Yurij Salmaniw (University of Oxford) – Bifurcation analysis of nonlocal aggregation-diffusion equations and systems

Abstract: Nonlocal aggregation-diffusion equations have emerged as a fundamental mean-field approximation for large interacting particle systems, with applications spanning, for example, opinion dynamics, statistical mechanics, physics, synchronisation, and mathematical biology. While much research has focused on their qualitative properties—existence, uniqueness, linear stability analysis, and long-time behavior—understanding their quantitative structure remains a key challenge: What patterns form? Under which conditions do they emerge? Are phase transitions continuous or discontinuous? How do these transitions influence other properties of the solution(s)? I am interested in the rigorous treatment of such questions from an analytical perspective, using tools from bifurcation theory.

In this talk, I will present a bifurcation analysis applied to some nonlocal aggregation-diffusion equations with applications in ecology, focusing on three recent efforts. The first couples a scalar nonlocal aggregation-diffusion equation (population dynamics) with an ordinary differential equation (a 'spatial map' for the population). In the second case, we study a similar problem that can be transformed from a two-equation PDE-ODE system into a three-equation parabolic-elliptic-ODE system while maintaining the stability and solution structure properties. Finally, I will present some preliminary results for a fully nonlocal two-species aggregation-diffusion system and some current progress toward understanding the global bifurcation structure of these problems from a numerical point of view. Together, these examples highlight the versatility of a robust bifurcation analysis in understanding precise solution behaviour of complex nonlocal PDE models and the different (but related) ways these tools can be applied depending on the problem of interest.