Effect Of Cross-diffusion In The Prey-predator Model With Holling ? Response Function

The phenomenon of cross-diffusion exists widely,such as population problem,infectious disease problem and so on.Therefore,many previous scholars have stud-ied the influence of cross-diffusion not only in the academic branch but also in many practical problems in life.Among them,the effect of cross-diffusion in the diffusion prey-predator model gets a lot of attention,scholars have also made valuable and classic achievements.These results have not only advanced the development of the branch of mathematics but also can reflect the nature of some phenomena.Thus in life it has the extremely high application value such as the study of population development and the prevention and control of virus pestsThis paper,in the Neumann boundary conditions,we study the effect of cross-diffusion in the nonlinear diffusion prey-predator model with Holling ? type re-sponse function and protected zone.We are talking about predator steady-state First,we discuss the existence of non-trivial positive solutions,by means of the maximum value principle and global bifurcation theorem,the prior estimates the coexistence region of positive solutions.Then we study the asymptotic behavior of the positive solution when the intrinsic rate of growth of predator ??? and the parameters in staggered diffusion term ??? and l??,then the growth trend of prey and predator is given in combination with biological significance.Finally,we will get the shadow system and explain the global bifurcation structure of the system roughly.