Spatiotemporal Dynamics of a Modified Leslie–Gower Model with Weak Allee Effect

In this paper, we consider a diffusive Leslie–Gower model with weak Allee effect in prey. We first study the global existence and non-negativity of the solutions by using the upper-lower solution method. We analyze the local stability of the positive constant steady state and prove the global attractivity by means of the LaSalle’s invariance principle. Additionally, we derive the parameter region that makes the positive constant steady state stable, and conclude that the boundary of this region contains a Hopf bifurcation curve and countable Turing curves. Thus, we get the existence of Turing–Hopf bifurcation and Turing–Turing bifurcation. Moreover, we calculate the normal form of the Turing–Hopf singularity on the center manifold. Our theoretical analysis shows that the system may produce a pair of spatially inhomogeneous steady states, spatially homogeneous periodic solutions, transient spatially inhomogeneous periodic solutions or even other solutions near the Turing–Hopf singularity. Finally, we carry out some numerical simulations for illustrating the analytical results.