Bifurcations and hydra effect in a patchy ecosystem with asymmetric dispersal

A patchy ecosystem is a biological community of interacting populations, where the subpopulations of (typically) the same species are distributed in spatially distinct habitat patches. Here, we consider a patchy predator–prey model where the prey population can access two distinct habitats; however, the predator population is confined within one habitat. The key ingredients of our model are the possibility of fragile prey extinction at low densities (Allee effect) and asymmetric dispersal between the patches of the prey. Our study examines two different aspects; first, the effects of asymmetric dispersal between the patches on the dynamics of the patch model and the persistence of the species; second, the impact of dispersal on the predator hydra effect that characterizes an increase in population size in response to greater mortality. Our results show that as the dispersal rate increases, the system experiences transitions from multiple coexistence states to unique coexistence state, to oscillatory coexistence state, to predator-free state. Additionally, for a suitable dispersal rate, the patch model shows multistability between different states, where the final state depends on the initial density of the population. The model exhibits the hydra effect at a stable state in the predator species with or without predator self-interference in the presence of dispersal. However, the predator hydra effect manifests only with predator self-interference in the absence of dispersal. The hydra effect is also observed in our model in a stable oscillatory coexistence state. Additionally, we observe multiple hydra effects in which two alternative stable states exhibit hydra effect. The range of the mortality that causes the hydra effect becomes shorter with increasing dispersal rate until it disappears. Thus, the dispersal rate plays a critical role in the dynamics of the multi-patch model and the appearance of the hydra effect.