Qualitative Analysis of a Predator-prey Systemwith Ratio-dependent and Modified Leslie-GowerFunctional Response

In this paper, a predator-prey model with ratio-dependent andmodified Leslie-Gower functional response subject to homogeneous Neumannboundary condition is considered. First, properties of the constant positive sta-tionary solution are shown, including the existence, nonexistence, multiplicityand stability. In addition, a comparatively characterization of the stability isobtained. Moreover, the existing result of global stability is improved. Finally,properties of nonconstant positive stationary solutions are further studied. Bya priori estimate and the theory of Leray-Schauder degree, it is shown thatnonconstant positive stationary solutions may exist when the system has twoconstant positive stationary solutions.