Hopf-Hopf bifurcation in a predator-prey model with nonlocal competition and refuge in prey
Yuxin Ma, Ruizhi Yang
Abstract
In this paper, a diffusive predator-prey model with nonlocal competition and prey refuge is considered. The influence of parameters on the existence, multiplicity and stability of nonhomogeneous steady-state solutions is studied. It is obtained that an unstable positive nonconstant steady state exists in the neighborhood of the positive constant steady state. Compared with the model without nonlocal competition, the model with nonlocal competition can generate Hopf-Hopf bifurcation under some conditions. Through the qualitative analysis, the normal form at the Hopf-Hopf bifurcation singularity is calculated to analyze the different dynamic properties exhibited by the model in different parameter regions. In order to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the nonlocal competition, numerical simulations are carried out. Through the numerical simulations, it is further shown that under certain conditions, the nonlocal competition will lead to the stablly spatially non-homogeneous periodic solutions and stablly spatially non-homogeneous quasi-periodic solutions.