Stationary Partterns For A Predator-prey Model With Difusion And Cross-difusion

In recent years, a lot of great achievements have been made on the research ofmathematical models in ecology, and the theoretical results have been successfullyapplied to the prevention and control of infectious diseases and species. Alongwith much further research, it has greatly interested mathematicians to solve theecological problems acquired from mathematical models. With realistic problemstaken as the background, the models have got a vast source of materials andtremendous practical significance. In this paper we mainly study the reactionof biochemistry in natural science, and the changes acting on the whole systemresulting from interactions of such factors as ecosystem. Our mathematical modelis exactly to describe these realistic problems in abstract mathematical language,and to study the nature of these issues by mathematical methods.The issue investigated in this paper is a predator-prey model with difusionand cross-difusion under the homogeneous Neumann boundary condition. First,the apriori estimate of positive solutions is given by the application of Harnackinequality and the maximum principle. Then, we use the integral properties toprove the non-existence of nontrivial solutions. At last, the existence of nontrivialsolutions is presented from Leray-Schauder degree theory.