Dynamic maximum entropy provides accurate approximation of structured population dynamics

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.Author summary: Complex processes in biology and physics have much in common. Collective motion of animals can be well described by models of interacting particles and emergent collective behavior can often be characterized as phase transitions of such a model. 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. It is thus not surprising that this method is in turn useful in understanding biological systems. However, realistic problems such as structured population dynamics, studied in our work, are set in dynamic environments, caused for instance by fluctuations in food supply. Therefore, 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. We focus on two processes (one in physics, one in biology) in our study: the motion of a charged particle and the dynamics of structured populations, both in a changing environment. We show that the method is extremely accurate even when the environmental changes are very fast, thus providing a powerful tool to study both biological and physical processes in changing environments.