This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \(\begin{aligned} \left\{ \begin{aligned}&\partial _tu=\Delta u - \chi \nabla \cdot (u\nabla v)+ u(a-bu),\quad&(t,x)\in {(0},\infty )\times \mathbb {R}^N, \\&{\tau \partial _tv}=\Delta v-uv,\quad&(t,x)\in {(0},\infty )\times \mathbb {R}^N, \end{aligned} \right. \end{aligned}\) where u represents the population density of a biological species, and v denotes the density of a chemical substance. We show that the presence of the chemical does not hinder the spreading of the species in general, and it does not accelerate the spreading speed under conditions that \(v(0,\cdot )\) decays spatially or the parameters satisfy \(0<-\chi \ll 1\) and \(\tau =1\) . In the proof, we establish a novel connection between u and v using the sup- and inf-convolution techniques for viscosity solutions. Additionally, our numerical simulations reveal a noteworthy phase transition in \(\chi \) : for \(v(0, \cdot )\) uniformly distributed across space, the spreading speed accelerates only when \(\chi \) surpasses a critical positive value.
