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This paper studies the asymptotic behavior of solutions of theparabolic-parabolic chemotaxis model with logistic-type sources inheterogeneous bounded domains: \begin{equation*}  \label{u-v-eq00} \begin{cases} u_t=\Delta u-\chi\nabla\cdot (u \nablav)+u\Big(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_{\Omega}u\Big),\quad x\in \Omega\cr\tau v_t=\Delta v-\lambda v +\mu u,\quad x\in \Omega \cr \frac{\p u}{\p\nu}=\frac{\p v}{\p \nu}=0,\quad x\in\p\Omega. \end{cases}\qquad(\ast)  \end{equation*} \noindent We find parameter regions in which the system has aunique positive entire solution, which is globally asymptotically stable. {Moreprecisely under suitable assumptions on the model's parameters, the system hasa unique entire positive solution $(u^*(x,t),v^*(x,t))$ such that for any%$t_0\in\RR$ and $u_0 \in C^0(\bar{\Omega}),$ $v_0 \inW^{1,\infty}(\bar{\Omega})$ with $u_0,v_0\ge 0$ and $u_0\not\equiv 0$, theglobal classical solution $(u(x,t;t_0,u_0,v_0)$, $v(x,t;t_0,u_0,v_0))$ of$(\ast)$ satisfies $$  \lim_{t \to \infty}\Big(\sup_{t_0 \in\mathbb{R}}\|u(\cdot,t;t_0,u_0,v_0)-u^*(\cdot,t)\|_{C^0(\bar\Omega)}+\sup_{t_0\in\mathbb{R}}\|v(\cdot,t;t_0,u_0,v_0)-v^*(\cdot,t)\|_{C^0(\bar\Omega)}\Big)=0. $$

Uniqueness and nonlinear stability of entire solutions in a  parabolic-parabolic chemotaxis models with logistic source on bounded  heterogeneous environments