Mathematical Sciences: Nonlinear Mathematical Transport Theory and Population Models

本文主要研究由数学输运理论和人口模型产生的非线性偏积分-微分方程组和积分差分方程组。前者研究分子或颗粒在气体中的碰撞、吸收和反射效应的一般输运过程，其动力学过程受非线性Boltzman方程的支配。文中将讨论大时间的存在性、唯一性、对初始数据的连续依赖性、光滑性、逼近和渐近性等问题。这些技巧将使用有序Banach空间和格中的抽象微分方程、强连续算子半群和路径积分的理论。与气体运动行为相关的具体模型将被考虑，其中碰撞受一般相互作用势支配。种群动力学的研究将利用生长-扩散过程的离散时间性质，其演化由一个动态系统描述，该系统模拟生活在同一栖息地的物种之间的竞争。相互作用假设有两种定性形式：和平竞争或捕食者-猎物竞争。这项工作的基本关注点是建立物种共存的条件，并获得单个物种的最优扩散策略。这项工作的数学背景是正算符的谱理论应用于Frechet导数，研究与增长和色散相关的Hammerstein方程的解。

This work will focus on the investigation of nonlinear partial integro-differential and integro-difference equations which arise from mathematical transport theory and population models. In the former, studies of the general transport process describing the effects of collision, absorbtion and reflection of molecules or particles in gases whose kinetics is governed by the nonlinear Boltzman equation are carried out. Problems of existence, uniqueness continuous dependence on initial data, smoothness, approximation and asymptotic behavior for large time will be undertaken. Techniques will employ the theory of abstract differential equations in ordered Banach spaces and lattices, strongly continuous operator semigroups and path integrals. Specific models associated with the kinetic behavior of gases where collisions are governed by general interaction potentials will be considered. Work on population dynamics will utilize the discrete time nature of a growth-dispersal process whose evolution is described by a dynamical system modeling competition among species living in the same habitat. Interaction assumptions have two qualitative forms: peaceful competition or predator-prey competition. Fundamental concerns in this work are the establishment of conditions for coexistence of species and obtaining optimal dispersion strategies for the individual species. The mathematical context for this work is the spectral theory of positive operators applied to Frechet derivatives, in studying solutions of Hammerstein equations which relate growth and dispersion.