  This paper is concerned with existence, non-existence and uniqueness of
positive (coexistence) steady states to a predator-prey system with
density-dependent dispersal. To overcome the analytical obstacle caused by the
cross-diffusion structure embedded in the density-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 density-dependent dispersal on the population dynamics for the first
time by showing that the density-dependent dispersal is a beneficial strategy
promoting the coexistence of species in the predator-prey system by increasing
the chance of predator's survival.
