Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b₀. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.