Stability and Hopf bifurcation analysis of Caputo time-fractional delayed reaction-diffusion models for sterile insect technology

Dasen Lin; Ahmadjan Muhammadhaji; Yimamu Maimaiti; Dasen Lin; Ahmadjan Muhammadhaji; Yimamu Maimaiti

doi:10.3934/nhm.2025062

Networks and Heterogeneous Media

2025, Volume 20, Issue 5: 1437-1465. doi: 10.3934/nhm.2025062

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Research article

Stability and Hopf bifurcation analysis of Caputo time-fractional delayed reaction-diffusion models for sterile insect technology

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 11 August 2025 Revised: 21 November 2025 Accepted: 12 December 2025 Published: 17 December 2025

Based on the theory of fractional differential equations, this paper studied a single-species contraception model with feedback control terms. By comprehensively applying the Mittag-Leffler function construction method, Laplace transform, and stability criteria for fractional-order systems, the existence, uniqueness, boundedness, and local stability of the model's equilibrium points were analyzed, and the key influencing factors of the system's steady-state behavior were identified. Furthermore, the mechanisms of Hopf bifurcation and Turing bifurcation were explored, the sufficient conditions for the existence of these two types of bifurcations were derived, and the theoretical framework for the system's dynamic behavior was improved. A numerical example was designed to verify the validity of the theoretical results, and the simulation results confirmed the correctness of the stability conclusions and bifurcation conditions. In the conclusion section, the numerical simulations were reviewed and correlated with the theory; future research directions were proposed in light of the limitations of the current study, providing references for subsequent explorations.
- pest control,
- fractional-order model,
- reaction-diffusion,
- Hopf bifurcation,
- turing instability
Citation: Dasen Lin, Ahmadjan Muhammadhaji, Yimamu Maimaiti. Stability and Hopf bifurcation analysis of Caputo time-fractional delayed reaction-diffusion models for sterile insect technology[J]. Networks and Heterogeneous Media, 2025, 20(5): 1437-1465. doi: 10.3934/nhm.2025062

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Abstract

Based on the theory of fractional differential equations, this paper studied a single-species contraception model with feedback control terms. By comprehensively applying the Mittag-Leffler function construction method, Laplace transform, and stability criteria for fractional-order systems, the existence, uniqueness, boundedness, and local stability of the model's equilibrium points were analyzed, and the key influencing factors of the system's steady-state behavior were identified. Furthermore, the mechanisms of Hopf bifurcation and Turing bifurcation were explored, the sufficient conditions for the existence of these two types of bifurcations were derived, and the theoretical framework for the system's dynamic behavior was improved. A numerical example was designed to verify the validity of the theoretical results, and the simulation results confirmed the correctness of the stability conclusions and bifurcation conditions. In the conclusion section, the numerical simulations were reviewed and correlated with the theory; future research directions were proposed in light of the limitations of the current study, providing references for subsequent explorations.

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