Global dynamics of a three-species spatial food chain model

In this paper, we study the following initial-boundary value problem of a three-species spatial food chain model{ut=d1Δu+u(1−u)−b1uv,x∈Ω,t>0vt=d2Δv−∇⋅(ξv∇u)+uv−b2vw−θ1v,x∈Ω,t>0wt=Δw−∇⋅(χw∇v)+vw−θ2w,x∈Ω,t>0 in a bounded domain Ω⊂R2 with smooth boundary and homogeneous Neumann boundary conditions, where all parameters are positive constants. By the delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data. Moreover by constructing Lyapunov functionals and using LaSalle's invariance principle, we establish the global stability of the prey-only steady state, semi-coexistence and coexistence steady states.