Topics in Mathematical Biology and Fluids

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

Transport and diffusion phenomena are ubiquitous in nature. For example, various important biochemical reactions take place in moving fluid flows. The reactant densities are transported by the flow and diffuse according to Fick's law. The principal investigator (PI) plans to develop a novel mathematical toolkit to describe the delicate interplay between transportation and diffusion in various physical and biological contexts. For instance, in specific scenarios, the ambient fluid flow can create small-scale structures in the densities involved and enhance their diffusion. A deeper understanding of this enhanced diffusion phenomenon has implications across various disciplines, ranging from stabilizing the chemotaxis process to improving communication efficiency in collective motions. Through detailed mathematical analysis, the PI plans to identify situations where this enhanced diffusion phenomenon plays a major role and to capture the interesting dynamics of the associated systems. The PI also plans to recruit talented undergraduate and graduate students to participate in this research project. Through this academic training, the PI hopes to equip the students with sufficient knowledge and skills to address future challenges that arise in science and technology. This project aims to develop novel mathematical tools to analyze the long-time behaviors of coupled biology-fluid systems and transport-type equations arising in biological phenomena. The project addresses three main topics. In the first project, the PI plans to explore the delicate interaction between biological phenomena and their ambient fluid flows. Fluid transport phenomena can alter the overall qualitative features of biological processes. For example, the introduction of strong fluid flows can mitigate certain chemotaxis-induced concentration effects. The PI plans to develop mathematical tools to describe delicate interactions within coupled biology-fluid systems. In the second project, biological experiments guide the mathematical analysis. In the ocean, marine animals such as abalone release eggs and sperm in the fluid stream. Eggs emit chemical attractants while sperm aggregate towards them via random walk and chemotaxis. Once the gametes meet, the fertilization happens. Given that these processes occur in fluid flows effectively sheared on the length scales involved, it is biologically intriguing to study the relationship between fertilization rate and shear rate. The PI plans to develop faithful mathematical models and provide a convincing explanation for the experimental data from marine scientists. The third project focuses on hydrodynamic stability and small-scale creation in fluid mechanics. The PI plans to explore the stabilization mechanisms of shear flows in Navier-Stokes systems and investigate non-local models related to the Euler equation. A deeper understanding of these systems might be helpful in understanding the coupled biology-fluid systems. 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.