The phenomenon of large population densities in a chemotaxis competition system with loop

Abstract

We study herein the initial boundary value problem for a two-species chemotaxis competition system with loop

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u_{1}=\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),&{}\quad x\in \Omega ,\quad t>0,\\ \partial _{t} u_{2}=\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla v_{2}) +\mu _{2}u_{2}(1-u_{2}-a_{2}u_{1}), &{}\quad x\in \Omega ,\quad t>0,\\ \partial _t v_1=\Delta v_{1}- v_{1}+u_{1}+u_{2}, &{}\quad x\in \Omega ,\quad t>0,\\ \partial _t v_2=\Delta v_{2}- v_{2}+u_{1}+u_{2}, &{}\quad x\in \Omega ,\quad t>0,\\ \end{array}\right. \end{aligned}$$

under the homogeneous Neumann boundary condition, where $\Omega \subset {\mathbb {R}}^{n} (n\ge 3)$ is a smooth and bounded domain, $\chi _{ij}>0$, $\mu _{i}>0$, $a_i>0$ $(i, j=1, 2)$. In the radial symmetric setting, for any $T>0$ and $L>0$, it is proved that there exists positive initial data such that the corresponding solution $(u_1, u_2, v_1, v_2)$ satisfies

$$\begin{aligned} u_1(x_{L}, t_{L})>L\quad \text {or}\quad u_2(x_{L}, t_{L})>L\quad \text {for some}\quad (x_{L}, t_{L})\in \Omega \times (0, T). \end{aligned}$$

Moreover, when $\chi _{11}=\chi _{12}, \chi _{21}=\chi _{22}$, $\mu =\max \{\mu _1, \mu _2\}\in (0, 1)$, one can find initial data $(u_{10}, u_{20}, v_{10}, v_{20})\in \left( C^0({\overline{\Omega }})\right) ^2\times \left( W^{1, \infty }(\Omega )\right) ^2$, which is irrelevant to $\mu$, such that for all $\mu \in (0, 1)$, the corresponding solution $(u_{1, \mu }, u_{2, \mu }, v_{1, \mu }, v_{2, \mu })$ fulfills

$$\begin{aligned} u_{1, \mu }(x_{\mu }, t_{\mu })>\frac{L}{\mu }\quad \text {or}\quad u_{2, \mu }(x_{\mu }, t_{\mu })>\frac{L}{\mu }\quad \text {for some}\quad (x_{\mu }, t_{\mu })\in \Omega \times (0, T). \end{aligned}$$

In particular, it is proved that blowup for one of the chemotactic species implies also blowup for the other one at the same time.