Simulation of Spatial Systems with Demographic Noise

TL;DR

This paper compares two operator-splitting techniques for simulating demographic noise in spatial population dynamics, identifying biases in one method and proposing a hybrid approach to improve accuracy and stability.

Contribution

It reveals a bias in the DCM scheme for simulating demographic noise and introduces a hybrid method that combines deterministic diffusion with stochastic simulation for better results.

Findings

DCM scheme exhibits a strong bias towards the active phase.

Treating diffusion deterministically removes the bias.

Hybrid approach improves simulation stability and accuracy.

Abstract

Demographic (shot) noise in population dynamics scales with the square root of the population size. This process is very important, as it yields an absorbing state at zero field, but simulating it, especially on spatial domains, is a non-trivial task. Here we compare the results of two operator-splitting techniques suggested for simulating the corresponding Langevin equation, one by Pechenik and Levine (PL) and the other by Dornic, Chat\'e and Mu\~noz (DCM). We identify an anomalously strong bias toward the active phase in the numerical scheme of DCM, a bias which is not present in the alternative scheme of PL. This bias strongly distorts the phase diagram determined via the DCM procedure for the range of time-steps used in such simulations. We pinpoint the underlying cause in the inclusion of the diffusion, treated as an on-site decay with a constant external source, in the stochastic part of the algorithm. Treating the diffusion deterministically is shown to remove this unwanted bias while keeping the simulation algorithm stable, thus a hybrid numerical technique, in which the DCM approach to diffusion is applied but the diffusion is simulated deterministically, appears to be optimal.