This paper deals with a forager–exploiter model involving nonlinear diffusions and volume-filling effects ut=∇·((u+1)m∇u)-∇·(S1(u)∇w),x∈Ω,t>0,vt=∇·((v+1)l∇v)-∇·(S2(v)∇u),x∈Ω,t>0,wt=Δw-(u+v)w-μw+r(x,t),x∈Ω,t>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot ((u+1)^m\nabla u)-\nabla \cdot (S_1(u)\nabla w), ~&x\in \Omega , t>0,\\&v_t=\nabla \cdot ((v+1)^l\nabla v)-\nabla \cdot (S_2(v)\nabla u), ~&x\in \Omega , t>0,\\&w_t=\Delta w-(u+v)w-\mu w+r(x,t), ~&x\in \Omega , t>0 \end{aligned} \right. \end{aligned}$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document} with n≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 1$$\end{document}, where μ>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mu >0$$\end{document}, m,l∈R\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m,l\in \mathbb R$$\end{document} and r∈C1(Ω¯×[0,∞))∩L∞(Ω×(0,∞))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$r\in C^1(\bar{\Omega }\times [0,\infty ))\cap L^\infty (\Omega \times (0,\infty ))$$\end{document} is a given nonnegative function, the initial data u0,v0,w0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$u_0, v_0, w_0$$\end{document} satisfy 0≤u0,v0≤1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0\le u_0, v_0\le 1$$\end{document} and w0≥0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$w_0\ge 0$$\end{document}. Volume-filling effects account for an ordinary form by taking S1(u)=u(1-u),S2(v)=v(1-v).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} S_1(u)=u(1-u), ~S_2(v)=v(1-v). \end{aligned}$$\end{document}It is proved that the corresponding initial-boundary value problem admits a unique global bounded classical solution. Furthermore, if r satisfies ∫tt+1∫Ωr→0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\int \limits _t^{t+1}\int \limits _\Omega r\rightarrow 0$$\end{document} as t→∞\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t\rightarrow \infty $$\end{document}, then the global bounded classical solution (u, v, w) that converges to 1|Ω|∫Ωu0,1|Ω|∫Ωv0,0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( \frac{1}{|\Omega |}\int \limits _\Omega u_0, \frac{1}{|\Omega |}\int \limits _\Omega v_0,0\right) $$\end{document} as t→∞\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t\rightarrow \infty $$\end{document}.
