  We study a spatial (two-dimensional) Rosenzweig-MacArthur model under the
following assumptions: $(1)$ prey movement follows a nonlinear diffusion, $(2)$
preys have a refuge zone (sometimes called "protection zone") where predators
cannot enter, (3) predators move following linear diffusion. We present a
bifurcation analysis for the system that shows the existence of positive
solutions at the steady state. We complement the theoretical results with
numerical computations and compare our results with those obtained in the case
of having linear diffusion for the prey movement. Our results show that both
models, with linear and nonlinear diffusion for the prey, have the same
bifurcation point and the positive solution curves are virtually the same in a
neighborhood of this point, but they get drastically different as the
bifurcation parameter approaches to zero.
