Spatial reacti on-diffusion models have been employed to describe many emergent phenomena in biology. The modellin g technique most commonly adopted is systems of p artial differential equations (PDEs)\, which assum es there are sufficient densities of particles th at a continuum approximation is valid. However\, the simulation of computationally intensive indivi dual-based models has become a popular way to inv estigate the effects of noise in reaction-diffusio n systems.

The specific stochastic model s with which we shall be concerned in this talk a re referred to as `compartment-based'\; or `on- lattice'\;. These models are characterised by a discretisation of the computational domain into a grid/lattice of `compartments'\;. Within eac h compartment particles are assumed to be well-mi xed and are permitted to react with other particle s within their compartment or to transfer between neighbouring compartments.

In this work we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compart ment-based model in the other. Rather than attemp ting to balance average fluxes\, our algorithms an swer a more fundamental question: `how are indivi dual particles transported between the vastly diff erent model descriptions?'\; First\, we presen t an algorithm derived by carefully re-defining t he continuous PDE concentration as a probability d istribution. Whilst this first algorithm shows ve ry strong convergence to analytic solutions of te st problems\, it can be cumbersome to simulate. Ou r second algorithm is a simplified and more effic ient implementation of the first\, it is derived i n the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by co mparing the averaged simulations to analytic solu tions of PDEs for mean concentrations.

Rel ated Links

- http://rsif.royalsocietypublishing.org/content /12/106/20150141 - First paper associated wit h this talk \;