  This work is the second of the series of three papers devoted to the study of
asymptotic dynamics in the chemotaxis system with space and time dependent
logistic source,$$\partial_tu=\Delta u-\chi\nabla\cdot(u\nabla
v)+u(a(x,t)-ub(x,t)),\quad 0=\Delta v-\lambda v+\mu u ,\ x\in\mathbb{R}^N, \
(1) $$where $N\ge1$ is a positive integer, $\chi,\lambda,\mu>0$, and the
functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the
series, we studied the phenomena of pointwise and uniform persistence, and
asymptotic spreading for solutions with compactly supported or front like
initials. In the second of the series, we investigate the existence, uniqueness
and stability of strictly positive entire solutions. In this direction, we
prove that, if $0\le\mu\chi<b_\inf$, then (1) has a strictly positive entire
solution, which is time-periodic (respectively time homogeneous) when the
logistic source function is time-periodic (respectively time homogeneous).
Next, we show that there is positive constant $\chi_0$, such that for every
$0\le\chi<\chi_0$, (1) has a unique positive entire solution which is uniform
and exponentially stable with respect to strictly positive perturbations. In
particular, we prove that $\chi_0$ can be taken to be $b_\inf/2\mu$ when the
logistic source function is either space homogeneous or the function
$b(x,t)/a(x,t)$ is constant. We also investigate the disturbances to Fisher-KKP
dynamics caused by chemotatic effects, and prove
that$$\sup_{0<\chi\le\chi_1}\sup_{t_0\in\mathbb{R},t\ge
0}\frac{1}{\chi}\|u_{\chi}(\cdot,t+t_0;t_0,u_0)-u_0(\cdot,t+t_0;t_0,u_0)\|_{\infty}<\infty$$for
every $0<\chi_1<b_\inf/\mu$ and every uniformly continuous initial function
$u_0$, with $u_{0\inf}>0$, where
$(u_\chi(x,t+t_0;t_0,u_0),v_\chi(x,t+t_0;t_0,u_0))$ denotes the unique
classical solution of (1) with $u_\chi(x,t_0;t_0,u_0)=u_0(x)$, for every
$0\le\chi<b_\inf$.
