Asymptotic behavior in a forager-exploiter model with nonlinear resource consumption with/without general logistic sources

Abstract

This work is concerned with a forager-exploiter model with nonlinear resource consumption with/without general logistic sources{ut=Δu−χ∇⋅(u∇w)+a1u−b1uα,x∈Ω,t>0,vt=Δv−ξ∇⋅(v∇u)+a2v−b2vβ,x∈Ω,t>0,wt=Δw−u+v(1+u+v)γw−μw+r(x,t),x∈Ω,t>0 in a bounded smooth domain Ω⊂Rn (n≥2) with homogeneous Neumann boundary conditions, where the parameters ai,bi≥0 (i=1,2), and χ,ξ,γ,μ are some given positive constants, and α,β>1, r∈C2(Ω¯×[0,∞))∩L∞(Ω×(0,∞)) is a given nonnegative function. The present work shows that, if α,β,γ satisfy{α,β>(n+2)(1−γ)2,if0<γ<nn+2,α,β>1,ifγ≥nn+2, then the corresponding initial-boundary value problem admits a unique global bounded classical solution. Furthermore, the paper also establishes the large time behavior of the classical solution to the above model with/without general logistic sources in the case of the function r is a positive constant or has suitable decay conditions.