Narrow Results

The diffusive Holling–Tanner predator–prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatially nonhomogeneous periodic solutions, which is induced by nonlocal prey competition. In particular, the constant positive steady state may lose the stability through Hopf bifurcation when the given parameter passes through some critical values, and the bifurcating periodic solutions near such values could be spatially nonhomogeneous and orbitally asymptotically stable.

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions.

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.