Chaos in a Tumor Growth Model with Delayed Responses of the Immune System
A simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth proc...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/891095 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849401534160306176 |
|---|---|
| author | M. Saleem Tanuja Agrawal |
| author_facet | M. Saleem Tanuja Agrawal |
| author_sort | M. Saleem |
| collection | DOAJ |
| description | A simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed. |
| format | Article |
| id | doaj-art-8ccaff6faff0433f80ebab288dc9844f |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-8ccaff6faff0433f80ebab288dc9844f2025-08-20T03:37:44ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/891095891095Chaos in a Tumor Growth Model with Delayed Responses of the Immune SystemM. Saleem0Tanuja Agrawal1Department of Applied Mathematics, Z. H. College of Engineering & Technology, A.M.U, Aligarh 202002, IndiaDepartment of Applied Mathematics, Z. H. College of Engineering & Technology, A.M.U, Aligarh 202002, IndiaA simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed.http://dx.doi.org/10.1155/2012/891095 |
| spellingShingle | M. Saleem Tanuja Agrawal Chaos in a Tumor Growth Model with Delayed Responses of the Immune System Journal of Applied Mathematics |
| title | Chaos in a Tumor Growth Model with Delayed Responses of the Immune System |
| title_full | Chaos in a Tumor Growth Model with Delayed Responses of the Immune System |
| title_fullStr | Chaos in a Tumor Growth Model with Delayed Responses of the Immune System |
| title_full_unstemmed | Chaos in a Tumor Growth Model with Delayed Responses of the Immune System |
| title_short | Chaos in a Tumor Growth Model with Delayed Responses of the Immune System |
| title_sort | chaos in a tumor growth model with delayed responses of the immune system |
| url | http://dx.doi.org/10.1155/2012/891095 |
| work_keys_str_mv | AT msaleem chaosinatumorgrowthmodelwithdelayedresponsesoftheimmunesystem AT tanujaagrawal chaosinatumorgrowthmodelwithdelayedresponsesoftheimmunesystem |