Bifurcation and spatiotemporal patterns of a Leslie-Gower model with diffusive

This paper is concerned with a diffusive Leslie-Gower predator-prey system subject to homogeneous Neumann boundary conditions. We derive a priori estimates for positive steady states by the maximum principle and establish the nonexistence of nonconstant positive steady states by energy method. Then the conditions for Turing instability of a unique positive constant steady state are obtained. Based on Leray-Schauder degree theory, the existence of nonconstant positive steady states is discussed. Additionally, a series of thorough qualitative examinations are carried out on the local bifurcation at both simple and double eigenvalues, and global structure and direction of the steady state bifurcations at simple eigenvalues are investigated. To conclude, numerical simulations are performed to validate and complement the analytical findings.