In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.
PING BI AND SHIGUI RUANresponses and grow into cancers) (Dunn et al. [16,17], Koebel et al. [25], Schreiber, Old, and Smyth [42]). The theoretical study of tumor-immune system interaction dynamics has a long history (Adam and Bellomo [1]). In an attempt to make the models closer to reality, more and more models have been developed (Arciero, Jackson, and Kirschner [2], de Pillis, Radunskaya, and Wiseman [10], Kirschner and Panetta [24], Kuznetsov et al. [26], Lejeune, Chaplaina, and Akili [28], Nani and Freedman [34], Owen and Sherratt [35]). We refer the reader to a recent survey by Eftimie, Bramson, and Earn [18] on spatially homogeneous mathematical models describing the interactions between a malignant tumor and the immune system. However, mathematical models for the interaction dynamics of the immune components with a target population are very idealized. It is almost impossible to construct realistic models due to the complexity of the processes involved; thus it is feasible to propose simple low dimensional models which are capable of displaying some of the essential immunological phenomena, in particular the two-dimensional ODE models for the interaction of tumor cells and effector cells of the immune system (d'Onofrio [11,12,13]). The basic modeling idea is to assume that effector cells attack tumor cells, and their proliferation is stimulated, in turn, by the presence of tumor cells. However, tumor cells also induce a loss of effector cells, and there is an influx of effector cells, whose intensity may depend on the size of the tumor.Delayed responses cannot be ignored for the tumor-immune system interaction, just as Asachenkov et al. [3] and Mayer, Zaenker, and an der Heiden [32] pointed out that the delays should be taken into account to describe the times necessary for molecule production, proliferation, differentiation of cells, transport, etc. In fact, tumor-immune system interaction models with delay have been studied extensively; see Asachenkov et al. [3], Byrne [5], Byrne and Gourley [6], d'Onofrio and Gandolfi [14], d'Onofrio et al. [15], Galach [19], Liu, Hillen, and Freedman [31], Mayer, Zaenker, and an der Heiden [32], Piotrowska and Foryś [37], Rordriguez-Perez et al. [39], Villasana and Radunskaya [47], and the references cited ...
