Dynamical Behavior Of A Class Of Three Species Predator-prey System

This paper is concerned with the following three species predator-prey system where x ∈ Ω, t>0.Firstly, we consider the stability of the positive equilibrium when Ω is a bounded domain and u1,u2,u3 satisfy We obtain the global stability of the positive equilibrium with non-negative initial-value by using the invariant domain.Secondly, we discuss the traveling wave solutions when Ω=R. By using the Schaud-er’s fixed point theorem and constructing upper and lower solutions, we prove the existence of non-negative traveling wave solutions connecting equilibrium (1,0,0) with positive e-quilibrium (k1,k2,k3) if c>c*, during which the asymptotic behavior of traveling wave solutions is confirmed by using the technique of contracting rectangles. Moreover the nonexistence of traveling wave solutions with c< c* is established by using the relevant conclusions of the asymptotic speed of spreading. Thus, we obtain the minimal wave speed of non-negative traveling wave solutions. From the viewpoint of population dy-namics, the traveling wave solutions discussed here can describe the following biological process:at any fixed location, there was only one kind of prey, and the predator and two preys will coexist after a long-term species interaction if we introduce a predator and another prey.