Stability and bifurcation of a ratio-dependent prey-predator system with cross-diffusion

This paper is purported to investigate a ratio-dependent prey-predator system with cross-diffusion in a bounded domain under no flux boundary condition. The asymptotical stabilities of nonnegative constant solutions are investigated to this system. Moreover, without estimating the lower bounds of positive solutions, the existence, multiplicity of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function and inverse function.