Interacting Particles Diffusing in Spatially Heterogeneous System

We propose and analyze a novel measure valued process which models the behaviour of interacting particles. The model is motivated by a modeling of complex intracellular reaction networks in spatially heterogeneous systems, but can be used for modeling other biological processes  affected by movement in continuous space, such as ecological competitions, epidemic dynamics, and some population genetics.

It models interaction dynamics between different molecular species and continuous movement of molecules in space. Interaction rates at a spatial location are proportional to the mass of different species present locally and to a location specific interaction rate, which may be a function of the local or global species mass as well.

We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types.
When the mass of all species scales the same way we get a deterministic limit, whose long-term behaviour depends on the mobility of types and localization of reactions. When the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.