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Co-dimension two bifurcations analysis of a delayed tumor model with Allee effect

  • Dai, Qinrui1
  • 1 Wuhan Huaxia University of Technology, Wuhan, China , Wuhan (China)
Published Article
Advances in Difference Equations
Springer International Publishing
Publication Date
Dec 07, 2021
DOI: 10.1186/s13662-021-03671-8
Springer Nature
  • Difference Equations, Special Functions and Orthogonal Polynomials


The mathematical model has become an important means to study tumor treatment and has developed with the discovery of medical phenomena. In this paper, we establish a delayed tumor model, in which the Allee effect is considered. Different from the previous similar tumor models, this model is mainly studied from the point of view of stability and co-dimension two bifurcations, and some nontrivial phenomena and conclusions are obtained. By calculation, there are at most two positive equilibria in the system, and their stability is investigated. Based on these, we find that the system undergoes Bautin bifurcation, zero-Hopf bifurcation, and Hopf–Hopf bifurcation with time delay and tumor growth rate as bifurcation parameters. The interesting thing is that there is a Zero-Hopf bifurcation, which is not common in tumor models, making abundant dynamic phenomena appear in the system. By using the bifurcation theory of functional differential equations, we calculate the normal form of these Co-dimension two bifurcations. Finally, with the aid of MATLAB package DDE-BIFTOOL, some numerical simulations have been performed to support our theoretical results. In particular, we obtain the bifurcation diagram of the system in the two parameter plane and divide its regions according to the bifurcation curves. Meanwhile, the phenomena of multistability and periodic coexistence of some regions can be also demonstrated. Combined with the simulation results, we can know that when the tumor growth rate and the delay of immune cell apoptosis are small, the tumor may tend to be stable, and vice versa.

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