Besançon meeting on Probability, Ecology & Evolution 2026

The workshop Probability, Ecology & Evolution 2026 will be held in Besançon from 9 to 11 September 2026. The program will feature seven 45-minute talks, two 3-hour minicourses, and a poster session.

The event is free of charge, but registration is mandatory. Please register here by June 12th.

The workshop will start at 14:00 on September 9 and end around 12:30 on September 11. It will take place in Amphi A at the UFR ST (Science Campus, Métrologie A – 🌍 map). The easiest way to reach the campus is by taking bus line L3 (direction: Crous Université or Pôle Temis). Buses run every 5–10 minutes from the city centre during peak hours.

We encourage early-career researchers to submit an abstract for the poster session and to request funding for the event (independently of the abstract submission), we will try to support as many people as we can and should be able to give you an answer by the end of June.

This event is hosted at the Laboratoire de mathématiques de Besançon and supported by RT Math Bio Santé, ANR project GARP, Fédération Bourgogne Franche-Comté Mathématiques and EIPHI graduate school.

Program

9/09		10/09		11/09
		09:00 – 10:30	Minicourse 1	09:00 – 10:30	Minicourse 2
		10:50 – 12:20	Minicourse 2	10:50 – 11:40	Talk
				11:40 – 12:30	Talk
			Lunch
14:00 – 15:30	Minicourse 1	14:00 – 14:50	Talk
		14:50 – 15:40	Talk
15:50 – 16:40	Talk	16:00 – 16:50	Talk
16:40 – 17:30	Talk	17:00 – 19:00	POSTERS
		20:00	Dinner

Abstracts

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Minicourses

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Mathematical introduction to multiomics integration

François Ged (Université de Bretagne Occidentale)

Modern biology measures several molecular layers on the same individuals, such as genome, transcriptome, proteome, and more. This yields several high-dimensional matrices driven by a common, unobserved biological signal. Multiomics integration aims to recover this shared structure together with what is specific to each layer.

The course is organized around a single object: linear latent-variable models. From this generative viewpoint, the classical multivariate methods, namely PCA, factor analysis, and canonical correlation analysis, form one family, differing only in their noise structure and in what they treat as shared across views. Two of the main modern multiomics integration methods then appear as natural descendants: MOFA from factor analysis, DIABLO from canonical correlation analysis.

Starting from the biological motivation, this mini-course develops this unifying viewpoint and uses it to make sense of the modern integration methods. If time allows, we will also discuss how known biological structure can be incorporated into such models. Back to top⤴️

Branching Brownian motion and F-KPP equations in a non-homogeneous medium

Lisa Hartung (Johannes-Gutenberg-Universität)

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Talks

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Genealogical structures under interactive neutral reproduction: Ancestral influence graph and Frankenstein process.

Ellen Baake (Bielefeld University)

We consider the two-type Moran model of population genetics with frequency-dependent neutral reproduction. Relying on the model's graphical representation in terms of a particle system, we establish a (factorial) moment dual. Moment duals are usually related to the genealogy of the population, but in this case, the connection is mysterious. This is because the natural ancestral graph of the model (the so-called ancestral influence graph, AIG) exhibits a complicated hierarchical structure, whereas the moment dual is a simple density-dependent branching process. We solve this mystery by starting from the fact that moment duality is a property in expectation; it need not hold pathwise. This provides the freedom to construct what we call the Frankenstein process by tracing back sample configurations and piecing them together across different realisations of the AIG. This leads to the dual process and resolves the mystery.

This is joint work with Fernando Cordero and Hannah Dopmeyer. Back to top⤴️

Multi-locus gene genealogies in fixed pedigrees

Matthias Birkner (Johannes-Gutenberg-Universität)

We study the joint genealogical structure of a finite sample at several genetic loci, embedded within the (fixed) pedigree generated by an exchangeable diploid bi-parental population model. In the finite variance case, the large population limit agrees with its pedigree-averaged version, namely Griffiths and Marjoram's ancestral recombination graph. In the case of highly skewed offspring distributions, when averaging over the pedigree, the limit is an ancestral recombination graph with simultaneous multiple mergers (a \(\Xi\)-ancestral recombination graph). Conditioned on the pedigree, we find a new multi-locus scaling limit, which is built using a realisation of a point process \(\Psi\) that encodes the timing and scale of simultaneous multiple mergers caused by generations with large individual progenies. In particular, in this case, the fixed pedigree structure impacts the distribution of multi-locus statistics.

