LEAPS-MPS:Research in Applied Analysis: Evolution Equations on Networks

This project investigates novel chemotaxis phenomena governing the directed movement of mobile agents in response to various types of stimuli. Chemotaxis is an important mechanism which enables biospecies, such as bacteria, cells, and other organisms to migrate, spread or localize within complex media. This mechanism is ubiquitous in nature as it facilitates a wide variety of fundamental processes including transport of immune cells to infected regions, movement of embryonic cells, tumor growth, and migration of biospecies toward food sources. The focus of this project is on the chemotaxis model and dynamics of organisms spatially confined to multiconnected networks such as, for example, fibers of dermal tissues, arteries in the vascular systems, or natural river systems. The principal objective is to provide new analytical insights into long-term dynamics and spreading phenomena of mobile agents on several types of networks. This goal is achieved through mathematical analysis of a system of partial differential equations on metric graphs. In addition, the PI will organize a research seminar at Auburn University and a summer school adjacent to the fourth Joint Alabama-Florida DEDS conference. These activities will provide ample opportunities for students to enter mathematical research in a paced and guided fashion. Furthermore, this project provides unique educational experience for students in Alabama and Florida and facilitates their entry into advanced-degree programs. The study of nonlinear waves and other coherent structures on graphs is a novel, rapidly developing, and intriguing field of research in applied dynamical systems and partial differential equations. This project concerns nonlinear waves arising in population dynamics driven by chemotaxis. The main focus is on the reaction-diffusion Fisher-KPP model and the reaction-advection-diffusion Keller-Segel model posed on metric graphs. Employing operator semigroup methods, a priori estimates for solutions of heat equation on graphs, and comparison principles, the principal investigator studies global well-posedness of relevant partial differential equations on unbounded networks and determines spatial spreading properties of biologically relevant solutions. In addition, the PI investigates, via the Maslov index, stability of traveling wave solutions for the Fisher-KPP equation in the moving environment. Located at the interface of applied dynamical systems and partial differential equations this proposal addresses several models originated in bio-engineering, ecology, and sociology. 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.