Mathematical Sciences: Mathematical Models of Structured Population Dynamics

韦伯 研究了结构种群动力学的线性和非线性偏微分方程的定性行为。 在这些方程中，结构变量如年龄、大小、成熟度或种群成员的其他物理属性将个体种群行为与总体种群行为联系起来。 研究方法主要是利用Banach空间中线性算子半群理论、Banach空间中线性算子谱理论、Banach格中正线性和非线性算子理论。 该项目将这些研究应用于（1）血液生产系统的模型和该系统中的异常，如再生障碍性贫血，（2）增殖细胞系染色体末端端粒缺失的模型和在这些缺失过程中正常细胞和癌细胞的区别，（3）AZT治疗AIDS的定性模型和治疗结果HIV耐药株的发展，（4）肿瘤细胞群的周期性化疗模型，（5）具有传染潜伏期和地理依赖性的流行病模型。 这个项目研究人口的行为方式，如果这种行为是由某种包含人口特征的方程描述的话。 这些特征包括人口成员的年龄、大小或成熟度。 数学结果被应用到模型的细胞群体，流行病，和MF治疗。 在这些不同的应用中，物理和生物现象的定性行为被建模为动态种群过程。 这项工作的意义在于发展数学方法来分析描述结构化群体的方程的解的定性行为，并将这些方法应用于生物技术过程的定性理解。

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.