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Applied Spectral Analysis in Population Dynamics, Biophysics, and Physical Chemistry

群体动力学、生物物理学和物理化学中的应用光谱分析

基本信息

批准号：
1714402
负责人：
Stanislav Molchanov
金额：
$ 23.26万
依托单位：
University of North Carolina at Charlotte
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714402&HistoricalAwards=false
关键词：
Applied Spectral Analysis Population Dynamics

项目摘要

This research project will carry out development and analysis of several mathematical models. The first set of questions originates in biotechnology. Fast progress in modern biophysics requires new theoretical results on protein solutions (suspensions), particularly in the studies of protein diffusion and of transitions between globular and diffusive states. Diffusion processes in polymer solutions differ from classical diffusion -- while the diffusion coefficient grows linearly with temperature in the classical setting, it decreases near the critical temperature for proteins. A mathematical model for phase transitions in polymers will be developed, allowing one to explain this fact and to explore other near-critical phenomena. One of the possible applications is the analysis of the diffusion of medications in living tissue. The second part of the project concerns population dynamics. Some of the most important questions in this area concern the description of states that are homogeneous in space and time and the study of their stability with respect to local and random perturbations of the environment. These questions will be studied for a wide class of models, including cases with immigration and heavy-tailed migration in the population. The project also includes a study of nonlinear partial differential equations describing the evolution of wave packets in liquids.The research will be based mostly on the spectral analysis of non-traditional Schrodinger type operators: nonlocal operators and operators on fractals and fractal graphs. A general spectral theory of convolution-type operators and their perturbations will be developed. New results on the inverse scattering problem are anticipated as a result of the last part of the project.

本研究项目将对几个数学模型进行开发和分析。第一组问题源于生物技术。现代生物物理学的快速发展需要关于蛋白质溶液(悬浮液)的新的理论结果，特别是在蛋白质扩散和球状和扩散状态之间的转变的研究中。聚合物溶液中的扩散过程不同于经典扩散过程--在经典环境中，扩散系数随温度线性增长，但在蛋白质的临界温度附近下降。将开发一个聚合物相变的数学模型，使人们能够解释这一事实，并探索其他近临界现象。其中一个可能的应用是分析药物在活组织中的扩散。该项目的第二部分涉及人口动态。这一领域中的一些最重要的问题涉及空间和时间上齐次的状态的描述以及它们相对于环境的局部和随机扰动的稳定性的研究。这些问题将在广泛的一类模型中进行研究，包括移民和人口中的重尾移民的情况。该项目还包括描述波包在液体中演化的非线性偏微分方程组的研究。该研究将主要基于非传统薛定谔算子的频谱分析：非局部算子以及关于分形图和分形图的算子。我们将发展卷积型算子及其扰动的一般谱理论。作为项目最后部分的结果，逆散射问题预计会有新的结果。