Based on joint work, partly in progress, with Frederic Alberti, Louis Fan, Fabian Freund, Maximillan Newman and John Wakeley. Back to top⤴️

TBA

Camille Coron (INRAE Paris-Saclay)

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Invasion dynamics for a quasi-critical birth-death process

Xavier Erny (Télécom SudParis)

We study the dynamics of invasion of populations which enjoy positive density-dependent effects. We start with a single individual and consider a single type birth and death process. The initial individual growth rate is asymptotically zero and increases due to positive density dependence. This can model, for instance, the dynamics of an invasive species in a new habitat, or the spread of a mutation in a resident population. Although the process remains asymptotically critical throughout the invasion phase, three distinct dynamical regimes emerge. We prove that the probability that the population reaches macroscopic levels decreases as \(K^{-1/2}\), where \(K\) is the scaling parameter. First, the process needs to escape the neighborhood of \(0\), and conditioning on survival, it grows linearly up to size \(\sqrt{K}\). The scaled process is approximated by a diffusion, as in the critical branching process, with an additional drift term coming from cooperation, which breaks the branching property. Second, at the intermediate scale \(\sqrt{K}\), we observe another diffusion, surviving with positive probability, without conditioning. Finally, beyond the \(\sqrt{K}\) scale, the process can be approximated by the classical deterministic ODE limit. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases rely on uniform approximations on polynomial time scales. Back to top⤴️

TBA

Raphaël Forien (INRAE Avignon)

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Selection of the fittest or selection of the luckiest

Zsofia Talyigas (BOKU University)

Natural selection is commonly assumed to become more effective as it becomes stronger. However, selection acts on phenotypes rather than directly on genotypes, and phenotypic success is inherently noisy. Here we study how this mismatch shapes long-term evolutionary dynamics. Using a minimal stochastic model in which individuals inherit genetic fitness while selection acts on noisy phenotypic expressions, we show that increasing selection strength accelerates adaptation only up to a critical threshold. Beyond this point, stronger selection paradoxically slows evolution and erodes genetic diversity by favoring the luckiest individuals rather than the genetically fittest.

We identify two distinct evolutionary regimes—selection of the fittest and selection of the luckiest—separated by a sharp transition. This transition corresponds to a previously unrecognized change in the structure of traveling fitness waves, from semipulled to fully pulled fronts, with profound consequences for adaptation speed and genealogical structure. Our results reveal a biological instance of Goodhart’s law: when phenotypic measures become overly optimized targets, they cease to reliably promote genetic improvement. These findings highlight intrinsic limits to the effectiveness of strong selection and suggest that optimal evolutionary outcomes require intermediate selection strength in noisy environments. Back to top⤴️

Stationary solutions of the FKPP equation are unique in dimension up to 6, but not necessarily in dimension at least 7.

Oliver Tough (University of Durham)

We consider stationary solutions of the FKPP equation, i.e. solutions of \[\Delta u+f(x,u)=0,\] with Dirichlet boundary conditions. We show that under a general KPP-type condition on \(f\), stationary solutions are unique in dimensions \(d\leq 6\) and exist if and only if the generalised principal eigenvalue of \(\Delta+f_s(x,0)\) is strictly positive, but counterexamples for both exist in dimension \(d\geq7\). We generalise this to a general condition on \(f\) including for instance \(f:=c(x)(u-u^{1+\gamma})\), where we show \(d\leq 2+4/\gamma\) is both a sufficient and sharp condition. Our conditions on the boundary of \(\Omega\) are very general.

We apply this to branching Brownian motion as follows. A branching process survives globally if it never dies out, and survives locally if it returns to an arbitrarily chosen ball at arbitrarily large times. We show that branching Brownian motion cannot survive globally without surviving locally (i.e. drifting off to infinity) in dimensions \(d\leq 6\), for very general domains with killing on the boundary, but we exhibit a simple counterexample on the whole space in dimensions \(d\geq 7\). Back to top⤴️

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Posters

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Markov processes forced on a subspace by a large drift, with applications to population genetics

Samuel Ayomide Adeosun (Albert-Ludwigs-Universität Freiburg)

We consider a sequence of Markov processes \(X^1,X^2,\ldots\) with state space \(E\), where \(X^N\) is subject to a strong drift towards a subset \(D\subseteq E\), and where \(\Phi(X^N)\) evolves on a slower timescale for a suitable map \(\Phi:E\to D\). Using the method of martingale problems, we prove a limit theorem showing that, as \(N\to\infty\), \(\Phi(X^N)\Rightarrow Z\) in the space of càdlàg paths, while \(X^N\) converges to \(X\) in measure. We apply this general limit result to models of copy number variation of genetic elements in a diploid Moran population of size \(N\). At time \(t\), the population is described by \(X_t^N\in\mathcal P(\mathbb N_0)\), where \(X_t^N(k)\) denotes the frequency of individuals with copy number \(k\), and \(\Phi:\mathcal P(\mathbb N_0)\to\mathbb R\) is the first-moment map. Back to top⤴️

