Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model
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 ha...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2015-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/354918 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832545824221954048 |
|---|---|
| author | Marluci Cristina Galindo Cristiane Nespoli Marcelo Messias |
| author_facet | Marluci Cristina Galindo Cristiane Nespoli Marcelo Messias |
| author_sort | Marluci Cristina Galindo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f62d27f71c50483ba0303395587601dc |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-f62d27f71c50483ba0303395587601dc2025-02-03T07:24:41ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/354918354918Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer ModelMarluci Cristina Galindo0Cristiane Nespoli1Marcelo Messias2Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista (UNESP), 19060-900 Presidente Prudente, SP, BrazilDepartamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista (UNESP), 19060-900 Presidente Prudente, SP, BrazilDepartamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista (UNESP), 19060-900 Presidente Prudente, SP, BrazilWe 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.http://dx.doi.org/10.1155/2015/354918 |
| spellingShingle | Marluci Cristina Galindo Cristiane Nespoli Marcelo Messias Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model Abstract and Applied Analysis |
| title | Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model |
| title_full | Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model |
| title_fullStr | Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model |
| title_full_unstemmed | Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model |
| title_short | Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model |
| title_sort | hopf bifurcation cascade of period doubling chaos and the possibility of cure in a 3d cancer model |
| url | http://dx.doi.org/10.1155/2015/354918 |
| work_keys_str_mv | AT marlucicristinagalindo hopfbifurcationcascadeofperioddoublingchaosandthepossibilityofcureina3dcancermodel AT cristianenespoli hopfbifurcationcascadeofperioddoublingchaosandthepossibilityofcureina3dcancermodel AT marcelomessias hopfbifurcationcascadeofperioddoublingchaosandthepossibilityofcureina3dcancermodel |