Spatio-Temporal Dynamics in a Predator–Prey Model with Generalist Predator and Two Nonlocal Delays

In this paper, a predator–prey system with generalist predator and two nonlocal delays is investigated. For the nondelayed system with nonlocal competition, we find that diffusion may induce both Turing bifurcation and Hopf bifurcation, leading to stability switching. For the system with one nonlocal delay, the stable and unstable regions for the positive equilibrium are identified, and the bifurcation curves are determined, including mode-1 and mode-0 Hopf bifurcations, as well as Turing bifurcation. Stability switching may arise from the combined effects of diffusion and nonlocal delay. In the system with two nonlocal delays, Hopf bifurcation curves are found. The explicit formula of normal form for the double Hopf bifurcation induced by two nonlocal delays is derived. It is shown that the model exhibits complex dynamical behaviors, including stability switching of coexisting equilibria, coexistence of spatially homogeneous and inhomogeneous periodic solutions, coexistence of two spatially inhomogeneous periodic solutions, and the emergence of quasi-periodic solutions.