Dynamics on semi-discrete Mackey-Glass model

Yulong Li; Long Zhou; Fengjie Geng; Yulong Li; Long Zhou; Fengjie Geng

doi:10.3934/math.2025130

AIMS Mathematics

2025, Volume 10, Issue 2: 2771-2807. doi: 10.3934/math.2025130

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Research article

Dynamics on semi-discrete Mackey-Glass model

School of Science, China University of Geosciences (Beijing), 100083, Beijing, China

Received: 18 December 2024 Revised: 17 January 2025 Accepted: 06 February 2025 Published: 14 February 2025
MSC : 39A28, 39A30, 39A60

Red blood cells play an extremely important role in human metabolism, and the study of hematopoietic models is of great significance in biology and medicine. A kind of semi-discrete hetmatopoietic model named Mackey-Glass Model was proposed and analyzed in this paper. The existences, stabilities, and local dynamics of the fixed points were discussed. By using bifurcation theory, we studied the Neimark-Sacker bifurcation, saddle-node bifurcation, and strong resonance of 1:4. The numerical simulations were presented to illustrate the results of theoretical analysis obtained in this paper, and complex dynamical behaviors were found such as invariant cycles, heteroclinic cycles and Li-Yorke chaos. In addition, a new periodic bubbling phenomenon was discovered in numerical simulations. These not only reflect the richer dynamical behaviors of the semi-discrete models, but also some reflect the complex metabolic characteristics of the hematopoietic system under environmental intervention.
- semi-discrete hetmatopoietic model,
- Neimark-Sacker bifurcation,
- saddle-node bifurcation,
- strong resonance of 1:4,
- numerical simulations
Citation: Yulong Li, Long Zhou, Fengjie Geng. Dynamics on semi-discrete Mackey-Glass model[J]. AIMS Mathematics, 2025, 10(2): 2771-2807. doi: 10.3934/math.2025130

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Abstract

Red blood cells play an extremely important role in human metabolism, and the study of hematopoietic models is of great significance in biology and medicine. A kind of semi-discrete hetmatopoietic model named Mackey-Glass Model was proposed and analyzed in this paper. The existences, stabilities, and local dynamics of the fixed points were discussed. By using bifurcation theory, we studied the Neimark-Sacker bifurcation, saddle-node bifurcation, and strong resonance of 1:4. The numerical simulations were presented to illustrate the results of theoretical analysis obtained in this paper, and complex dynamical behaviors were found such as invariant cycles, heteroclinic cycles and Li-Yorke chaos. In addition, a new periodic bubbling phenomenon was discovered in numerical simulations. These not only reflect the richer dynamical behaviors of the semi-discrete models, but also some reflect the complex metabolic characteristics of the hematopoietic system under environmental intervention.

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