Coarse graining the Cyclic Lotka-Volterra Model: SSA and local maximum likelihood estimation

TL;DR

This paper develops a method to approximate the deterministic drift component of a stochastic simulation of the Cyclic Lotka-Volterra model using local diffusion models fitted via maximum likelihood, enabling analysis of dynamic features.

Contribution

It introduces a novel approach combining SSA simulations with local maximum likelihood estimation of diffusion models for analyzing complex stochastic systems.

Findings

Successful fitting of local diffusion models to SSA data

Enhanced ability to perform stability and steady-state analysis

Framework for guiding further SSA experiments

Abstract

When the output of an atomistic simulation (such as the Gillespie stochastic simulation algorithm, SSA) can be approximated as a diffusion process, we may be interested in the dynamic features of the deterministic (drift) component of this diffusion. We perform traditional scientific computing tasks (integration, steady state and closed orbit computation, and stability analysis) on such a drift component using a SSA simulation of the Cyclic Lotka-Volterra system as our illustrative example. The results of short bursts of appropriately initialized SSA simulations are used to fit local diffusion models using Ait-Sahalia's transition density expansions \cite{ait2,aitECO,aitVEC} in a maximum likelihood framework. These estimates are then coupled with standard numerical algorithms (such as Newton-Raphson or numerical integration routines) to help design subsequent SSA experiments. A brief discussion of the validity of the local diffusion approximation of the SSA simulation (a jump process) is included.