Boundedness of solutions in a predator–prey system with density-dependent motilities and indirect pursuit–evasion interaction

Abstract

This paper is concerned with the predator–prey system with density-dependent motilities and indirect pursuit–evasion interaction ut=Δ(d1(w)u)+u(λ−u+av),x∈Ω,t>0,vt=Δ(d2(z)v)+v(μ−v−bu),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0in a smooth bounded domain Ω⊂Rn(n≥1) with homogeneous Neumann boundary conditions, where the parameters λ,μ,a and b are positive constants. The global existence and the boundedness of the classical solution are established in two dimensions if the motility functions d1(w) and d2(z) satisfy the following hypotheses•d1(w)∈C3[0,∞), d1(w)>0, d1′(w)<0 for all w≥0,•d2(z)∈C3[0,∞), d2(z)>0, d2′(z)>0 and d2′(z) is bounded for all z≥0. However, further assume that the motility function d2(z) fulfills more rigorous conditions•d2(z)∈C3[0,∞), 0<d2(z)<η, d2′(z)>0 and d2′′(z)<0 for all z≥0 with η>0.It was also proved that the global solutions are uniformly bounded with respect to time in higher dimensions.