Global existence and stability of three species predator-prey system with prey-taxis

Abstract

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e., under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn(n ≧ 1) with smooth boundary, where the parameters d, η, r, µ, χ1, χ2, ai > 0, i = 1,..., 6. We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle.