Dynamics of a delayed reaction–diffusion predator–prey model with nonlocal competition and double Allee effect in prey

Abstract

In this paper, we study the effects of the nonlocal competition and double Allee effect in prey on a diffusive predator–prey model. We investigate the local stability of coexistence equilibrium in the predator–prey model by analyzing the eigenvalue spectrum. We study the consequence of double Allee effect on the prey population. Also, we discuss the existence of Hopf bifurcation under different parameters by using the gestation time delay of predators as a bifurcation parameter and analyzing the distribution of eigenvalues. By utilizing the normal form method and center manifold theorem, we have given some conditions that could determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Through our research, we obtain that the double Allee effect can affect the coexistence of prey and predator and induce periodic oscillations of the densities of prey and predator. In addition, the nonlocal competition can also affect the stability and homogeneity of the solutions, but may receive the consequences of the Allee effect as well as time delay. In the numerical simulation part, we further demonstrate the correctness of this conclusion by comparison.