Abstract
We study individual-based dynamics in finite populations, subject to randomly switching environmental conditions. These are inspired by models in which genes transition between on and off states, regulating underlying protein dynamics. Similarly, switches between environmental states are relevant in bacterial populations and in models of epidemic spread. Existing piecewise-deterministic Markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching. Here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear-noise approximation. Specifically, we derive the stationary distributions of a number of model systems, in good agreement with simulations. This improves existing approaches which are limited to the regimes of fast and slow switching.
Highlights
There is a broad consensus that noise plays a crucial role in most dynamical systems in biology, chemistry, and in the social sciences
Kramers-Moyal and van Kampen expansions. The latter is known as the linear-noise approximation (LNA) [1,2]
Existing approaches either disregard intrinsic fluctuations and only account for environmental noise or they focus on cases in which there is a clear separation of time scales between the population dynamics and the dynamics of the environment
Summary
There is a broad consensus that noise plays a crucial role in most dynamical systems in biology, chemistry, and in the social sciences. Much of this work focuses on processes between discrete interacting individuals, which can be members of a population in epidemiology [4,5,6], atoms or molecules in chemical reaction systems [7,8], or proteins in the context of gene regulatory networks [9,10] Many of these models are Markovian and their natural description is in terms of a master equation which describes the time evolution of the underlying probability distribution of microstates. The second-order term in the expansion introduces some stochastic effects of noise in the population but approximates the individual-level dynamics by a simpler Gaussian process on a continuum domain [13]. In models with switching environments the lowest-order expansion in the strength of the intrinsic noise leads to a “piecewise-deterministic Markov process” (PDMP) [42,43] In this approximation the dynamics of the population of individuals is described by deterministic rate equations between stochastic switches of the environment.
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