Kenny Xie L&S Math & Physical Sciences
Mean Field Theory for Carrying Capacity Dynamics for Carrying Capacity
How do we let mathematical models of evolution feel the squeeze of competition? Current phylogenetic “birth-death” frameworks treat every lineage as if it lives alone; they falter once growth slows because space or resources run out. My project tackles the open challenge of mean-field interacting multi-type birth-death processes: proving that those models still behave predictably when each individual’s birth and death rates grow in proportion to the average population size instead of being artificially capped. Achieving this will require replacing classical Lipschitz bounds with ones that depend on Wasserstein distances, weaving together probability, optimal-transport geometry, and fixed-point analysis. A successful proof will extend the powerful “propagation-of-chaos” machinery to density-dependent interactions and open the door to fast, likelihood-based inference for real-world problems—from how antibodies evolve inside germinal centres to how viruses spread through communities—giving evolutionary trees the mathematics they need to breathe under crowding.
Message To Sponsor
Thank you for investing in my SURF project this summer. Your support allows me to explore how evolutionary dynamics can be harnessed to understand competition and cooperation among learning systems—work I’m deeply passionate about. The opportunity to dive into this question at Berkeley will sharpen my research skills and, I hope, contribute fresh insights to the field. I’m grateful for your generosity in making this experience possible.