Stability and spatiotemporal dynamics in a diffusive predator–prey model with nonlocal prey competition

Abstract

In this paper, we investigate the influence of the nonlocal intraspecific competition of the prey on the dynamics of the diffusive Rosenzweig–MacArthur model with Holling type II functional response. Using the linear stability analysis, the conditions for the positive constant steady state to remain stable and to undergo Turing–Hopf bifurcation have been studied under the Neumann boundary conditions. We find that the introduction to the nonlocal term can produce Turing patterns, which cannot occur in the original model. Furthermore, we are interested in the interaction of Turing bifurcation and Hopf bifurcation. We also develop the algorithm of the normal form of the Turing–Hopf bifurcation for the model with nonlocality. By applying the developed normal form, the dynamical classification near the Turing–Hopf bifurcation point can be analytically determined. The stable spatially inhomogeneous steady states, stable spatially inhomogeneous periodic solutions and unstable spatially inhomogeneous periodic solutions are found. Especially, we find that two stable spatially inhomogeneous steady states and one stable spatially inhomogeneous periodic solution can coexist for appropriate parameters and that there are transitions from one unstable solution to another stable one.