Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. Thus, the variance that segregates within families is not perturbed by selection, and can be predicted from the variance components. This does not necessarily imply that the trait distribution across the whole population should be Gaussian, and indeed selection or population structure may have a substantial effect on the overall trait distribution. One of our main aims is to identify some general conditions on the allelic effects for the infinitesimal model to be accurate. We first review the long history of the infinitesimal model in quantitative genetics. Then we formulate the model at the phenotypic level in terms of individual trait values and relationships between individuals, but including different evolutionary processes: genetic drift, recombination, selection, mutation, population structure, …. We give a range of examples of its application to evolutionary questions related to stabilising selection, assortative mating, effective population size and response to selection, habitat preference and speciation. We provide a mathematical justification of the model as the limit as the number M of underlying loci tends to infinity of a model with Mendelian inheritance, mutation and environmental noise, when the genetic component of the trait is purely additive. We also show how the model generalises to include epistatic effects. We prove in particular that, within each family, the genetic components of the individual trait values in the current generation are indeed normally distributed with a variance independent of ancestral traits, up to an error of order 1∕M. Simulations suggest that in some cases the convergence may be as fast as 1∕M.

We investigate a new model for populations evolving in a spatial continuum. This model can be thought of as a spatial version of the Λ-Fleming-Viot process. It explicitly incorporates both small scale reproduction events and large scale extinction-recolonisation events. The lineages ancestral to a sample from a population evolving according to this model can be described in terms of a spatial version of the Λ-coalescent. Using a technique of Evans (1997), we prove existence and uniqueness in law for the model. We then investigate the asymptotic behaviour of the genealogy of a finite number of individuals sampled uniformly at random (or more generally 'far enough apart') from a two-dimensional torus of sidelength L as L → ∞. Under appropriate conditions (and on a suitable timescale) we can obtain as limiting genealogical processes a Kingman coalescent, a more general Λcoalescent or a system of coalescing Brownian motions (with a non-local coalescence mechanism).

Many organisms have sex chromosomes with large nonrecombining regions that have expanded stepwise, generating “evolutionary strata” of differentiation. The reasons for this remain poorly understood, but the principal hypotheses proposed to date are based on antagonistic selection due to differences between sexes. However, it has proved difficult to obtain empirical evidence of a role for sexually antagonistic selection in extending recombination suppression, and antagonistic selection has been shown to be unlikely to account for the evolutionary strata observed on fungal mating-type chromosomes. We show here, by mathematical modeling and stochastic simulation, that recombination suppression on sex chromosomes and around supergenes can expand under a wide range of parameter values simply because it shelters recessive deleterious mutations, which are ubiquitous in genomes. Permanently heterozygous alleles, such as the male-determining allele in XY systems, protect linked chromosomal inversions against the expression of their recessive mutation load, leading to the successive accumulation of inversions around these alleles without antagonistic selection. Similar results were obtained with models assuming recombination-suppressing mechanisms other than chromosomal inversions and for supergenes other than sex chromosomes, including those without XY-like asymmetry, such as fungal mating-type chromosomes. However, inversions capturing a permanently heterozygous allele were found to be less likely to spread when the mutation load segregating in populations was lower (e.g., under large effective population sizes or low mutation rates). This may explain why sex chromosomes remain homomorphic in some organisms but are highly divergent in others. Here, we model a simple and testable hypothesis explaining the stepwise extensions of recombination suppression on sex chromosomes, mating-type chromosomes, and supergenes in general.

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Abstract. We survey a class of models for spatially structured populations which we have called spatial Λ-Fleming-Viot processes. They arise from a flexible framework for modelling in which the key innovation is that random genetic drift is driven by a Poisson Point Process of spatial 'events'. We demonstrate how this overcomes some of the obstructions to modelling populations which evolve in two (and higher) dimensional spatial continua, how its predictions match phenomena observed in data, and how it fits with classical models. Finally we outline some directions for future research.

We construct a measure-valued equivalent to the spatial Λ-Fleming-Viot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the state-space of the SLFV and to derive several path properties of the measure-valued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a look-down construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.

The success of filamentous fungi in colonizing most natural environments can be largely attributed to their ability to form an expanding interconnected network, the mycelium, or thallus, constituted by a collection of hyphal apexes in motion producing hyphae and subject to branching and fusion. In this work, we characterize the hyphal network expansion and the structure of the fungus Podospora anserina under controlled culture conditions. To this end, temporal series of pictures of the network dynamics are produced, starting from germinating ascospores and ending when the network reaches a few centimeters width, with a typical image resolution of several micrometers. The completely automated image reconstruction steps allow an easy post-processing and a quantitative analysis of the dynamics. The main features of the evolution of the hyphal network, such as the total length L of the mycelium, the number of "nodes" (or crossing points) N and the number of apexes A, can then be precisely quantified. Beyond these main features, the determination of the distribution of the intrathallus surfaces (S i ) and the statistical analysis of some local measures of N, A and L give new insights on the dynamics of expanding fungal networks. Based on these results, we now aim at developing robust and versatile discrete/continuous mathematical models to further understand the key mechanisms driving the development of the fungus thallus.Due to their paramount role in most ecosystems and in different sectors of human economy, filamentous fungi have attracted a lot of attention in the last decades. Indeed, these organisms are able to grow in complex environments, leading to their ubiquitous occurrence 1,2 . The success of filamentous fungi in colonizing most natural environments can be largely attributed to their ability to form hyphae, whose width typically ranges from 2 to 20 µm, and which are usually composed of multiple cells separated by septa 3 . At the edge of the fungal colony, apical cells explore new territories in search for food and are mainly involved in nutrient acquisition through the production of extracellular enzymes for the decomposition of organic matter, as well as through the sensing of their local environment 4 . Sub-apical cells generate new hyphae by lateral branching, thus increasing the density and surface area of the colony. In ascomycetes and basidiomycetes, the branches also mediate the hyphal fusion events, or anastomosis, that occur in the thallus and appear to be crucial for the exchange of nutrients and signals