Predator density-dependent prey dispersal in a patchy environment with a refuge for the prey

In this article, we examine a two-patch predator-prey model which incorporates a refuge for the prey. We suppose that prey migration is dependent on predator density, according to a general function. We consider two different time scales in the dynamics of the model, a fast one describing patch to patch migration, and a slow one involving local prey and predator interaction. We take advantage of the time scales to reduce the dimension of the model by use of methods of aggregation of variables, and thereby examine the effect of predator density-dependent migration of prey on the stability of the predator-prey system. We establish a simple criterion of viability, namely, the existence of a positive and globally stable equilibrium, and show that density dependence has beneficial effects on both species by providing larger equilibrium densities.