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Boundedness in a Two-Species Chemotaxis System with Nonlinear Resource Consumption

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Abstract

This paper deals with a two-species chemotaxis system with nonlinear resource consumption

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\nabla \cdot (u\nabla w)+\mu _1 u(1-u-a_1v),\quad &{}x\in \Omega ,\quad t>0,\\ v_t=\Delta v-\nabla \cdot (v\nabla w)+\mu _2 v(1-a_2u-v),\quad &{}x\in \Omega ,\quad t>0,\\ w_t=\Delta w-\displaystyle {\frac{(u+v)w}{(1+u+v)^\theta }},\quad &{}x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^n(n\ge 1)\), where the parameters \(\mu _1,\mu _2, a_1,a_2>0\). For all suitably regular initial data, we prove that if \(\theta >0\), the system possesses a unique global bounded classical solution.

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Acknowledgements

The authors are very grateful to the anonymous reviewers for their carefully reading and valuable comments that lead to a substantial improvement of this manuscript. This work is supported by Natural Science Foundation of Chongqing (No. cstc2021jcyj-msxmX0412).

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  1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Houzuo Ou & Liangchen Wang

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  1. Houzuo Ou
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LW provided the main ideas, HO wrote the main manuscript text.

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Ou, H., Wang, L. Boundedness in a Two-Species Chemotaxis System with Nonlinear Resource Consumption. Qual. Theory Dyn. Syst. 23, 14 (2024). https://doi.org/10.1007/s12346-023-00873-1

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