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Bifurcation Analysis of a Tumour-Immune Model with Nonlinear Killing Rate as State-Dependent Feedback Control

Likun Guan

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China

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Jin Yang

https://orcid.org/0000-0001-7529-5649

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China

E-mail Address: seehom@126.com

Corresponding author.

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Yuanshun Tan

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China

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Zijian Liu

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China

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, and

Robert A. Cheke

Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK

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https://doi.org/10.1142/S0218127422501553Cited by:0 (Source: Crossref)

Abstract

Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincaré map of the proposed model and then address the existence and orbital asymptotically stability of the model’s tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincaré mapping, we investigate the model’s transcritical and subcritical pitchfork bifurcations with respect to the key parameter.

Keywords:

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