The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic-elliptic chemotaxis system with space and time dependent logistic source,where N ≥ 1 is a positive integer, χ, λ and µ are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series [45], we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initials, and the asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series [46], we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. a(x, t) ≡ a(t) and b(x, t) ≡ b(t)), we proved in [46] that the unique spatially homogeneous strictly positive entire solution (u * (t), v * (t)) of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when 0 < 2χµ < inf t∈R b(t).In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0, 0) and (u * (t), v * (t)) in the case of space homogeneous logistic source. We show that for every χ > 0 with χµ 1 + sup t∈R a(t) inf t∈R a(t) < inf t∈R b(t), there is a positive constant c * χ such that for every c > c * χ and every unit vector ξ, (0.1) has a transition front solution of the form (u(x, t), v(x, t)) = (U (x · ξ − C(t), t), V (x · ξ − C(t), t)) satisfying that C ′ (t) = a(t)+κ 2 κ for some positive number κ, lim inf t−s→∞ C(t)−C(s) t−s = c, and lim x→−∞ sup t∈R |U (x, t) − u * (t)| = 0 and lim x→∞ sup t∈R | U (x, t) e −κx − 1| = 0.Furthermore, we prove that there is no transition front solution (u(x, t), v(x, t)) = (U (x · ξ − C(t), t), V (x · ξ − C(t), t)) of (0.1) connecting (0, 0) and (u * (t), v * (t)) with least mean speed less than 2 √ a, where a = lim inf t−s→∞ 1 t−s t s a(τ )dτ .
