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Bifurcation and pattern formation of a tumor–immune model with time-delay and diffusion

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  • Jia, Yunfeng

Abstract

A tumor–immune model with time-delay and diffusion is considered. Firstly, the local stability of equilibria and the existence of Hopf bifurcation are studied. Secondly, the direction and stability of Hopf bifurcation are discussed. Finally, the numerical simulations are used to verify the effectiveness of the theoretical results. It is found that the time-delay can destroy the stability of positive equilibrium and then affect the occurrence of Hopf branch. Specifically, the equilibrium is stable if the model is without delay or with small delay, and so there is no bifurcation; Conversely, when the delay is large, it induces the instability of equilibrium and the Hopf bifurcation occurs, the model then exhibits rich spatiotemporal dynamics.

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  • Jia, Yunfeng, 2020. "Bifurcation and pattern formation of a tumor–immune model with time-delay and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 92-108.
    • Handle: RePEc:eee:matcom:v:178:y:2020:i:c:p:92-108
    DOI: 10.1016/j.matcom.2020.06.011
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    1. Jaiswal, Shubham & Chopra, Manish & Das, S., 2019. "Numerical solution of non-linear partial differential equation for porous media using operational matrices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 138-154.
    2. Chuangxia Huang & Jie Cao & Fenghua Wen & Xiaoguang Yang, 2016. "Stability Analysis of SIR Model with Distributed Delay on Complex Networks," PLOS ONE, Public Library of Science, vol. 11(8), pages 1-22, August.
    3. Rihan, F.A. & Abdel Rahman, D.H. & Lakshmanan, S. & Alkhajeh, A.S., 2014. "A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 606-623.
    4. Liu, Peng & Liu, Xijun, 2017. "Dynamics of a tumor-immune model considering targeted chemotherapy," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 7-13.
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    Cited by:

    1. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    2. Che, Han & Wang, Yu-Lan & Li, Zhi-Yuan, 2022. "Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 149-163.
    3. Liang, Yuqin & Jia, Yunfeng, 2022. "Stability and Hopf bifurcation of a diffusive plankton model with time-delay and mixed nonlinear functional responses," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).

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