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

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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.