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This paper deals with the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source, (0.1) u t = Δ u − χ ∇ ⋅ (u v ∇ v) + u (a (t , x) − b (t , x) u) , x ∈ Ω 0 = Δ v − μ v + ν u , x ∈ Ω ∂ u ∂ n = ∂ v ∂ n = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, a (t , x) and b (t , x) are positive smooth functions, and χ , μ and ν are positive constants. In recent years, it has been drawn a lot of attention to the question of whether logistic kinetics prevents finite-time blow-up in various chemotaxis models. In the very recent paper (Kurt and Shen, 2021), we proved that for every given nonnegative initial function 0 ⁄ ≡ u 0 ∈ C 0 (Ω ̄) and s ∈ R , (0.1) has a unique globally defined classical solution (u (t , x ; s , u 0) , v (t , x ; s , u 0)) with u (s , x ; s , u 0) = u 0 (x) , which shows that, in any space dimensional setting, logistic kinetics prevents the occurrence of finite-time blow-up even for arbitrarily large χ. In Kurt and Shen (2021), we also proved that globally defined positive solutions of (0.1) are uniformly bounded under the assumption (0.2) a inf > μ χ 2 4 if 0 < χ ≤ 2 μ (χ − 1) if χ > 2. In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption (0.2). Among others, we provide some concrete estimates for ∫ Ω u − p and ∫ Ω u q for some p > 0 and q > 2 N and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a "rectangular" type bounded invariant set (in L q) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution (u ∗ (t , x) , v ∗ (t , x)) , which is periodic in t if a (t , x) and b (t , x) are periodic in t and is independent of t if a (t , x) and b (t , x) are independent of t. [ABSTRACT FROM AUTHOR]