Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals

By Youshan Tao, Michael Winkler. 2015

134 citations recorded by the source corpus.

We consider a model for two species interacting through chemotaxis in such a way that each species produces a signal which directs the respective motion of the other. Specifically, we shall be concerned with nonnegative solutions of the Neumann problem, posed in bounded domains $\Omega\subset \mathbb{R}^n$ with smooth boundary, for the system $$\begin{cases} u_t= \Delta u - \chi \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v-v+w, & x\in \Omega, \, t>0, \qquad (\star)\\ w_t= \Delta w - \xi \nabla \cdot (w\nabla z), & x\in \Omega, \, t>0, \\ 0=\Delta z-z+u, & x\in \Omega, \, t>0, \end{cases}$$ with parameters $\chi \in \{\pm 1\}$ and $\xi\in \{\pm 1\}$, thus allowing the interaction of either attraction-repulsion, or attraction-attraction, or repulsion-repulsion type. It is shown that $\bullet$ in the attraction-repulsion case $\chi=1$ and $\xi=-1$, if $n\le 3$ then for any nonnegative initial data $u_0\in C^0(\bar{\Omega})$ and $ w_0\in C^0 (\bar{\Omega})$, there exists a unique global classical solution which is bounded; $\bullet$ in the doubly repulsive case when $\chi=\xi=-1$, the same holds true; $\bullet$ in the attraction-attraction case $\chi=\xi=1$, $-$ if either $n=2$ and $\int_\Omega u_0 + \int_\Omega w_0$ lies below some threshold, or $n\ge 3$ and $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$ are sufficiently small, then solutions exist globally and remain bounded, whereas $-$ if either $n=2$ and $m$ is suitably large, or $n\ge 3$ and $m>0$ is arbitrary, then there exist smooth initial data $u_0$ and $w_0$ such that $\int_\Omega u_0 + \int_\Omega w_0=m$ and such that the corresponding solution blows up in finite time. In particular, these results demonstrate that the circular chemotaxis mechanism underlying ($\star$) goes along with essentially the same destabilizing features as known for the classical Keller-Segel system in the doubly attractive case, but totally suppresses any blow-up phenomenon when only one, or both, taxis directions are repulsive.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/105c11cf418ba66cf97689494171bc4794ba9e80

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Citations in the corpus

On the weakly competitive case in a two-species chemotaxis model (2016, background): “Let us finally mention that also (parabolic-elliptic) Keller-Segel type systems of two species and two chemicals have been studied where the signal for each species is produced by the other ([41]).”
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction (2017, background): “In essence, the proof follows the idea from [20].”