Dynamics of a general reaction-diffusion system with spatial memory and functional responses containing two delays

Spatial memory benefits populations by enabling more efficient foraging, reducing predation risk, and improving habitat selection, which in turn enhances survival and reproduction. In this paper, a novel diffusive two-species system incorporating spatial memory delay and functional responses with two discrete delays is investigated. The primary focus is on analyzing how the memory-based diffusion coefficient and delays influence the system’s dynamic behaviors. By choosing the spatial memory and two functional response delays as bifurcation parameters, we show that delays can destabilize the positive equilibrium and induce both spatially homogeneous and spatially heterogeneous periodic solutions through Hopf bifurcation when the memory-based diffusion coefficient remains in a range excluding Turing instability. Furthermore, we characterize the associated stability switching curves generated by the combined effects of the two delay parameters. To illustrate these theoretical results, we apply the general framework to a delayed predator-prey system with Holling type-II functional response. Numerical simulations reveal rich spatiotemporal patterns and show how the stability switching curves partition the parameter space into regions of stable and oscillatory dynamics.