Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well-documented to intersperse much more tightly than traditionally assumed, especially in communities of microorganisms. Advancing the development of mathematical and computational approaches to shed novel light onto eco-evolutionary problems is a challenge of utmost relevance. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times—i.e., the times that individual bacteria from the community remain in a dormant state to cope with stress—evolves its average value to approximately fit the antibiotic-exposure time. Moreover, the distribution develops right-skewed heavy tails, revealing the presence of individuals with anomalously large lag times. Here, we develop a parsimonious individual-based model mimicking the actual demographic processes of the experimental setup. Individuals are characterized by a single phenotypic trait: their intrinsic lag time, which is transmitted with variation to the progeny. The model—in a version in which the amplitude of phenotypic variations grows with the parent’s lag time—is able to reproduce quite well the key empirical observations. Furthermore, we develop a general mathematical framework allowing us to describe with good accuracy the properties of the stochastic model by means of a macroscopic equation, which generalizes the Crow-Kimura equation in population genetics. Even if the model does not account for all the biological mechanisms (e.g., genetic changes) in a detailed way—i.e., it is a phenomenological one—it sheds light onto the eco-evolutionary dynamics of the problem and can be helpful to design strategies to hinder the emergence of tolerance in bacterial communities. From a broader perspective, this work represents a benchmark for the mathematical framework designed to tackle much more general eco-evolutionary problems, thus paving the road to further research avenues.

Multiple ecological forces act together in shaping the composition of microbial communities. Phyloecology approaches ---which combine phylogenetic relationships with community ecology--- have the potential to disentangle such forces, but are often hard to connect with quantitative predictions from theoretical models. On the other hand, macroecology, which focuses on statistical patterns of abundance and diversity, provides natural connections with theoretical models but often neglects inter-speficic correlations and interactions. Here, we propose a unified framework combining both such approaches to microbial communities. In particular, by using both cross-sectional and longitudinal abundance metagenomic data, we reveal the existence of a novel empirical macroecological law establishing that correlations in species-abundance fluctuations across communities decay from positive to null values as a function of phylogenetic similarity in a consistent manner across ecologically distinct microbiomes. We formulate three mechanistic models ---relying on alternative ecological forces--- that produce distinct predictions. We conclude that the empirically observed macroecological pattern can be quantitatively explained as a result of fluctuating shared resources, i.e. environmental filtering and not e.g. as a result of species competition. Finally, we also show that the macroecological law is also valid for temporal data of a single community, and that the properties of delayed temporal correlations are reproduced by our model.

The ecological and evolutionary dynamics of large sets of individuals can be naturally addressed from a theoretical perspective using ideas and tools from statistical mechanics. This parallelism has been extensively discussed and exploited in the literature, both in the context of population genetics and in phenotypic evolutionary or adaptive dynamics. Following this tradition, here we construct a framework allowing us to derive ‘macroscopic’ evolutionary equations from a rather general ‘microscopic’ stochastic dynamics representing the processes of reproduction, mutation and selection in a large community of individuals, each one characterized by its phenotypic features. In this set up, ecological and evolutionary timescales are intertwined, which makes it particularly suitable to describe microbial communities, a timely topic of utmost relevance. Our framework leads, even in the limit of infinitely large populations, to a probabilistic description of the distribution of individuals in phenotypic space, as encoded in what we call ‘generalized Crow-Kimura equation’ or ‘generalized replicator-mutator equation’. We discuss the limits in which such an equation reduces to the theory of ‘adaptive dynamics’ (i.e. the standard approach to evolutionary dynamics in phenotypic space, which usually focuses on the mean or at most the variance of such distributions). Moreover, we emphasize the aspects of the theory that are beyond the reach of standard adaptive dynamics. In particular, working out, as an example, a simple model of a growing and competing population, we show that the resulting probability distribution can possibly exhibit dynamical phase transitions changing from unimodal to bimodal —by means of an evolutionary branching— or to multimodal, in a cascade of evolutionary branching events. Furthermore, our formalism allows us to rationalize these events as a cascade of ‘dynamical phase transitions’, 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/adaptive dynamics and paves the way to more systematic studies of e.g. microbial communities as well as to future developments including theoretical analyses of the evolutionary process from the perspective of non-equilibrium statistical mechanics.

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