Blow-up, quenching, aggregation and collapse in a chemotaxis model with reproduction term

In this paper, we consider the following chemotaxis model with ratio-dependent logistic reaction term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} \tfrac{{\partial u}} {{\partial t}} = D\nabla (\nabla u - u\tfrac{{\nabla w}} {w}) + u(a - b\tfrac{u} {w}), (x,t) \in Q_T , \hfill \\ \tfrac{{\partial u}} {{\partial t}} = \beta u - \delta w, (x,t) \in Q_T , \hfill \\ u\nabla \ln (\tfrac{u} {w}) \cdot \vec n = 0, x \in \partial \Omega 0 < t < T, \hfill \\ u(x,0) = u_0 (x) > 0, x \in \bar \Omega , \hfill \\ w(x,0) = w_0 (x) > 0, x \in \bar \Omega , \hfill \\ \end{gathered} \right.$$\end{document} It is shown that the solution to the problem exists globally if b + β ≥ 0 and will blow up or quench if b + β < 0 by means of function transformation and comparison method. Various asymptotic behavior related to different coefficients and initial data is also discussed.