Effect of dispersal in two-patch prey-predator system with positive density dependence growth of preys

Prey-predator systems in patchy environment, connected through dispersal between patches is a very common phenomenon observed in nature, which have a significant impact in ecology, species persistence and extinction, etc. In the present paper, we consider a two patch prey-predator system where the patches are connected through dispersal between preys populations only. We consider positive density dependence growth for preys population. In addition, we consider the time scale difference (different life span) between preys and predator populations. From our study, we can conclude that dispersal can save both the populations from extinction, when in a single patch initial preys density is lower the Allee threshold. Also, time difference can increase the basin of attraction of the coexistence equilibrium of our two-patch model. Time scale difference also can help to reach the steady state faster than the without time scale difference, and it also causes the amplitude death when populations are in limit cycle oscillation. We also analyze our model by considering the time delay in dispersal dynamics, and we show that delay induced dispersal can stabilize the system and cause the amplitude death when individual populations are in the limit cycle, without dispersal. In addition, dispersal in non-identical patches can stabilize at its interior equilibrium even if the environment is harsh for both the populations in both the individual patches.