报告摘要：

      Biological systems often evolve on different time scales or take place on various length scales. For instance, predators disperse faster than the prey; vectors (mosquitoes) have shorter lifespan than hosts (humans); some species exhibit more rapid evolution, and so on. Geometric singular perturbation theory is a very powerful tool in analyzing physical and biological systems with different time scales. Relaxation oscillations, typically occur in dynamical systems with multiple time scales, are periodic orbits with slow and fast segments. In this talk, I will review Fenichel's theory on geometric singular perturbation theory and introduce a new criterion for the existence of relaxation oscillations based on extending the so-called entry-exit function to multi-dimensional slow-fast systems. Various multi-scale biological systems, such as predator-prey systems, epidemic models, rapid evolution systems with switching prey, and eco-evolutionary systems which exhibit slow-fast dynamics and relaxation oscillations, will be presented. (Based on a joint paper with Ting-Hao Hsu).

报告人简介：

       Professor Ruan Shigui，1992年在加拿大阿尔伯特(Alberta)大学数学系获博士学位，现为美国迈阿密大学(University of Miami)数学系终身教授。主要研究领域是动力系统和微分方程及其在生物和医学中的应用，在包括《PNAS》、《Lancet Infect Dis》、《Memoirs Amer Math Soc》、《J Functional Analysis》、《J Math Pures Appl》、《Math Ann》、《SIAM J Math Anal》等学术期刊上发表了约200篇学术论文，受到了国内外同行的关注与大量引用，2014 和2015年连续被汤森路透集团列为全球高被引科学家。担任了一些重要学术期刊如《BMC Infectious Diseases》、《Bulletin of Mathematical Biology》、《DCDS-B》、《Mathematical Biosciences》等的编委，是《Mathematical Biosciences and Engineering》的主编（数学）。