Coexistence of heterogeneous predator-prey systems with prey-dependent dispersal

This paper is concerned with existence, non-existence and uniqueness of positive (coexistence) steady states to a predator-prey system with prey-dependent dispersal. To overcome the analytical obstacle caused by the cross-diffusion structure embedded in the prey-dependent dispersal, we use a variable transformation to convert the problem into an elliptic system without cross-diffusion structure. The transformed system and pre-transformed system are equivalent in terms of the existence or non-existence of positive solutions. Then we employ the index theory alongside the method of the principle eigenvalue to give a nearly complete classification for the existence and non-existence of positive solutions. Furthermore we show the uniqueness of positive solutions and characterize the asymptotic profile of solutions for small or large diffusion rates of species. Our results pinpoint the positive role of prey-dependent dispersal on the population dynamics for the first time by showing that the prey-dependent dispersal in the predator-prey system is a beneficial strategy increasing the chance of predator's survival and hence promoting the coexistence of species.