Journal of Applied Mathematics
Volume 2012 (2012), Article ID 891095, 16 pages
Research Article

Chaos in a Tumor Growth Model with Delayed Responses of the Immune System

Department of Applied Mathematics, Z. H. College of Engineering & Technology, A.M.U, Aligarh 202002, India

Received 3 November 2011; Revised 22 December 2011; Accepted 29 December 2011

Academic Editor: Mohamad Alwash

Copyright © 2012 M. Saleem and Tanuja Agrawal. 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.


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.