Intermittency as metastability: A predictive approach to evolution in rugged landscapes

Max Planck Institute for Mathematics in the Sciences - Leipzig, Germany

^(a) smerlak@mis.mpg.de (corresponding author)

Received: 3 December 2020
Accepted: 26 February 2021

Abstract

Many systems across the sciences evolve through the interaction of multiplicative growth and diffusive transport. In the presence of disorder, these opposing forces can generate localized structures and bursty dynamics, a phenomenon known as “intermittency” in non-equilibrium physics and as “punctuated equilibrium” in evolutionary theory. This behaviour is difficult to forecast; in particular there is no general principle to locate the regions where the system will settle, how long it will stay there, or where it will jump next. Here I introduce a Markovian representation of growth-transport dynamics that closes these gaps. I show that any autonomous positive linear system can be mapped onto a generalization of the “maximal entropy random walk”, a Markov process with non-local transition rates. From this transformation follow a characterization of localization islands as minima of an effective potential, a general scheme to coarse-grain the state space along the basins of attraction of these minima, and an entropic Lyapunov function. These results unify the concepts of linear intermittency and metastability, and provide a generally applicable method to reduce, and predict, the dynamics of disordered linear systems. Applications range from Zeld’dovich's parabolic Anderson model to Eigen's quasispecies model of molecular evolution.