Global dynamics of an indirect prey-taxis system with an anti-predation mechanism

In this paper, we shall study the following model:{Nt=dNΔN+λN−N2−NP,x∈Ω,t>0,Pt=Δ(d(S)P)+μP−P2+γNP,x∈Ω,t>0,St=dSΔS+τN−ηS,x∈Ω,t>0,∂N∂ν=∂P∂ν=∂S∂ν=0,x∈∂Ω,(N,P,S)(x,0)=(N0,P0,S0)(x),x∈Ω, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary. Based on semigroup estimates and Moser iteration, we first establish the existence of the global classical solution with a uniform-in-time bound. Moreover, we build the global stabilization of constant steady states by constructing suitable Lyapunov functionals and give the decay rates of solutions. Furthermore, we find that the anti-predation mechanism will generate steady state bifurcation. We further prove the global existence of non-constant positive steady state solutions under certain conditions. Finally, we numerically demonstrate spatially inhomogeneous coexistence patterns (i.e., non-constant positive steady state solutions) for the predator, the prey and the signal.