| Research Abstract |
The purpose of this research is to study the effect of time delays on persistence, which is one of the most fundamental concepts in mathematical biology, to analyze chaotic phenomena and to establish the analytic methods for mathematical models in biology with time delays. In particular, we considered (1) the chemostat models with time delays effect in growth process of the species and the recycling process of the materials ; (2) the epidemiological models with time delays in the process for susceptible individuals to become infectious after they are infected ; (3) the mathematical models with time delays in medical science for the delivery of drug directly to the macrophage by using the phagocytosis of senescent red blood cells.
We investigated the effects of time delays on persistence of the above models and established the analytic methods for the global stability of the mathematical models in biology with time delays. The results obtained in this research are as follows : (1) We det
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ermined an initial age distribution of red blood cells so that the duration of the therapeutic effect is the maximum (paper 2). (2) We proved some difference-differential inequalities and applied them for the stability analysis of nonlinear retarded or neutral functional differential systems (paper 3, 4, 6). In particular, we transformed retarded systems to neutral ones and analyzed population dynamics (paper 1, 7, 8). (3) The method obtained in 2 was applied for (2)(paper 5, 9) and its extended model (paper 10, 11, 20). (4) The same method was applied for a 2-dimensional neural network and the effect of time delays on its stability (paper 12). (5) We analyzed the dynamic behaviors of mathematical models in population dynamics (difference systems) and studied the conditions for chaos to occur. We proved the system evolves to chaotic system from stable one (paper 14,18). (6) We obtained some conditions for the non-existence of periodic solutions in the dynamical systems (paper 15-17). (7) We proposed the time delayed models with the effect of the environmental hormone and analyzed their persistence and stability (paper 19). (8) We considered persistence and catastrophe for lattice models (paper 13,19). Less
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