Statistical mechanics of phenotypic eco-evolution: From adaptive dynamics to complex diversification

Abstract

The ecological and evolutionary dynamics of large populations can be addressed theoretically using concepts and methodologies from statistical mechanics. This approach has been extensively discussed in the literature, both within the realm of population genetics, which focuses on genes or “genotypes,” and in adaptive dynamics, which emphasizes traits or “phenotypes.” Following this tradition, here we construct a theoretical framework allowing us to derive “macroscopic” evolutionary equations from a general “microscopic” stochastic dynamics representing the fundamental processes of reproduction, mutation, and selection in a large community of individuals, each one characterized by its phenotypic features. Importantly, in our setup, ecological and evolutionary timescales are intertwined, which makes it particularly suitable to describe microbial communities, a timely topic of utmost relevance. The framework leads to a probabilistic description—even in the case of arbitrarily large populations—of the distribution of individuals in phenotypic space as encoded in what we call the “generalized Crow-Kimura equation” or “generalized replicator-mutator equation.” We discuss the limits in which such an equation reduces to the (deterministic) theory of “adaptive dynamics,” i.e., the standard approach to evolutionary dynamics in phenotypic space. Moreover, we emphasize the aspects of the theory that are beyond the reach of standard adaptive dynamics. In particular, by developing a simple model of a growing and competing population as an illustrative example, we demonstrate that the resulting probability distribution can undergo “dynamical phase transitions.” These transitions may involve shifts from a unimodal distribution to a bimodal distribution, generated by an evolutionary branching event, or to a multimodal distribution through a cascade of evolutionary branching events. Furthermore, our formalism allows us to rationalize these cascades using the parsimonious approach of Landau’s theory of phase transitions. Finally, we extend the theory to account for finite populations and illustrate the possible consequences of the resulting stochastic or “demographic” effects. Altogether, the present framework extends and/or complements existing approaches to evolutionary and adaptive dynamics and paves the way to more systematic studies of microbial communities as well as to future developments including theoretical analyses of the evolutionary process from the general perspective of nonequilibrium statistical mechanics.