Resumen de Turing-Hopf Bifurcation in a Predator-Prey Model with Nonlocal Competition and General Group Defence

Ali Rehman, Biao Liu, Ranchao Wu

• This paper investigates the dynamics of a prey-predator model with group defence subjected to nonlocal intra-specific competition, extending the classical diffusion model to incorporate more realistic ecological interactions. The nonlocal effect leads to the formation of complex spatiotemporal patterns. It is shown that the presence of nonlocal competition destabilizes the equilibrium point and induces Turing instability, which cannot occur in the original model. Through stability analysis, we not only derive the conditions for instability but also find the conditions under which the system exhibits Turing bifurcation, Hopf bifurcation and codimension-2 Turing-Hopf bifurcation. Furthermore, we focus on the analysis of Turing-Hopf bifurcation and obtain the normal form using the method of center manifold reduction for nonlocality. To give details of the spatiotemporal dynamics near the Turing-Hopf bifurcation point, the parameter plane is divided into six regions, with the help of the normal form at the Turing-Hopf bifurcation point, spatially homogeneous and inhomogeneous periodic solutions are shown to exist. Finally, numerical simulations are provided to support the theoretical results.