Abstract
In this paper, we investigate the dynamical behaviors of the cross-diffusion Leslie–Gower predator–prey system with a strong Allee effect, and mainly explore the Turing instability of spatially homogeneous periodic solutions for the system. First, by analyzing the corresponding ODE system, we derive the existence and stability conditions of Hopf bifurcation, and prove the existence of periodic solutions in the perturbed system. Second, by applying Floquet theory and the implicit function theorem, we derive the sufficient conditions under which cross-diffusion coefficients destabilize the periodic solutions and induce Turing instability. Theoretical analysis shows that when the cross-diffusion coefficients and self-diffusion coefficients satisfy a certain inequality relationship, the system undergoes Turing instability at periodic solutions and forms spatial patterns. Finally, we conduct numerical simulations to confirm and illustrate the theoretical findings and reveal the effect of different diffusion mechanisms on the formation of spatial patterns in the system. This study not only deepens the understanding of the spatiotemporal dynamics of predator–prey systems with Allee effects and cross-diffusion but also provides a theoretical basis for the mechanisms of population distribution and pattern formation in ecology.