Bifurcation analysis in a diffusive predator-prey model with spatial memory of prey, Allee effect and maturation delay of predator

In this paper, we formulate a spatial model with memory delay of the prey, Allee effect and maturation delay with delay-dependent coefficients of predators. We first explore the model without delays and diffusions, and find that it can undergo a saddle-node bifurcation when the intensity of Allee effect is at the tipping point. Then for the scenario of stability of the coexistence steady state without delays, we obtain the crossing curves on the delays plane. The model can undergo Hopf bifurcation when delays pass through these crossing curves from a stable region to an unstable one. We further calculate the normal form of Hopf bifurcation and hence obtain the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. It is shown that the model can possess multiple stability switches and a stable spatially heterogeneous periodic solution with mode-4 as delays vary.