Sylvie Méléard is Professor at Ecole polytechnique, France, and senior Fellow of the Institut Universitaire de France. She is a specialist in stochastic processes, measure-valued processes, quasi-stationary distributions and is interested in their applications in biology, particularly in evolutionary ecology, bacterial biology and hematology. She is the head of the French MMB (Mathematical Modeling and Biodiversity) network between biologists and mathematicians, and PI of the SINGER project (Stochastic dynamics of single cells: growth, emergence and resistance). She gave an invited lecture at ICM2022. She was Editor-in-Chief of the journal Stochastic Processes and their Applications from 2018 to 2021 and Professor at University Paris Nanterre, France, from 1992 to 2006.

This Medallion Lecture will be given at the INFORMS/APS meeting in Nancy, France, June 28–30.

Multiscale eco-evolutionary models: from individuals to populations

Since Darwin’s revolutionary work on evolution and natural selection, many mathematicians have worked on modeling his theories. Different schools of thought have developed, involving different classes of mathematical objects. Ecological models of structured population dynamics usually rely on deterministic models in large populations, such as dynamical systems (as the famous Lotka–Volterra system) and partial differential equations. Population genetics are more interested in random fluctuations of gene frequencies in small populations (like in the Wright–Fisher model) and therefore make extensive use of probabilistic tools. A few decades ago, eco-evolutionary models emerged, seeking to link these two approaches. Our work is placed in this framework. Our point of view consists in focusing on stochastic individual behaviors, taking into account demographic parameters (birth and death rates), evolutionary parameters (mutations, gene transfer), and ecological parameters (interactions between individuals), all these parameters depending on the genetic or phenotypic characteristics of the individual. This point of view is strongly reinforced by the ability of biologists to obtain more and more individual data, for example, for bacteria, thanks to single cell microscopes or microfluidic techniques. There are three main sources of randomness in eco-evolutionary mechanisms which happen at different time and size scales: at the molecular level (errors in DNA replication or genetic information exchanges), at the individual level (division time, life span, contacts, access to resources), and at a macroscopic level (environmental variations). Mathematically, it is very exciting that all the parameters we have mentioned have their own scales, which can be different according to the species considered and also can vary according to the environment. Depending on these scales, the mathematical models and the associated mathematical questions can be of different nature and challenging and open new fields of investigation.

We consider bacteria or cell populations. The ability of an individual to survive or divide depends on phenotypic or genetic parameters whose quantitative expression (real or vectorial) is called a trait. The evolution of the trait distribution results from different main mechanisms. The heredity is the vertical transmission of the ancestral trait to offspring, except when a mutation occurs. The mutations generate trait variability in the population. Although their reproduction is asexual, bacteria or cells can horizontally exchange genetic information during their life. The selection process takes place at two levels. The variability in traits allows an individual with a higher probability of survival or a better ability to reproduce to create a subpopulation of offspring that will invade the population (genetic selection). In addition, selection also favors those individuals best able to survive in competition with others (ecological selection). Horizontal gene transfer is obtained by direct contact between cells, either by the transfer of small parts of chromosomal DNA or by the transfer of plasmids, small circular double-stranded DNA structures which can be very costly for the cell in terms of energy used. Gene transfer plays an essential role in the evolution, maintenance, transmission of virulence and antibiotic resistance.

One is faced with the fundamental question: how to describe and quantify the successive invasions of favorable mutants? The individual behaviors are described by the mean of a stochastic measure-valued process. We study different long time asymptotic behaviors depending on the assumptions on mutation size and frequency and on horizontal transmission rate. In some cases, simulations indicate that these models should exhibit surprising asymptotic behaviors such as cyclic behaviors. We explore these behaviors on a simple model where population and time sizes are on a log-scale. Explicit criteria are given to characterize the possible asymptotic behaviors. The impact of the time and size scales on macroscopic approximations is also investigated, leading to Hamilton-Jacobi equations.