Effects Of Cross-Diffusion And Spatial Heterogeneity On Positive Steady-State Solutions

It is well known that the ecological population model is one of the most impor-tant tools for us to understand and investigate the nature, moreover, the property of its positive steady-state solutions is one of the most important topics of the ap-plied mathematicians all the time. Due to the existence and important effects of cross-diffusion and spatial heterogeneity, more and more researchers have begun to study the effects of the two on the positive steady-state solutions. But the non-linearity of cross-diffusion and the appearance of nonconstant coefficients make the investigation very difficult. In particular, the work concerning the investigation of positive steady-state solutions of cross-diffusive system in heterogeneous environ-ment is scarce. This paper is mainly divided into four parts, which are devoted to studying the impact of cross-diffusion and spatial heterogeneity on the positive steady-state solutions.Firstly, this paper discusses the positive steady-state solutions of a predation model with fractional cross-diffusion in homogeneous environment. By the max-imum principle and the Harnack inequality, we obtain a priori estimates of the positive steady-state solutions. By the energy method and Leray-Schauder degree theory, nonexistence and existence of nonconstant positive steady-state solutions are shown respectively. The result reveals that cross-diffusion can create nonconstant positive steady-state solutions.Secondly, we consider the positive steady-state solutions of a predation model with cross-diffusion and a protection zone. By a priori estimate of the positive steady-state solutions and the eigenvalue theory,nonexistence of the positive steady-state solutions is proved under suitable conditions. Then the bifurcation structure of the positive steady-state solutions is given by the bifurcation theory. Finally, the asymptotic behavior of the positive steady-state solutions is given, moreover, the uniqueness and stability of the positive steady-state solutions under suitable conditions are shown. The result implies that cross-diffusion in heterogeneous envi- ronment can change the bifurcation structure of the positive steady-state solutions essentially.Thirdly, this paper considers a cross-diffusive Lotka-Volterra cooperative system in heterogeneous environment, and shows the properties of the positive steady-state solutions under weak cooperation. By the maximum principle, a priori estimate of the positive steady-state solutions is obtained, which further deduces the nonexis-tence of the positive steady-state solutions. Then by the combination of the bifurca-tion theory and the Lyapunov-Schmidt reduction, detailed structure of the positive steady-state solution set is obtained as the cross-diffusion is very large. Finally, by the linearization principle and Hopf bifurcation theory for parabolic equations, the stability of the positive steady-state solutions is determined when d1/d2is sufficiently small and sufficiently large. The result reveals that under large cross-diffusion, the spatial heterogeneity has important effects on both of the existence and the stability of the positive steady-state solutions.Finally, the positive steady-state solution of a cross-diffusive Lotka-Volterra competition system with a protection zone is considered. The local bifurcation theory yields the local bifurcation of the positive steady-state solutions. Then by the combination of the bifurcation theory and the Lyapunov-Schmidt reduction, the detailed structure of the positive steady-state solution set is obtained when the cross-diffusion is very large. In addition, the asymptotic behavior of the positive steady-state solutions is shown by the elliptic theory. The structure of the positive solutions of the limiting system is also shown. The result reveals that cross-diffusion is beneficial for the existence of positive steady-state solutions, which together with spatial heterogeneity can cause much more complicated stationary patterns.