Abstract:

In this paper, a nonlocal Leslie-Gower predator-prey model with delay and diffusion is investigated. Firstly, the local stability of the steady states in the model is studied with aid of the zeros in the characteristic equations. Besides, the existence of Hopf bifurcation is explored by taking $\tau$ as the varying parameter. Hopf-Hopf bifurcation singularity in the model was analyzed by choosing $tau$ and the capture rate $m$ as dual variable parameters. Furthermore, the normal form near the Hopf-Hopf singularity of the model is derived by using the central manifold theory. Finally, numerical simulations are carried out to illustrate the obtained theoretical results.The study reveals that the joint effect of two varying parameters can lead to stable spatially non-homogeneous periodic solutions in the system, which indicates that the dynamical behavior near the Hopf-Hopf singularity plays an important role in the formation and evolution of spatiotemporal patterns in the Leslie-Gower system.

Key words: nonlocal, Leslie-Gower model, delay, Hopf-Hopf bifurcation, normal form.