Refuge versus dispersion in the logistic equation

In this paper we consider a logistic equation with nonlinear diffusion arising in population dynamics. In this model, there exists a refuge where the species grows following a Malthusian law and, in addition, there exists also a non-linear diffusion representing a repulsive dispersion of the species. We prove existence and uniqueness of positive solution and study the behavior of this solution with respect to the parameter λ, the growth rate of the species. Mainly, we use bifurcation techniques, the sub-supersolution method and a construction of appropriate large solutions.