Relevance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Last modified: 2014-06-09

#### Abstract

Position-jump processes are used for the mathematical

modelling of spatially extended chemical and biological systems with

increasing frequency. A large subset of the literature concerning such

processes is concerned with modelling the effect of stochasticity on

reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well mixed within each of these voxels and are allowed to react with other

molecules within the same voxel or to jump to neighbouring voxels with predefined transition rates.

For a variety of reasons implementing position-jump processes on

irregular grids is becoming increasingly important. However, it is not

immediately clear what form an appropriate irregular partition of the

domain should take if it is to allow the derivation of mean molecular

concentrations that agree with a given partial differential equation for

molecular concentrations. I will demonstrate that the Voronoi domain

partition is the appropriate method with which to divide the

computational domain, under the assumption of a fixed functional form for the transition rates.

In this talk, I will investigate theoretically the propriety of the

Voronoi domain partition as an appropriate method to partition domains for position-jump models and provide simulations of diffusion processes in order to corroborate our results.

modelling of spatially extended chemical and biological systems with

increasing frequency. A large subset of the literature concerning such

processes is concerned with modelling the effect of stochasticity on

reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well mixed within each of these voxels and are allowed to react with other

molecules within the same voxel or to jump to neighbouring voxels with predefined transition rates.

For a variety of reasons implementing position-jump processes on

irregular grids is becoming increasingly important. However, it is not

immediately clear what form an appropriate irregular partition of the

domain should take if it is to allow the derivation of mean molecular

concentrations that agree with a given partial differential equation for

molecular concentrations. I will demonstrate that the Voronoi domain

partition is the appropriate method with which to divide the

computational domain, under the assumption of a fixed functional form for the transition rates.

In this talk, I will investigate theoretically the propriety of the

Voronoi domain partition as an appropriate method to partition domains for position-jump models and provide simulations of diffusion processes in order to corroborate our results.