As a a Ph.D. student at the University of Arizona, I worked with Ibrahim Fatkullin, and was generally interested in:
- Stochastic differential equations with singular coefficients
- The Bessel process
- Interacting stochastic particle systems which exhibit swarming/aggregation
- The hydrodynamic limits of such systems
- Models of collective behavior
At CWRU, I'm interested in applying some of these interests in a more ecological context.
In my PhD, I worked on a system of singularly-interacting stochastic particles of varying masses, which experience inelastic ("sticky") collisions. This system can be viewed as 2D overdamped gravity. A preprint is available:
- Coalescing particle systems and applications to nonlinear Fokker-Planck equations, with Ibrahim Fatkullin. (Accepted to CMS; preprint on arXiv.)
In addition to describing this particle system, it also provides some results concerning blow-up in the multispecies Keller-Segel model.
When scaled appropriately, the hydrodynamic limit of the empirical mass density converges to the solution of a nonlinear Fokker-Planck equation, such as the Keller-Segel model of chemotaxis, or its generalization, the multispecies Keller-Segel model. These PDEs blow up in finite time, and form Dirac-type singularities. Such blow-ups correspond to coalescence in the particle system.
By analyzing collision dynamics, I developed an efficient scheme for detecting and simulating collisions, despite the singularities which are in involved. For example, here's a coarsening system of ~40k pairwise interacting particles: