A PDE approach to Collective Behavior in Biological Systems: Nonlocal Interactions and Free Boundaries

Understanding how large groups of cells or other organisms organize into collective patterns such as clustering, swarming, and tissue expansion is a fundamental challenge in applied sciences. These behaviors play an important role in biology and medicine, for example in wound healing, tumor growth, and the spread of microbial colonies. A basic difficulty is that simple mechanisms at the level of individuals -- chemical signaling, attraction/repulsion, growth, and crowding -- can produce complicated large-scale outcomes, including sharp boundaries, moving fronts, and abrupt transitions between different regimes. Such phenomena can be difficult to predict reliably with existing mathematical models. The investigator will build and analyze new models to explain how the emergence of these phenomena occur and the principles that govern them. This includes the introduction of a new class of mathematical models for the range expansion of immotile cell colonies, motivated by experimental observations that show that classical models, the so-called reaction-diffusion equations, do not accurately capture some important biological features. This will advance the mathematical foundations of collective behavior and will provide a framework for connecting experiments to theory, thereby strengthening prediction in the life sciences. The project also supports education and training of students and postdocs working at the interface of partial differential equations, analysis, and mathematical biology. The investigator will study partial differential equations (PDEs) that combine proliferation or aggregation with congestion/saturation effects. One direction concerns Patlak-Keller-Segel type chemotaxis models with nonlinear diffusion and nonlocal interactions. Using multiscale analysis and long-time asymptotics, the investigator will derive free-boundary problems that describe the dynamics after aggregation and will connect these limits to geometric interface motions, including flows driven by fractional mean curvature or surface diffusion. A second direction concerns kinetic swarming models with nonlocal attraction-repulsion interactions and the derivation of hydrodynamic models on moving domains to describe the collective dynamics after aggregation occurs. The main analytical tools include energy methods, compactness arguments, and relative-entropy ideas to obtain stability and convergence in singular limits. A unifying theme of the proposal is how growth and congestion shape collective behaviors: the investigator will analyze a tumor model coupling growth-fragmentation dynamics with mechanical congestion, with the goal of describing long-time dynamics through traveling-wave solutions. In each application area, the mathematical tools - free-boundary limits, interface dynamics, hypocoercive/entropy methods, and nonlocal PDE analysis - are used to pass from individual interaction rules to macroscopic predictions, including front speeds, interface laws, and regime transitions, that can be compared with experiments. 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.