Turing Bifurcation in a Leslie–Gower-Type Predator–Prey Model with Smith Growth and B–D Functional Response

In this paper, a Leslie–Gower-type reaction–diffusion predator–prey system with Smith growth rate of prey and Beddington–DeAngelis functional response is considered and Turing bifurcation of the positive constant steady-state solution is explored. The local asymptotic stability and detailed Hopf bifurcation of the positive equilibrium of the associated ODE system is investigated first according to the bifurcation theory of ODEs. Then, the existence of Turing bifurcation of the positive constant steady-state solution of reaction–diffusion system is established and the corresponding normal forms in the critical and perturbed cases for the prey diffusion coefficient to be taken as the bifurcation parameter are derived in virtue of the normal form method and the center manifold theorem for parabolic partial differential equations. Finally, numerical simulations are also carried out when the spatial domain is taken as [Formula: see text] and [Formula: see text] in order to verify our theoretical predications.