Asymptotic Behavior Analysis Of Positive Steady-State Solutions For A Predator-Prey System With A Protection Zone In Inhomogeneous Space

As early as the twenties of the twentieth century,scholars began to use mathematical models to study biological populations,prey-predator model is one of them,population size has always been a concern for biologists in this model.With further research,in order to more accurately describe the situation of predator species chasing prey species and prey species hiding from predator species,cross-diffusion term is introduced in the model.With the development of biological populations,in order to protect endangered species,scholars have established protection zones in the model,this has important implications for the conservation of endangered species.In this paper studies asymptotic behavior problem of positive steady-state solutions for a predator-prey system with a protection zone in inhomogeneous space,where the interaction is affected by a Holling type Ⅱ functional response.Firstly,it revolves around the coexistence region of positive solutions,and the prior estimates of positive solutions are obtained through the maximum value principle,the non-existence regions of positive steady states are obtained by the comparison principle of eigenvalues,then the existence results of positive solutions are obtained through the bifurcation theorem.We next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species μ→∞.The conclusion shows that positive solution(uμ,vμ)of the system uμ converges to θλ in Ω0,vμ converges to 0 in Ω1 and vμ tends to infinity in Ω1.It shows that when the growth rate of predators is large,outside the protection zone,prey species extinction and the number of predators will be infinite;inside the protection zone,prey species stabilize at a bounded coefficient.Finally,we discuss the stability of trivial and semitrivial solutions of system.