In this article, a parabolic-elliptic system of partial differential equations arising in chemotaxis with non-constant monotone chemotactic sensitivity is analyzed. Let \(\Omega \) be a bounded and regular domain, u the density of a biological species and v the concentration of a chemical satisfying the parabolic-elliptic system \(\begin{aligned} \left\{ \begin{array}{l} \displaystyle u_{t} - \Delta u = - div (u\chi (v) \nabla v) + \mu u (1- u), \; \; t>0, \; x\in \Omega , \\ \displaystyle -\Delta v+ v = u, \; \; t>0, \; x\in \Omega \end{array}\right. \end{aligned}\) under Neumann boundary conditions, and bounded and positive initial data. We study the asymptotic behaviour of solutions under suitable assumptions in \(\chi \chi >0; \; \; \chi ^{\prime } \le 0; \; \; \; \chi ^{\prime \prime } \ge 0\) when \(\mu \) is sufficiently large for a given initial data \(u_0\) . The result is obtained by using the system of ordinary differential equations: \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{d \overline{u}}{dt} = \chi (\underline{u}) (\overline{u} -\underline{u})\overline{u}- \chi ^{\prime } (\underline{u}) c^2_{\Omega } ( 2+ \max \{1, \Vert u_0\Vert _{L^{\infty }(\Omega )} \}) ( \overline{u} - \underline{u} ) \overline{u} + \mu \overline{u}(1-\overline{u}), & t>0, \\ \displaystyle \frac{d\underline{u}}{dt} = \chi (\underline{u}) (\underline{u} -\overline{u}) \underline{u} + \mu \underline{u}(1-\underline{u}), & t>0; \end{array} \right. \end{aligned}\) and a comparison method to obtain \(\begin{aligned} \underline{u}(t)< u(t,x)<\overline{u}(t); \; \; \underline{u}(t)< v(t,x) < \overline{u}(t), \text{ a.e. } x\in \Omega , \; t>0. \end{aligned}\) The asymptotic behaviour of the system is also analyzed to obtain \(\begin{aligned} \lim _{t \rightarrow +\infty } \Vert u-1\Vert _{L^{\infty }(\Omega )} + \Vert v-1\Vert _{L^{\infty }(\Omega )}=0. \end{aligned}\)
