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This work is the second of the series of three papers devoted to the study ofasymptotic dynamics in the chemotaxis system with space and time dependentlogistic source,$$\partial_tu=\Delta u-\chi\nabla\cdot(u\nablav)+u(a(x,t)-ub(x,t)),\quad 0=\Delta v-\lambda v+\mu u ,\ x\in\mathbb{R}^N, \(1) $$where $N\ge1$ is a positive integer, $\chi,\lambda,\mu>0$, and thefunctions $a(x,t), b(x,t)$ are positive and bounded. In the first of theseries, we studied the phenomena of pointwise and uniform persistence, andasymptotic spreading for solutions with compactly supported or front likeinitials. In the second of the series, we investigate the existence, uniquenessand stability of strictly positive entire solutions. In this direction, weprove that, if $0\le\mu\chi<b_\inf$, then (1) has a strictly positive entiresolution, which is time-periodic (respectively time homogeneous) when thelogistic source function is time-periodic (respectively time homogeneous).Next, we show that there is positive constant $\chi_0$, such that for every$0\le\chi<\chi_0$, (1) has a unique positive entire solution which is uniformand exponentially stable with respect to strictly positive perturbations. Inparticular, we prove that $\chi_0$ can be taken to be $b_\inf/2\mu$ when thelogistic source function is either space homogeneous or the function$b(x,t)/a(x,t)$ is constant. We also investigate the disturbances to Fisher-KKPdynamics caused by chemotatic effects, and provethat$$\sup_{0<\chi\le\chi_1}\sup_{t_0\in\mathbb{R},t\ge0}\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$$forevery $0<\chi_1<b_\inf/\mu$ 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 uniqueclassical solution of (1) with $u_\chi(x,t_0;t_0,u_0)=u_0(x)$, for every$0\le\chi<b_\inf$.

Parabolic-elliptic chemotaxis model with space-time dependent logistic  sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of  strictly positive entire solutions