肿瘤是指人体在各种致瘤因子作用下局部组织细胞异常增生所形成的新生物，恶性肿瘤细胞引起的癌症，作为世界第二大死亡率疾病，成为近百年来医学研究的重点。本文分别介绍了科学家们在细胞、组织、分子水平上建立的模型，分别从数量、大小、密度三个不同的角度对肿瘤细胞的生长增殖机制进行研究分析。

第一类利用常微分方程在细胞水平上分析肿瘤细胞数量。从最初数量指数型增长的Malthus模型，到后期数量趋于定值的Logistic模型、改进的Gompertz模型、完善的一般模型，层层推进得到普适化的最佳拟合规律，但不能清楚地解释其生物学原理。

第二类利用偏微分方程在组织水平上分析肿瘤细胞大小。假设肿瘤细胞群构成一个球体，结合反应扩散原理，构建关于营养物浓度和抑制物浓度的模型，分别考虑无核和有核的情况，通过对其整体解存在唯一性的判定和平衡解渐近稳定性的判定，得到肿瘤大小变化的发展规律。

第三类利用偏微分方程在分子水平上分析肿瘤细胞密度。模拟肿瘤细胞在静止、增殖状态下的动态变化，构建关于细胞密度的模型，通过对其平衡解渐近稳定性的判定，得到肿瘤密度变化的发展规律。

文章最后详细介绍了第三类模型中平凡平衡解的渐近稳定性的判定证明，数值模拟和对医用药物使用量的实际意义。

Tumor is a new organism formed by abnormal hyperplasia of local tissue cells under the action of various carcinogen. Cancer caused by malignant tumor cells, as the world's second largest mortality disease, has become the focus of medical research in the past 100 years. In this paper, we introduce the models established by scientists at three levels: cellular, tissue and molecular, and analyze the growth and proliferation mechanisms of tumor cells from the point of quantity, size and density respectively.

The first class models analyze the number of tumor cells at the cellular level by using ordinary differential equations. From the initial Malthus model, to the Logistic model, the improved Gompertz model and the perfect general model, the best fitting law is obtained through successive progression of models, but the biological principle cannot be clearly explained by these models.

The second class models analyze the size of tumor cells at the tissue level by using partial differential equations. Assuming that the tumor cell group constitutes a sphere, models about nutrient concentration and inhibitor concentration are constructed based on the principle of reaction diffusion. We consider the nuclear-free and nucleate conditions respectively, and predicate the existence and uniqueness of the overall solution and the asymptotic stability of the steady state, which can determine the development law of tumor size.

The third class models analyze the density of tumor cells at the molecular level by  using partial differential equations. We simulate the dynamic changes of tumor cells in quiescent state and proliferating state, and construct the models about the density of tumor cells, and judge the asymptotic stability of the steady state, which can obtain the development law of tumor density.

At last, the asymptotic stability of the trivial steady state in the third class model is proved in detail, and some parameters are simulated numerically.