IS32: Stochastic eco-evolutionary models

date: 7/14/2025, time: 14:00-15:30, room: IM HS

Organizer: Nicolas Champagnat (Inria Center at Université de Lorraine)

Chair: Nicolas Champagnat (Inria Center at Université de Lorraine)

The gene’s eye-view of quantitative genetics

Emmanuel Schertzer (University of Vienna)

Modelling the evolution of a continuous trait in a biological population is one of the oldest problems in evolutionary biology, which led to the birth of quantitative genetics. With the recent development of GWAS methods, it has become essential to link the evolution of the trait distribution to the underlying evolution of allelic frequencies at many loci, co-contributing to the trait value. The way most articles go about this is to make assumptions on the trait distribution, and use Wright’s formula to model how the evolution of the trait translates on each individual locus. Here, we take a gene’s eye-view of the system, starting from an explicit finite-loci model with selection, drift, recombination and mutation, in which the trait value is a direct product of the genome. We let the number of loci go to infinity under the assumption of strong recombination, and characterize the limit behavior of a given locus with a McKean-Vlasov SDE and the corresponding Fokker-Planck IPDE. In words, the selection on a typical locus depends on the mean behaviour of the other loci which can be approximated with the law of the focal locus. Results include the independence of two loci and explicit stationary distribution for allelic frequencies at a given locus (under some hypotheses on the fitness function). We recover Wright’s formula and the breeder’s equation as special cases. This is joint work with P. Courau and A. Lambert.

Convergence of a general structured individual-based model with possibly unbounded growth, birth and death rates

Virgile BRODU (Université de Lorraine, Inria Nancy Grand-Est)

First, we introduce càdlàg measure-valued processes, with biological motivations. We focus on a construction with Poisson point measures, with useful martingale properties. Then, we present a general convergence result for these measure-valued processes. In the literature, such a convergence result is proven with bounded growth, birth and death rates (\([1]\),\([2]\),\([3]\),\([4]\),\([5]\)). We present a more general framework with unbounded rates, where the convergence still holds true in a weighted space of measures. This is joint work with Nicolas Champagnat and Coralie Fritsch.

Bibliography

\([1]\) Nicolas Fournier and Sylvie Méléard. "A microscopic probabilistic description of a locally regulated population and macroscopic approximations.” The Annals of Applied Probability, vol. 14, no. 4, 2004, pp. 1880-1919.

\([2]\) Nicolas Champagnat, Régis Ferrière and Sylvie Méléard. "Individual-based probabilistic models of adaptive evolution and various scaling approximations.” Seminar on Stochastic Analysis, Random Fields and Applications V: Centro Stefano Franscini, Ascona, May 2005, 2008, pp. 75-113.

\([3]\) Viet Chi Tran. "Large population limit and time behaviour of a stochastic particle model describing an age-structured population." ESAIM: PS, vol. 12, 2008, pp. 345-386.

\([4]\) Fabien Campillo, Nicolas Champagnat and Coralie Fritsch. "Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models." Journal of Mathematical Biology, vol. 73, no. 6-7, 2016, pp. 1781-1821.

\([5]\) Josué Tchouanti. "Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat." Stochastics and Partial Differential Equations: Analysis and Computations, vol .12, no. 1, 2024, pp. 466-524.

Origin and persistence of polymorphism in loci targeted by disassortative preference: a general model

Manon Costa (Université de Toulouse)

The emergence and persistence of polymorphism within populations generally requires specific regimes of natural or sexual selection. Here, we develop a unified theoretical frame- work to explore how polymorphism at targeted loci can be generated and maintained by either disassortative mating choice or balancing selection due to, for example, heterozygote advantage. In this talk I will develop the result obtained in \([1]\). Our theoretical study of the model confirms that the conditions for the persistence of a given level of allelic polymorphism depend on the relative reproductive advantages among pairs of individuals. Interestingly, equilibria with unbalanced allelic frequencies were shown to emerge from successive introduction of mutants. We then investigate the role of the function linking allelic divergence to reproductive advantage on the evolutionary fate of alleles within the population. Our results highlight the significance of the shape of this function for both the number of alleles maintained and their level of genetic divergence.

Bibliography

\([1]\) Coron, C., Costa, M. Leman, H, Llaurens, V. and Smadi, C."Origin and persistence of polymorphism in loci targeted by disassortative preference: a general model". Journal of Mathematical Biology, 86(4) (2023).