Exploring threshold dynamics in a spatially heterogeneous ecosystem with memory-based diffusion and hunting cooperation on predators

Recognizing the threshold dynamics of highly developed animals with memory is significant for the governance of species within a specific domain. To investigate how the memory threshold affects population behavior, we formulate a spatially heterogeneous predator-prey system with memory-based diffusion and hunting cooperation on predators. In homogeneous environments, the occurrence conditions of Turing bifurcation and spatially inhomogeneous Hopf bifurcations respectively induced by the memory-based diffusion coefficient and the average memory period at coexistence constant steady states are investigated. Then, in heterogeneous environments, the stability of predator-free steady state is studied by the variational characterization of the principal eigenvalue, and the explicit expression of coexistence steady states is established by the implicit function theorem. In both homogeneous and heterogeneous environments, as the average memory period is beyond the thresholds, the spatially inhomogeneous periodic solutions occur by numerical simulations. Moreover, the increase of cooperative hunting can improve the predation rate of predators, thereby leading to the emergence of periodic solutions. It is worth noting that the introduction of heterogeneous environments results in a transition in spatial patterns from predator-free steady states to spatially inhomogeneous solutions, which biologically indicates that the heterogeneous environments are more conducive to predator invasion than homogeneous ones.