A Partial-differential Approximation for Spatial Stochastic Process Algebra

We study a spatial framework for process algebra with ordinary differential equation (ODE) semantics. We consider an explicit mobility model over a 2D lattice where processes may walk to neighbouring regions independently, and interact with each other when they are in same region. The ODE system size will grow linearly with the number of regions, hindering the analysis in practice. Assuming an unbiased random walk, we introduce an approximation in terms of a system of reaction-diffusion partial differential equations, of size independent of the lattice granularity. Numerical tests on a spatial version of the generalised Lotka-Volterra model show high accuracy and very competitive runtimes against ODE solutions for fine-grained lattices.