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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 354918, 11 pages
http://dx.doi.org/10.1155/2015/354918
Research Article

Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista (UNESP), 19060-900 Presidente Prudente, SP, Brazil

Received 27 June 2014; Accepted 13 October 2014

Academic Editor: Yongli Song

Copyright © 2015 Marluci Cristina Galindo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.