and zero Neumann boundary condition. We first analyze the existence, local asymptotic stability and Hopf bifurcation behavior of the coexistence equilibria of the local ODE system. It is shown that a Hopf bifurcation of the local ODE at the coexistence equilibrium is possible only when the predator's capture rate exceeds the sum of one and its intraspecific competition coefficient. When the coexistence equilibrium of the local ODE system does undergo a Hopf bifurcation, we further examine the emergence of a spatially homogeneous Hopf bifurcation of the diffusive system. Then we determine the direction of the Hopf bifurcation and the stability of the resulting periodic solutions by using the normal form theory and center manifold reduction. In particular, within the parameter region where the coexistence equilibrium of the local ODE is stable, we not only establish the existence of both Turing and Turing-Turing bifurcations for the diffusive system, but also derive the corresponding normal forms. In order to validate the theoretical findings, numerical simulations are performed for the diffusive system on a one-dimensional bounded spatial domain with appropriately selected parameter values. AMS Subject Classification (2020): 35B35; 35B40; 35K57; 92D40.