In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

中文翻译：

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时滞与肿瘤-免疫系统相互作用模型的非线性动力学

在本文中，我们回顾了有关延迟微分方程模型的非线性动力学的一些最新结果，这些模型描述了肿瘤细胞与免疫系统效应细胞之间的相互作用，其中延迟代表了分子产生，增殖，细胞分化，转运所需的时间。首先，我们考虑具有单个延迟的肿瘤-免疫系统相互作用模型，并给出关于延迟存在变化时平衡模型的存在性和局部稳定性以及模型中Hopf分叉的存在性的结果。其次，我们研究了具有两个延迟的肿瘤-免疫系统相互作用模型，并显示该模型经历了各种可能的分叉，包括Hopf，Bautin，Fold-Hopf（零-Hopf）和Hopf-Hopf分叉。最后，我们讨论了具有三个延迟的肿瘤-免疫系统相互作用模型，并证明该模型表现出更复杂的行为，包括混沌。提供数值模拟以说明延迟的肿瘤-免疫系统相互作用模型的非线性动力学。在讨论部分中给出了有关建模和分析肿瘤免疫动力学的更有趣的问题。