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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.

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 [

The competition for nutrients, the effects of growth factors, and the actions of the immune system stand out among possible interactions.

The immune system has the function of recognizing internal and external threats to the organism, reacting to eliminate, neutralize, or tolerate such threats. The recognition of tumour cells by the immune system can happen in distinct and complementary ways. According to Chammas et al. [

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 [

The equations of system (

In [

In this paper we analyse the local stability of the equilibria of system (

The parameter

We believe that the bifurcation analysis and the biological considerations presented here complement the results and the analysis of system (

The paper is organized as follows. In this introductory section, we present the studied model, describing the variables and parameters involved, and state the main results obtained. In Section

Considering

These equilibria are called biologically admissible if their three coordinates are greater than or equal to zero. The equilibrium points

For the equilibrium point

if

if

if

if

if none of the previous inequalities are satisfied,

As the equilibria

the equilibrium

The Jacobian matrix of system (

Equilibrium

Equilibrium

Equilibrium

Equilibrium

Equilibrium

Equilibrium

We observe that the equilibrium point

Consider the cubic polynomial

Then (Routh-Hurwitz Criterion)

If

If

Applying the previous lemma to the polynomial (

This section is a brief review of the projection method described in [

Consider the differential equation

Suppose that

Let

The first Lyapunov coefficient

A

A Hopf point is called

Based on these results, a numerical algorithm for calculating the first Lyapunov coefficient of a system

By using the Hopf bifurcation theorem stated in [

Consider system (

Considering the characteristic polynomial of the Jacobian matrix

The necessary condition for the point

If

To finish the proof, the transversality condition remains to be checked. Using software MAPLE, we calculate the derivative of the Jacobian matrix

Solving the equation

This ends the proof of Theorem

The numerical simulations presented in Figures

Phase portrait of system (

Phase portrait of system (

The existence of a stable limit cycle implies the occurrence of a periodic behaviour of system (

In the next section, by studying the continuation of this limit cycle when the parameter

For

Periodic orbit of period

Period-doubling bifurcation occurred with the periodic solution created in the Hopf bifurcation at the equilibrium

(a) Periodic orbit of period

Chaotic attractor created through the period-doubling bifurcations. Initial conditions:

Figure

A bifurcation diagram confirming the occurrence of the period-doubling cascade mentioned above, culminating in chaos, is shown in Figure

Bifurcation diagram for system (

The chaotic dynamics occurs when the value of

In turn, for

In Figure

Phase portrait of system (

In this work, using the concepts of qualitative theory, bifurcations, and chaos in dynamical systems, we analyze a three-dimensional cancer model, describing the interactions among healthy cells, tumour cells, and immune system cells. The analyzed model was proposed in [

In a first glance one could argue that, from the biological point of view, the chaotic dynamics could be related to the process of uncontrolled growth of tumour cells. However, we surprisingly observed that, for the set of parameter values considered the period-doubling bifurcations which occur to the cancer cells lead to the growth of effector cells, since their growth is proportional to the growth of tumour cells. In this way, it seems that, for the particular values of the parameter considered here, the growth of both types of cells occurs simultaneously and is represented mathematically by the period-doubling cascade, which leads to the creation of a chaotic attractor. Furthermore, supposing the increasing values of the parameter

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by PROPG/UNESP. The third author is supported by CNPq-Brazil, under the Project 308315/2012-0 and by FAPESP-Brazil under the Process 2012/18413-7. The authors would like to thank the referees for their valuable comments and suggestions, which helped them to improve the presentation of the paper.