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

本研究项目将开展几个数学模型的开发和分析。第一组问题源于生物技术。现代生物物理学的快速发展要求对蛋白质溶液（悬浮液），特别是在蛋白质扩散和球形和扩散状态之间的转换的研究新的理论结果。聚合物溶液中的扩散过程与经典扩散不同-虽然在经典设置中扩散系数随温度线性增长，但在蛋白质的临界温度附近它会降低。聚合物中的相变的数学模型将被开发，允许一个解释这一事实，并探索其他近临界现象。可能的应用之一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。在这一领域的一些最重要的问题涉及的状态是均匀的空间和时间的描述和研究他们的稳定性相对于本地和随机扰动的环境。这些问题将研究广泛的一类模型，包括移民和重尾迁移人口的情况下。该项目还包括研究描述液体中波包演化的非线性偏微分方程，研究将主要基于非传统薛定谔型算子的谱分析：非局部算子和分形及分形图上的算子。将发展卷积型算子及其扰动的一般谱理论。作为该项目最后一部分的结果，预计逆散射问题的新结果。

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.