Individual-based models under various time-scales

TL;DR

This paper reviews recent mathematical results on the scaling limits of individual-based models, linking biological time-scales with model parameters and highlighting the role of probability in understanding evolutionary dynamics.

Contribution

It provides a unified viewpoint on classical biological time-scales through mathematical scaling limits of individual-based models, emphasizing the role of probability theory.

Findings

Deterministic limiting behavior in many models

Multiple interconnected time-scales for selection and demography

Probability theory explains lineage randomness and mutation fixation

Abstract

This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encompass the timescale typical of the life-expectancy of a single individual, the longer one wherein a population can be characterized through its demographic dynamics, and at least four interconnected ones wherein selection occurs. The limiting behavior is generally deterministic. Yet, since there are selective effects on randomness in the history of lineages, probability theory is shown to be a key factor in understanding the results. Besides, randomness can be maintained in the limiting dynamics, for instance to model rare mutations fixing in the population.