Dynamic maximum entropy provides accurate approximation of structured population dynamics.

Katarína Bod’Ová,Enikő Szép,Nicholas H Barton,John P Barton

doi:10.1371/journal.pcbi.1009661

Katarína Bod’Ová, Enikő Szép + Show 2 more

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https://doi.org/10.1371/journal.pcbi.1009661

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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 with 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 in a dynamic environment.

Highlights

Conceptual understanding of realistic problems in applied sciences is often hindered by the curse of complexity, with quantities of interest coupling to finer features
When the system is settled to a steady state, statistical physics connects random fluctuations of the process with key macroscopic quantities, using the maximum entropy method
We use the dynamical maximum entropy approximation, which allows us to reduce the full problem to simpler dynamics, without the need to track microscopic details

Summary

Introduction

Conceptual understanding of realistic problems in applied sciences is often hindered by the curse of complexity, with quantities of interest coupling to finer features. Due to their multiscale character even simple questions lead to exploration of the full complexity of the system. Statistical mechanics provides a clever way to understand complex multiscale problems by linking processes on different scales through the parsimony principle—the method of maximum entropy (ME), introduced by [1]. The most straightforward approach is to use ME on dynamic trajectories, forcing constraints on the dynamical features (Fig 1B)

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