Boundedness of Solutions to a Fully Parabolic Indirect Pursuit–Evasion Predator–Prey System with Density-Dependent Diffusion in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{R}}}^2$$\\end{document}

Abstract

This paper deals with a fully parabolic indirect pursuit–evasion predator–prey system with density-dependent diffusion ut=Δ(ψ1(w)u)+u(λ-u+αv),vt=Δ(ψ2(z)v)+v(μ-v-βu),wt=Δw-w+v,zt=Δz-z+u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{t}=\\Delta (\\psi _1(w)u)+u(\\lambda -u+\\alpha v), v_{t}=\\Delta (\\psi _2(z) v)+v(\\mu -v-\\beta u), w_{t}=\\Delta w -w+v, z_{t}=\\Delta z-z+u$$\\end{document} under a smooth bounded domain Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb{R}}^2$$\\end{document} with homogeneous Neumann boundary conditions, where the parameters λ,μ,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda , \\mu , \\alpha$$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} are assumed to be positive. Through the establishment of appropriate conditions for the density-dependent diffusion functions ψ1(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _1(w)$$\\end{document} and ψ2(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _2(z),$$\\end{document} it is revealed that a unique classical solution exists for the corresponding initial-boundary problem, which remains uniformly bounded over time.