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The current work is the third of a series of three papers devoted to thestudy of asymptotic dynamics in the space-time dependent logistic sourcechemotaxis system, $$ \begin{cases} \partial_tu=\Deltau-\chi\nabla\cdot(u\nabla v)+u(a(x,t)-b(x,t)u),\quad x\in R^N,\cr 0=\Deltav-\lambda v+\mu u ,\quad x\in R^N, \end{cases} (0.1) $$ where $N\ge 1$ is apositive integer, $\chi, \lambda$ and $\mu$ are positive constants, thefunctions $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of theseries, we studied the phenomena of persistence, and the asymptotic spreadingfor solutions. In the second of the series, we investigate the existence,uniqueness and stability of strictly positive entire solutions.  In the current part of the series, we discuss the existence of transitionfront solutions of (0.1) connecting $(0,0)$ and $(u^*(t),v^*(t))$ in the caseof space homogeneous logistic source. We show that for every $\chi>0$ with$\chi\mu\big(1+\frac{\sup_{t\in R}a(t)}{\inf_{t\in R}a(t)}\big)<\inf_{t\inR}b(t)$, there is a positive constant $c^{*}_{\chi}$ such that for every$\underline{c}>c^{*}_{\chi}$ and every unit vector $\xi$, (0.1) has atransition front solution of the form$(u(x,t),v(x,t))=(U(x\cdot\xi-C(t),t),V(x\cdot\xi-C(t),t))$ satisfying that$C'(t)=\frac{a(t)+\kappa^2}{\kappa}$ for some number  $\kappa>0$, $\liminf_{t-s\to\infty}\frac{C(t)-C(s)}{t-s}=\underline{c}$,and$$\lim_{x\to-\infty}\sup_{t\in R}|U(x,t)-u^*(t)|=0 \quad \text{and}\quad\lim_{x\to\infty}\sup_{t\in R}|\frac{U(x,t)}{e^{-\kappa x}}-1|=0.$$Furthermore,we prove that there is no transition front solution$(u(x,t),v(x,t))=(U(x\cdot\xi-C(t),t),V(x\cdot\xi-C(t),t))$ of (0.1) connecting$(0,0)$ and $(u^*(t),v^*(t))$ with least mean speed less than$2\sqrt{\underline{a}}$, where  $\underline{a}=\liminf_{t-s\to\infty}\frac{1}{t-s}\int_{s}^{t}a(\tau)d\tau$.

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. III. Transition fronts