Evolutionary dynamics from a variational principle

Abstract

We demonstrate with a thought experiment that fitness-based population dynamical approaches to evolution are not able to make quantitative, falsifiable predictions about the long-term behavior of evolutionary systems. A key characteristic of evolutionary systems is the ongoing endogenous production of new species. These novel entities change the conditions for already existing species. Even {\em Darwin's Demon}, a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time, could not pre-state the impact of these novelties on established populations. We argue that fitness is always {\it a posteriori} knowledge -- it measures but does not explain why a species has reproductive success or not. To overcome these conceptual limitations, a variational principle is proposed in a spin-model-like setup of evolutionary systems. We derive a functional which is minimized under the most general evolutionary formulation of a dynamical system, i.e. evolutionary trajectories causally emerge as a minimization of a functional. This functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean-field approximation. We test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves. The model is further able to reproduce stylized facts of timeseries from several man-made and natural evolutionary systems. Light will be thrown on how species and their fitness landscapes dynamically co-evolve.Comment: 13 pages, 3 figures, 2 table