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Discrete Dynamics in Nature and Society
Volume 2006, Article ID 58463, 13 pages

Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response

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

Received 20 December 2005; Accepted 11 April 2006

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.


This paper is devoted to the study of the stability of limit cycles of a system of nonlinear delay differential equations with a discrete delay. The system arises from a model of population dynamics describing the competition between tumor and immune system with negative immune response. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions, using the methods presented by Diekmann et al.