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.

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Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions and fruitful comments which lead to significant improvement of this work. This work is funded by Chongqing Post-doctoral Innovative Talent Support program, China Postdoctoral Science Foundation under Grant 2020M673102, the Fundamental Research Funds for the Central Universities under Grants XDJK2020C054, 2020CDJQY-Z001 and 2019CDJCYJ001, the NSFC under Grants 11771062, 11971393 and 11971082, the Natural Science Foundation of Chongqing, China under Grant cstc2020jcyj-bshX0071.

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Tu, X., Tang, CL. & Qiu, S. The phenomenon of large population densities in a chemotaxis competition system with loop. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00650-6

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Mathematics Subject Classification

  • 35B44
  • 35K55
  • 92C17

Keywords

  • Chemotaxis with loop
  • Two species and two stimuli
  • Lotka–Volterra-type competition
  • Simultaneous blowup