Qualitative Analysis Of Two Predator-prey Models With Cross-diffusion

Predator-prey model is a very important biological model to depict the material changes of real world.Due to highly practical value,it has been studied by the researchers all over the world.With the increasing investigation of predator-prey model in recent years,experts have discovered that interaction of the interspecies plays a significant role in the population diffusion.Therefore,a growing number of researchers have focus on the influence of biological model with various cross diffusion term.This article is mainly talk about the properties of the solutions two kinds of predator-prey model with cross diffusion.One kind is predator-prey model with cross diffusion and Monod-Haldane functional response functionThe other one is predator-prey model with cross diffusion and Holling-Tanner functional response functionFor the first model,firstly,we obtain a priori estimates of positive solutions for model(1)by mean of the maximum principle;Furthermore,taking the growth rate of prey as a bifurcation parameter,we give the existence of local bifurcation positive solutions by using local bifurcation theory;Finally,we extend the local bifurcation solution to the global bifurcation solution and make the corresponding numerical simulation for model(1)using matlab software.For the second model,firstly,we obtain a priori estimates of positive solutions for model(2)by means of the maximum principle,On that basis the existence of positive solutions is proved by using the topological degree theory.