Periodic dynamics of a switching model with multiple stable states and linear harvest

Qing Zhu; Yu Li; Huaqin Peng; Qing Zhu; Yu Li; Huaqin Peng

doi:10.3934/era.2026133

Electronic Research Archive

2026, Volume 34, Issue 5: 2926-2946. doi: 10.3934/era.2026133

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Periodic dynamics of a switching model with multiple stable states and linear harvest

School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Received: 19 January 2026 Revised: 08 March 2026 Accepted: 19 March 2026 Published: 03 April 2026

Fishing moratoriums are vital for fishery management, restoring fish stocks, safeguarding ecosystems, and ensuring sustainable development. In this study, we explored and studied a canonical model with a Holling type-Ⅲ functional response consisting of two sub-equations switching each other. If the length of the closed season $ \bar{T}\leq T^{\ast} $, the trivial steady state is globally asymptotically stable. For $ \bar{T} > T^{\ast} $, we provided a complete analysis of the existence of a unique periodic solution and three exact periodic solutions. We also provided sufficient conditions for the existence of a globally asymptotically stable periodic solution. Finally, numerical examples were provided to illustrate the theoretical results.
- periodic solutions,
- switching dynamical systems,
- Holling type-Ⅲ functional response,
- global asymptotic stability
Citation: Qing Zhu, Yu Li, Huaqin Peng. Periodic dynamics of a switching model with multiple stable states and linear harvest[J]. Electronic Research Archive, 2026, 34(5): 2926-2946. doi: 10.3934/era.2026133

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Abstract

Fishing moratoriums are vital for fishery management, restoring fish stocks, safeguarding ecosystems, and ensuring sustainable development. In this study, we explored and studied a canonical model with a Holling type-Ⅲ functional response consisting of two sub-equations switching each other. If the length of the closed season $ \bar{T}\leq T^{\ast} $, the trivial steady state is globally asymptotically stable. For $ \bar{T} > T^{\ast} $, we provided a complete analysis of the existence of a unique periodic solution and three exact periodic solutions. We also provided sufficient conditions for the existence of a globally asymptotically stable periodic solution. Finally, numerical examples were provided to illustrate the theoretical results.

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