Bifurcations in a predator-prey Model with Diffusion and Memory

In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allee effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.