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

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay are 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. Galach ( 2003), which are observed in reality by Kirschner and Panetta ( 1998).