Global stability for a Lotka-Volterra competition system with symmetric diffusion matrices

Yizheng Hu; Xinze Lian; Zhengyi Lu; Yong Luo; Yizheng Hu; Xinze Lian; Zhengyi Lu; Yong Luo

doi:10.3934/mbe.2026032

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 4: 800-812. doi: 10.3934/mbe.2026032

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Research article

Global stability for a Lotka-Volterra competition system with symmetric diffusion matrices

1.
College of mathematics and physics, Wenzhou University, Wenzhou 325035, China
2.
School of Data Science and Artificial Intelligence, Wenzhou University of Technology, China
3.
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China

Received: 22 December 2025 Revised: 28 January 2026 Accepted: 09 February 2026 Published: 27 February 2026

The global stability of a two-species Lotka-Volterra competition n-patch system with symmetric diffusion matrices is a conjecture proposed by Hofbauer, So, and Takeuchi. The cases n = 3, 4, 5 have been demonstrated to hold validity. In this article, it is shown that the Hofbauer-So-Takeuchi conjecture holds true for general n. A concrete example is given to display the global stability for one of the equilibria of the system depending on the magnitude of the parameters. Furthermore, an algebraic curve problem in the system's parameter space for determining the stability regions is proposed.
- Lotka-Volterra,
- competition system,
- symmetric diffusion,
- type K monotone,
- global stability
Citation: Yizheng Hu, Xinze Lian, Zhengyi Lu, Yong Luo. Global stability for a Lotka-Volterra competition system with symmetric diffusion matrices[J]. Mathematical Biosciences and Engineering, 2026, 23(4): 800-812. doi: 10.3934/mbe.2026032

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Abstract

The global stability of a two-species Lotka-Volterra competition n-patch system with symmetric diffusion matrices is a conjecture proposed by Hofbauer, So, and Takeuchi. The cases n = 3, 4, 5 have been demonstrated to hold validity. In this article, it is shown that the Hofbauer-So-Takeuchi conjecture holds true for general n. A concrete example is given to display the global stability for one of the equilibria of the system depending on the magnitude of the parameters. Furthermore, an algebraic curve problem in the system's parameter space for determining the stability regions is proposed.

References

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