Mathematical Sciences: Mathematical Models of Structured Population Dynamics

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Webb The investigtor studies the qualitative behavior of the linear and nonlinear partial differential equations of structured population dynamics. In these equations structure variables such as age, size, maturity, or other physical attributes of members of the population relate individual population behavior to total population behavior. The methods of the research employ the theory of semigroups of linear operators in Banach spaces, the spectral theory of linear operators in Banach spaces, and the theory of positive linear and nonlinear operators in Banach lattices. The project appliies these researches to (1) models of the blood production system and abnormalities in this system such as aplastic anemia, (2) models of loss of telomeres in ends of chromosomes in proliferating cell lines and the distinction of normal and cancer cells in these loss processes, (3) qualitative models of AZT treatment of AIDS and the development of resistant strains of HIV as a result of treatment, (4) models of periodic chemotherapy of tumor cell populations, and (5) models of epidemics with infection incubation periods and geographical dependence. This project studies the ways in which populations may behave, if the behavior is described by certain kinds of equations that incorporate features of the populations. Such features include the age, size, or maturity of members of the population. The mathematical results are applied to models of populations of cells, of epidemics, and mf treatments. In these various applications the qualitative behaviors of physical and biological phenomena are modelled as dynamical population processes. The significance of this work is in the development of mathematical methods to analyze the qualitative behavior of solutions of the equations that describe structured populations, and to apply these methods to the qualitative understanding of biotechnological processes.