Accelerated stochastic and hybrid methods for spatial simulations of reaction–diffusion systems

Spatial distributions characterize the evolution of reaction–diffusion models of several physical, chemical, and biological systems. We present two novel algorithms for the efficient simulation of these models: Spatial τ -Leaping ( S τ -Leaping), employing a unified acceleration of the stochastic simulation of reaction and diffusion, and Hybrid τ -Leaping ( H τ -Leaping), combining a deterministic diffusion approximation with a τ -Leaping acceleration of the stochastic reactions. The algorithms are validated by solving Fisher’s equation and used to explore the role of the number of particles in pattern formation. The results indicate that the present algorithms have a nearly constant time complexity with respect to the number of events (reaction and diffusion), unlike the exact stochastic simulation algorithm which scales linearly.