Boundedness criterion and global solvability for the three-species food chain model with taxis mechanisms

In this paper, we shall investigate a three-species food chain model with taxis mechanisms including prey-taxis and alarm-taxis in a smooth bounded domain Omega subset of Rn(n >= 1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}<^>{n}(n\ge 1)$$\end{document} with homogeneous Neumann boundary conditions. More precisely, we first establish the boundedness criterion for a general food chain model with various taxis mechanisms for arbitrary spatial dimensions by using the semigroup estimates and coupled energy estimates. With the boundedness criterion, we prove the global boundedness of the solution with the general functional response functions under some smallness assumptions on the taxis coefficients by using the weighted energy estimates. On the other hand, for some special functional response functions including Beddington-DeAngelis type, ratio-dependent type and Harrison type, we also obtain the global existence of the solution with uniform-in-time bound without any smallness assumptions on the taxis coefficients or initial data.