Global boundedness of a diffusive prey-predator model with indirect prey-taxis and predator-taxis

Abstract

This paper considers a general reaction-diffusion predator-prey model with indirect prey-taxis and predator-taxis{ut=dΔu−χ∇⋅(u∇m)+cϕ(u,v)−g(u),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇w)+f(v)−ϕ(u,v),x∈Ω,t>0,τwt=Δw+βu−γw,x∈Ω,t>0,τmt=Δm+δv−ρm,x∈Ω,t>0, under homogeneous Neumann boundary condition in a smooth bounded domain Ω∈Rn. Here the parameters d,χ,ξ,c,β,γ,δ,ρ are positive, τ∈{0,1}, and the functions f(v), g(u), ϕ(u,v) satisfy some general assumptions. For the system with τ=1, under the assumption on the spatial dimension n≤3 and appropriate conditions on the parameters, the system possesses a classical solution which is global in time and bounded. Furthermore, for the system with τ=0, under the assumption on the arbitrary spatial dimension n and appropriate conditions on the parameters, the globally bounded solution is also obtained.