Existence On A Class Of A Cross-diffusion Predator-prey System With Modified Holling Ⅱ Response Function

Space environment has important theoretical significance for ecological popula-tion.In ecosystem,predators always migrate from high-density areas to low-density areas.In high-density areas of predators,predator population pressure decreases.Therefore,adding cross-diffusion items to predator-prey model can make the model more reasonable.When predator and prey species are added,the predator-prey model will become more reasonable.When the number of groups is too large,the competition for resources will intensify,which is not conducive to the protection of the ecological environment.Therefore,the Holling Ⅱ type response function is added.Taking into account the constraints between the two species,the HollingⅡ type response function is improved,and the modified Holling Ⅱ type response function is studied.In this paper,we study the existence and nonexistence of coexistence solutions for a staggered diffusive predator-prey system with a modified Holling Ⅱ response function under the Dirichlet boundary condition.A priori estimate is made for the positive solution of the steady-state system,and the range of the system solution is given by using the principle of comparison and the maximum principle.Then the index of the fixed point under different conditions is calculated from the range of the model solution,and the existence and non-existence of the coexistence solution are discussed by using the degree theory.Finally,the asymptotic behavior of the coexistence solution about the capture coefficient is proved.