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Hopf bifurcation in differential equations with delay for tumor-immune system competition model. (English) Zbl 1156.34066
A model that refers to the competition between the immune system and an aggressive host such as a tumor is introduced. 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 bifurcation parameter, 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, 395–406 (2003; Zbl 1035.92019)], which are observed in reality [see D. Kirschner and J. C. Panetta, J. Math. Biol. 37, 235–252 (1998; Zbl 0902.92012)].
MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
92C37Cell biology