Abstract
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.
Highlights
Cancer, known as malignant tumor, grows fast and is very difficult to cure, which seriously affects people’s life and health [1,2,3,4]
7 Conclusion and discussion Co-dimension two bifurcations are a common bifurcation phenomenon in delay differential equations, which leads to complex dynamic behavior of the system
A delayed tumor model is established, and the Allee effect is considered in this model
Summary
Known as malignant tumor, grows fast and is very difficult to cure, which seriously affects people’s life and health [1,2,3,4]. Most of them study the model from the perspective of stability and bifurcation, and consider the delay [8, 9] and other effects in the model. In [11], a three-dimensional ordinary differential equation model is established to explore the growth law of tumor cells, and its bifurcation and multistability are studied. These models have different definitions of tumor growth, most of which are exponential growth and logistic growth. Considering the delay of apoptosis of immune cells and the Allee effect of tumor cell growth, the following functional differential equation model is established:. A brief conclusion and expectation section completes the paper
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