In this note we present a study of the solutions associated to a particular spatial extension of the Rosenzweig-MacArthur model for predator and prey. The analysis presented here shows that positive steady state solutions emerge via a transcritical bifurcation mechanism, in accordance with the insight obtained from previous numerical and analytical results. In the model under discussion, prey is assumed to move avoiding crowds via a density-dependent diffusion and also incorporates the existence of a refuge zone, where predators cannot consume prey. Saturation in prey consumption is also included through a Holling type II functional response.