Qualitative Analysis Of Predator-prey Model With Cross-diffusion Term In Spatially Heterogeneous Environment

In many scientific experiments,the phenomenon that ecosystems are profoundly affected by spatial heterogeneity has been observed.Hence,it is very important and significative to study the heterogeneous effects of the environments on population dynamics.On the other hand,the prey species moves more rapidly to try to avoid the predator species when the predator species is around,but the predator species moves more slowly to try to stay near the prey species in the presence of the prey species.This effect is described by the cross-diffusion term.In this paper,we study the stationary problems for the predator-prey model with nonlinear cross-diffusion in a spatially heterogeneous environment,where the interaction is governed by a Beddington-Deangelis type functional response.Firstly,the prior estimate and the nonexistent region of the positive solution are obtained by the maximum principle and Poincare inequality,respectively.The sufficient conditions for the existence of positive solutions are obtained by Leray Schauder degree theory.Then the bifurcation structure of the positive solution is given by bifurcation theory.Finally,the nonlinear effect of cross-diffusion coefficient α or β on the coexistence region is derived.The coexistence region becomes larger as the cross-diffusion β is sufficiently large,Whereas the coexistence region becomes smaller as the cross-diffusionα is sufficiently large.In addition,when α→∞,we prove that there is only one type of positive solution set,while β→∞,there are two types of positive solution set.