Abstract
We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.
Highlights
Mathematical models describing cancer tumour growth have been extensively studied in the literature in order to understand the mechanism of disease and to predict its future behaviour; see [1] and references therein as a brief review on the subject
Trying to analyse this dynamical behaviour from the biological point of view, we observed that, after the occurrence of chaotic dynamics, both the tumour cells and the immune system cells (represented by the coordinates x(t) and z(t)) vanish and only the healthy cells remain positive and their amount tends to the carrying capacity y(t) = 1
As λ1 > 0, the equilibrium E is unstable, which could be possible from the biological point of view, since this equilibrium represents the extinction of healthy cells and the persistence of tumour cells and immune system cells
Summary
Mathematical models describing cancer tumour growth have been extensively studied in the literature in order to understand the mechanism of disease and to predict its future behaviour; see [1] and references therein as a brief review on the subject. Aiming to contribute to the understanding of these types of model we perform a bifurcation analysis of a cancer tumour growth model of three competing cell populations (tumour, healthy, and immune cells), which was proposed in [6]. Numerically studying the continuation of this stable limit cycle by increasing the parameter a from the Hopf point, we numerically found the occurrence of a cascade of period-doubling bifurcations which leads to the formation of the chaotic attractor shown to exist in [6] Trying to analyse this dynamical behaviour from the biological point of view, we observed that, after the occurrence of chaotic dynamics, both the tumour cells and the immune system cells (represented by the coordinates x(t) and z(t)) vanish and only the healthy cells remain positive and their amount tends to the carrying capacity y(t) = 1.
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