Mathematical Sciences: Nonlinear Mathematical Transport Theory and Population Models

这项工作将集中在非线性偏积分-微分方程和积分-差分方程的研究，这些方程产生于数学传输理论和人口模型。在前者中，研究了描述气体中分子或粒子的碰撞、吸收和反射效应的一般输运过程，其动力学由非线性玻尔兹曼方程控制。研究了存在性、唯一性、对初始数据的连续依赖、光滑性、近似性和大时间渐近性等问题。技术将采用有序巴拿赫空间和格中的抽象微分方程理论，强连续算子半群和路径积分。将考虑与气体动力学行为有关的特定模型，其中碰撞由一般相互作用势控制。种群动力学的研究将利用生长-扩散过程的离散时间性质，该过程的进化是由生活在同一栖息地的物种之间的动态系统模拟竞争来描述的。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。本工作的基本问题是建立物种共存的条件和获得最佳的分散策略。这项工作的数学背景是应用于Frechet导数的正算子的谱理论，用于研究与增长和色散有关的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.