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Discrete Dynamics in Nature and Society
Volume 2006, Article ID 95296, 9 pages
http://dx.doi.org/10.1155/DDNS/2006/95296

Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

Département de Mathématiques et Informatique, Faculté des Sciences, Université Chouaib Doukkali, BP 20, El Jadida 2400, Morocco

Received 26 August 2005; Accepted 28 November 2005

Copyright © 2006 Radouane Yafia. 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

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Gałach (2003), which are observed in reality by Kirschner and Panetta (1998).