This paper studies the quasilinear parabolic–elliptic–ODE chemotaxis–haptotaxis system with logistic source \(\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (u \nabla v)-\xi \nabla \cdot (u \nabla w)+\mu u(1-u-w),\,\,\, & x\in \Omega ,\,\,\, t>0,\\ 0=\Delta v-v+u,\,\,\, & x\in \Omega ,\,\,\, t>0,\\ w_{t}=-vw,\,\,\, & x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}\) under homogeneous Neumann boundary conditions in a bounded domain \(\Omega \subset \mathbb {R}^{n}\) , \(n\ge 3\) , with smooth boundary. \(\chi >0\) , \(\xi >0\) and \(\mu >0\) , D(u) is supposed to satisfy \(D(u)\ge (u+1)^{\alpha }\) with \(\alpha >0\) . Under appropriate regularity assumptions on the initial data \((u_0, v_0,w_0)\) , we prove that the system possesses a global and bounded classical solution for the critical case \(\mu =\frac{n-2}{n}(\chi +\xi )\) . Furthermore, if \(\mu >\frac{\chi ^{2}}{8}\) , it is shown that the corresponding solution exponentially and uniformly stabilizes to the constant stationary solution (1, 1, 0).
