Hopf bifurcation in differential equations with delay for tumor-immune system competition model

This paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a system of differential equations with one delay. It is shown that the dynamics depends crucially on the time delay parameter. By using the time delay as a parameter of bifurcation, the analysis is focused on the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state. The obtained results depict the oscillations, given by simulations (see [M. Galach, Int. J. Appl. Math. Comput. Sci., 13 (2003), pp. 395-406]), which are observed in reality (see [D. Kirschner and J. C. Panetta, J. Math. Biol., 37 (1998), pp. 235-252]). It is suggested to examine by laboratory experiments how to employ these results for control of tumor growth.