Nontrivial Solutions Of Prey-predator Systems With Small Cross-diffusion

In this paper,a predator-prey system with strongly coupled nonlinear reactiondiffusion terms is considered.First,some priori estimates for steady state solutions are indicated.Secondly,we give a necessary condition and a sufficient condition for the existence of positive steady state solutions,then a local bifurcation result is given. Finally,we discuss the structure of solutions with the diffusion coefficient as bifurcation parameters.