The spreading of global solutions of chemotaxis systems with logistic source and consumption on RN

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \{ {aligned} &∂_tu=Δ u - χ∇·(u∇ v)+ u(a-bu), &(t,x)∈ [0,∞)^N, \\ &{τ ∂_tv}=Δ v-uv, & (t,x)∈ [0,∞)^N, {aligned} . \] 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^*=2√ a (equalling the speed when v≡ 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,·) decays spatially or 0<-χ 1 and τ=1. Additionally, our numerical simulations reveal a noteworthy phase transition in χ: for v(0, ·) uniformly distributed across space, the spreading speed accelerates only when χ surpasses a critical positive value.