The spreading of global solutions of chemotaxis systems with logistic source and consumption on ${\mathbb{R}}^N$

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: $$\{\begin{array}{rlr}{\displaystyle }& {\displaystyle {\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+u(a-bu),}& {\displaystyle (t,x)\in [0,\mathrm{\infty })\times {\mathbb{R}}^N,}\\ {\displaystyle }& {\displaystyle \tau {\mathrm{\partial }}_{t}v=\mathrm{\Delta }v-uv,}& {\displaystyle (t,x)\in [0,\mathrm{\infty })\times {\mathbb{R}}^N,}\end{array}$$ where $u(t,x)$ represents the population density of a biological species, and $v(t,x)$ denotes the density of a chemical substance. Key findings of this study include: (i) the species spreads at least at the speed $c^{\ast }=2\sqrt{a}$ (equalling the speed when $v\equiv 0$), suggesting that the chemical substance does not hinder the spreading; (ii) the chemical substance does not induce infinitely fast spreading of $u$; (iii) the spreading speed remains unaffected under conditions that $v(0,\cdot )$ decays spatially or $0<-\chi \ll 1$ and $\tau =1$. 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.