From the Pain in the Torus to a Repulsion-Diffusion Equation

The phenomenon that underlies what Felsenstein famously dubbed “the pain in the torus” in 1975 can loosely be described as follows: In one and two dimensions, spatial population models with independent critical branching, and diffusive spatial motion, concentrate in increasingly few well-separated clumps, until they eventually die out. We illustrate this phenomenon with simulations, and sketch a proof in the setting of superBrownian motion.
Real populations do not behave in this way, and one of the reasons is that individuals migrate away from overcrowded areas. This effect can be incorporated into superBrownian motion as a pairwise repulsion between individuals. We explain how this relates to a certain (deterministic) repulsion-diffusion equation, for which we show well-posedness and give conditions on the strength of the repulsion that ensure global boundedness of solutions.