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

TL;DR

This paper studies how a biological species modeled by a chemotaxis system with logistic growth spreads in space, showing that the chemical does not slow or speed up the spread unless a critical chemotactic sensitivity is exceeded.

Contribution

It establishes the minimal spreading speed for the system and shows the chemical's influence on spreading speed, including a phase transition phenomenon based on chemotactic sensitivity.

Findings

Species spreads at least at speed 2√a, unaffected by the chemical.

Chemical does not cause infinite spreading speed.

Spreading speed remains stable under certain initial decay and parameter conditions.

Abstract

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=\Delta u - \chi\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{\tau \partial_tv}=\Delta v-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] 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 $c^*=2\sqrt a$ (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.