Positive solutions for predator-prey equations with nonlinear diffusion rates

In this paper, we will investigate the existence of positive solutions to the predator-prey interacting system $$ {-\varphi(x, u)\Delta u = uf(x, u, \upsilon) in \Omega {-\psi(x, \upsilon)\Delta\upsilon = \upsilon g(x, u, \upsilon) {\frac{\partial n}{\partial u} + ku = 0 on \partial\Omega {\frac{\partial n}{\partial\upsilon} + \sigma\upsilon = 0. $$ in a bound region $\Omega$ in $R^n$ with smooth boundary, where $\varphi$ and $\psi$ are strictly positive functions, serving as nonlinear diffusion rates, and $k, \sigma > 0$ are constants.