The spreading of global solutions of chemotaxis systems with logistic source and consumption on $\mathbb{R}^{N}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909775170633728 |
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| author | Hassan, Zulaihat Shen, Wenxian Zhang, Yuming Paul |
| author_facet | Hassan, Zulaihat Shen, Wenxian Zhang, Yuming Paul |
| contents | This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=Δu - χ\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{τ\partial_tv}=Δv-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] 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 $0<-χ\ll 1$ and $τ=1$. Additionally, our numerical simulations reveal a noteworthy phase transition in $χ$: for $v(0, \cdot)$ uniformly distributed across space, the spreading speed accelerates only when $χ$ surpasses a critical positive value. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spreading of global solutions of chemotaxis systems with logistic source and consumption on $\mathbb{R}^{N}$ Hassan, Zulaihat Shen, Wenxian Zhang, Yuming Paul Analysis of PDEs 35B40, 35K57, 35Q92, 92C17 This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=Δu - χ\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{τ\partial_tv}=Δv-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] 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 $0<-χ\ll 1$ and $τ=1$. Additionally, our numerical simulations reveal a noteworthy phase transition in $χ$: for $v(0, \cdot)$ uniformly distributed across space, the spreading speed accelerates only when $χ$ surpasses a critical positive value. |
| title | The spreading of global solutions of chemotaxis systems with logistic source and consumption on $\mathbb{R}^{N}$ |
| topic | Analysis of PDEs 35B40, 35K57, 35Q92, 92C17 |
| url | https://arxiv.org/abs/2405.19428 |