Dynamic maximum entropy provides accurate approximation of structured population dynamics

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.

Exponential Families with External Parameters

“…This section is motivated by a problem in quantitative genetics which has been dealt with in a series of papers [2][3][4]. See also Appendix A for a brief account.…”

“…which is equation (5.2) in [3] or equation (12) in [4]. It can be given normal form since g αβ is invertible.…”

“…We give a brief account of a classical problem in dynamics of quantitative traits whose approach to equilibrium described by a Fokker-Planck equation has been investigated in [2][3][4]. See also [37] for a gentle introduction to dynamics of populations.…”

“…Section 6 is motivated by a model problem in quantitative genetics (briefly recalled in Appendix A) where the dynamics of the system is given by a Fokker-Planck equation with gradient drift and the equilibrium or stationary distribution is an exponential family with external parameters. We recast the dynamic approximation procedure exposed in [2][3][4] in the framework of exponential family with external parameters and we give a generalization of the ODE that drives the approximating dynamics. We think that the consideration from the present point of view of the problem exposed in [2][3][4] may shed light on some still poorly understood aspects of the model.…”

“…We recast the dynamic approximation procedure exposed in [2][3][4] in the framework of exponential family with external parameters and we give a generalization of the ODE that drives the approximating dynamics. We think that the consideration from the present point of view of the problem exposed in [2][3][4] may shed light on some still poorly understood aspects of the model.…”