Stationary, non-stationary and invasive patterns for a prey-predator system with additive Allee effect in prey growth

This study deals with the pattern formation scenarios of a reaction-diffusion system of prey-predator interactions with Allee effect in prey growth, Holling type-II functional response and density dependent death rate for predator. The necessary conditions, such as the presence of Hopf and Turing bifurcations, to get pattern formation are rendered and exhaustive numerical simulations are performed to validate the analytical results. Numerical simulations affirm the existence of stationary patterns such as hot spot, cold spot, labyrinthine and mixture of spot and stripe patterns along with the non-stationary patterns such as spatiotemporal chaotic and quasi-periodic patterns in presence of both weak and strong Allee effects in prey growth. Numerical simulations also reveal the coexistence of both species (either stationary or chaotic in nature depending on the values for diffusion coefficients) in contrast to the extinction scenario for the corresponding temporal model. Further, we investigate the invasion of exotic species for the considered model for strong Allee effect. In this case, we find out the patchy invasion scenario apart from the invasion through propagation of continuous population fronts which is a common invasion scenario for deterministic prey-predator models. In the patchy invasion regime, we observe a striking patchy spatial distribution of species in the wake of an expanding population front as well as complete patchy invasion.