IS32: Stochastic Eco-Evolutionary Models

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

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).

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.