Dynamics of a predator-prey system with three species

This paper is concerned with the dynamics of a predator-prey system with three species. When the domain is bounded, the global stability of positive steady state is established by contracting rectangles. When the domain is R, we study the traveling wave solutions implying that one predator and one prey invade the habitat of another prey. More precisely, the existence of traveling wave solutions is proved by combining upper and lower solutions with a fixed point theorem, and the asymptotic behavior of traveling wave solutions is obtained by the idea of contracting rectangle. Moreover, we show the nonexistence of traveling wave solutions by applying the theory of the asymptotic spreading.