Dynamic maximum entropy provides accurate approximation of structured population dynamics

TL;DR

This paper revisits a dynamic maximum entropy method that simplifies complex stochastic population models into low-dimensional deterministic equations, accurately capturing key macroscopic behaviors even in rapidly changing environments.

Contribution

It explains and demonstrates how the dynamic maximum entropy method effectively approximates structured population dynamics in non-stationary conditions.

Findings

Exact recovery of Ornstein-Uhlenbeck process dynamics

High accuracy in stochastic island model with migration

Effective in rapidly changing environments

Abstract

Realistic models of biological processes typically involve interacting components on multiple scales, driven by changing environment and inherent stochasticity. Such models are often analytically and numerically intractable. We revisit a dynamic maximum entropy method that combines a static maximum entropy and a quasi-stationary approximation. This allows us to reduce stochastic non-equilibrium dynamics expressed by the Fokker-Planck equation to a simpler low-dimensional deterministic dynamics, without the need to track microscopic details. Although the method has been previously applied to a few (rather complicated) applications in population genetics, our main goal here is to explain and to better understand how the method works. We demonstrate the usefulness of the method for two widely studied stochastic problems, highlighting its accuracy in capturing important macroscopic quantities even in rapidly changing non-stationary conditions. For the Ornstein-Uhlenbeck process, the method recovers the exact dynamics whilst for a stochastic island model with migration from other habitats, the approximation retains high macroscopic accuracy under a wide range of scenarios for a dynamic environment.