Topics in Mathematical Biology and Fluid Mechanics

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This project focuses on several aspects of mathematical biology and fluid mechanics. Many critical biological phenomena involve fluid-cell interaction or fluid-like behaviors. For example, fertilization of marine animal eggs relies on the transportation of the ambient fluid stream and sardines form large fluid-like swarms to evade predators. The project's principal focus is to develop novel mathematical tools to analyze the various fluid-related or fluid-like effects in these biological phenomena. There are three main topics in this project. The first topic concerns the interaction between the fluid flow and biological phenomena, which is motivated by two experiments in marine animal fertilization and cell embryology. In the first experiment researchers observed optimal fluid speed to maximize the fertilization rate of abalone. The second experiment confirms that the cellular flow helps organ formation at the early stages of the embryos of Drosophila. Fluid mixing effect and fast-spreading effect play important roles in the analysis. These two effects introduce another 'fast diffusion time scale' into the biological system and significantly change the long-time behavior. The principal investigator (PI) plans to develop new mathematical tools that capture the interplay between these fluid phenomena and the biology involved. The project will provide research training experience for undergraduate students.A modified hypocoercivity functional, which was applied in the paper by J. Bedrossian and the PI, and detailed spectral analysis, are crucial to derive the 'fast diffusion time scale' in the nonlinear setting. Applying these tools to the biologically relevant models provides deeper understandings of the experiments. The second topic covers the single and multi-species hydrodynamic models of flocking behavior. In these models, the flock of agents converges to a limiting structural state through sharing information among individuals. However, if the agents only share information with their direct neighbors, the flock's limiting behavior is highly nontrivial. The PI plans to develop new tools and bring ideas from spectral graph theory and random graph theory to analyze it. The third topic concerns the growth/blow-up mechanism in fluid mechanics. The PI will study two non-local models related to the generation of small-scale blowup in the fluid motivated by the singular Hou-Luo scenario for the 3D-Euler equation. The tools developed here may turn out to be useful in some biological models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.