title: Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays.
abstract: In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical values and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations.
authors.affiliation: Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, China.
authors.forename: Ping
authors.initials: P
authors.lastname: Bi
authors.affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124-4250, USA.
authors.forename: Shigui
authors.initials: S
authors.lastname: Ruan
authors.affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China.
authors.forename: Xinan
authors.initials: X
authors.lastname: Zhang
mesh_terms: D003198:Computer Simulation; D006801:Humans; D007107:Immune System / Q000473:pathology*; D008954:Models, Biological; D009369:Neoplasms / Q000473:pathology*; D017711:Nonlinear Dynamics; D009716:Numerical Analysis, Computer-Assisted; D010507:Periodicity; D013997:Time Factors