Effects of time delays on persistence and global stability of mathematical biological models

本研究的目的是研究生物学中一些具体的数学模型中引入的时滞对其持久性和全局稳定性的影响，这是数学生物学中最基本的概念。在此基础上，建立了分析具有时滞的一般生物学数学模型的方法。具体的模型有：1.在物种的生长过程和物质的回收过程中都具有时滞效应的恒化器模型；2.在易感个体感染的过程中具有时滞的传染病模型；3.巨噬细胞吞噬红细胞的时滞医学模型。结果如下：1994年：在足够小的时滞下保证了恒化器模型的全局稳定性(文献1)；另外，当传染病模型的总体规模不变时，其全局稳定性也是有保证的(文献2)。1995：修正和推广了文献1的一致稳定性(文献3)；对于人口规模不同的传染病模型，指的是确保解收敛于地方病状态的一组初始数据。(文7)；对于医学模型，解的控制问题的结果在(文4)中出现。1996：证明了一个非线性微分差分不等式，并将其应用于具有无限时滞的非线性时滞或中立型微分差分系统的稳定性分析(文5，6)。该方法不仅适用于生物学中的数学模型，而且适用于一般模型。

The purpose of this research is to study the effct of time delays introduced in some concrete mathematical models in biology on their persistence and global stability, which are the most fundamental concept in mathematical biology. Further, the research is aimed at the establishment of the method to analyze the general mathematical models in biology with time delays.The concrete models are :1. the chemostat models with time delay effect both in the growth process of the species and the recycling process of the materials ;2. the epidemic models with time delays in the process that the susceptible individuals become infectious ;3. the medical models with time delays in the phagocytosis of red blood cells by the macrophages.The results are as follows :1994 : The global stability of chemostat models is ensured under the sufficiently small time delays (paper 1) ; Also the global stability of epidemic models is guaranteed if they have a constant population size (paper 2).1995 : The paper 1 is revised and extended for uniform stability (paper 3) ; for epidemic models with varying population size, the set of initial data which ensures for the solution to converge to the endemic state. (paper 7) ; for medical models, the results on control problem of the solution are appeared in (paper 4).1996 : A nonlinear differential difference inequality is proved and applied for stability analysis of nonlinear retarded or neutral differential difference systems with infinite delays (papers 5,6). The method can be applied not only for mathematical models in biology but also for general models.