The properties of solutions for predator-prey system with cross-diffusion and fear effects
The steady-state problem of a nonlinear cross-diffusion predator-prey model with fear factor,Allee effect and protection zone under Neumann boundary conditions is studied.Firstly,a prior es-timate of the steady-state solutions is obtained by using the maximum principle,elliptic regularity theory and Sobolev embedding theorem.Secondly,the local stability of trivial solutions and the semi-trivial solu-tions of predator extinction is given by linear stability analysis.Thirdly,by using the local bifurcation the-ory,it is proved that the system has transcritical bifurcation at the boundary equilibrium point of predator extinction.Finally,by applying uniateral global bifurcation theory,the global bifurcation of the system from the boundary equilibrium point of predator extinction is proved.The influence of Allee effect constant and prey diffusion coefficient on the asymptotic behavior of the equilibrium positive solutions of the system is further discussed.The results show that the nonlinear cross-diffusion term,fear effect and Allee effect can jointly promote the stable coexistence of prey and predators.
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