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Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems

Author

Listed:
  • Junbo Jia

    (Key Laboratory of Systems Health Science of Zhejiang Province, School of Life Science, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China)

  • Pan Yang

    (School of Mathematical Sciences, Changsha Normal University, Changsha 410100, China)

  • Huaiping Zhu

    (Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada)

  • Zhen Jin

    (Complex Systems Research Center, Shanxi University, Taiyuan 030051, China)

  • Jinqiao Duan

    (Department of Mathematics and Department of Physics, Great Bay University, Dongguan 523000, China)

  • Xinchu Fu

    (Department of Mathematics, Shanghai University, Shanghai 200444, China)

Abstract

Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the two attractors, and also obtain the possible components and their configuration. Moreover, we find that there is always a separatrix or cycle separatrix, which separates the two attractors. Finally, the biological meanings and implications of these structures are given and discussed.

Suggested Citation

  • Junbo Jia & Pan Yang & Huaiping Zhu & Zhen Jin & Jinqiao Duan & Xinchu Fu, 2023. "Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems," Mathematics, MDPI, vol. 11(19), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4139-:d:1251807
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    References listed on IDEAS

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    1. Xiaoqin Wang & Yongli Cai & Huihai Ma, 2013. "Dynamics of a Diffusive Predator-Prey Model with Allee Effect on Predator," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-10, February.
    2. Zongmin Yue & Xiaoqin Wang & Haifeng Liu, 2013. "Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, February.
    3. Shuling Yan & Xinze Lian & Weiming Wang & Youbin Wang, 2013. "Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-11, February.
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