Global dynamics of a three-species spatial food chain model with alarm-taxis and logistic source

Abstract

This work considers the following food chain model with alarm-taxis and logistic source ut=d1Δu+u(1−u)−b1uv,x∈Ω,t>0,vt=d2Δv−ξ∇⋅(v∇u)+uv−b2vw+θ1(v−v2),x∈Ω,t>0,wt=Δw−χ∇⋅(w∇(uv))+vw+θ2(w−w2),x∈Ω,t>0in a smoothly bounded domain Ω⊂RN(N≥3) with zero-flux boundary conditions, where the parameters di, bi, θi(i=1,2), ξ and χ are positive constants. If the positive parameters θ1 and θ2 are large enough, we show that this model possesses a globally bounded classical solution. Moreover, by constructing the Lyapunov functionals, we prove the global stability of the semi-coexistence steady states and the coexistence steady state under suitable conditions on parameters.