SIR model with immune-escape mutations: recurrent emergence and persistence of the epidemic

Denis Grange (Télécom-Paris)

We study an individual based SIR-model where the pathogen acquires immune-escape mutations at rate \(N^{-\alpha}\) per capita, where \(N\) is the population size and alpha in \((0,1)\). In this regime, even when the population is closed and immunity to a strain is permanent, the epidemic can persist thanks to the accumulation of sufficiently many mutations. We derive conditions for persistence of the pathogen in the large population limit, and observe emergence of interesting phenomena such as the presence of shadow variants, infecting only a negligible proportion of individuals, but that are necessary for the epidemic to overcome herd immunity and persist. Upon persistence, the epidemic sometimes reaches a stationary traveling wave dynamics, which, in simple cases, can be explicitly described. Back to top⤴️

Heuristics on the speed of invasion in an advancing population

Paul de Lambert (ENS Lyon)

Several studies have investigated systems of reaction-diffusion partial differential equations (PDEs) in expanding domains with two distinct propagation speeds, one faster than the other. While the faster speed is straightforward to compute, determining the slower one remains challenging. Several PDEs works rigorously establish the so-called ”following travelling wave” speed. The objective this work is to argue that the speeds of the PDEs do not coincide with the speeds of a system of particles which converges to those PDEs. In other words, the limits in time and in \(N\) do not commute. Back to top⤴️

Slow-fast dynamics of stochastic particle systems arising in telomere biology

Juan Mardomingo-Sanz (Univ. Lorraine - Inria Nancy)

The ends of linear chromosomes, called telomeres, shorten at each cell replication, eventually driving the cells to a senescent state when they become too short. The enzyme telomerase, present in cancerous cells and some unicellular organisms, elongates the telomeres and allows cells to continue replicating. Recent experiments show that if this enzyme is inactivated some rare survivors (ALT), which elongate their telomeres without telomerase, will appear and will eventually invade the cultures. To model the emergence and invasion of the ALT cells we study the large population limit of a simple stochastic particle system under appropriate scaling assumptions corresponding to rare mutations and different division rates depending on the cell type. This limit involves classical results on density-dependent Markov processes together with a sharp control of the mutant process, which starts with a single individual. Back to top⤴️

On a Branching Annihilating Random Walk

Janine Piesold (Johannes-Gutenberg-Universität Mainz)

We consider a discrete-time branching annihilating random walk (BARW). Within a time step, each particle, before dying, produces a random number of offspring which are then randomly and independently displaced in space. If, after the displacement, a site is occupied by several particles, all particles at that site are annihilated. This can be thought of as a very strong form of local competition and entails that the system is not monotone. On the poster, we will discuss the BARW and its long-time asymptotics on \(Z^d\) as well as on the complete graph. Furthermore, we will explore its connection to a classical urn occupancy problem and to a randomly disturbed deterministic iteration. Back to top⤴️

Besançon meeting on Probability, Ecology & Evolution 2024

On 10 December 2024, we hosted a one-day meeting at the MSHE in Besançon, bringing together researchers from Austria, France, Germany, and Switzerland for a workshop on stochastic models in biology. We thank all the participants for joining us and contributing to the success of our first event.

This event was supported by LmB, RDI-BMB, Chrono-environnement and EUR EIPHI (RaySynMath project).

Detailed program.

10:00 — 10:30

Welcome coffee 🥐

10:30 — 11:15

Céline Bonnet
Site frequency spectrum of a rescued population under rare resistant mutations

11:15 — 12:00

Florin Boenkost
Dust solutions and the nested Kingman coalescent

12:00 — 13:30

Lunch Break 🍴
vegetarian / vegan buffet

13:30 — 14:15

Martin Möhle
On multi-type Cannings models and their multi-type limiting coalescents

14:15 — 15:00

Madeleine Kubasch
Epidemic dynamics on large multi-level contact networks including households and workplaces

15:00 — 15:45

Coffee break ☕

15:45 — 16:30

Nicolás Zalduendo
The multi-type bisexual Galton–Watson process

16:30 — 17:15

Vincent Bansaye
Some growing, living networks and branching processes

Participants are welcome to join for drinks after the conference, at their own expense.

A dinner will be organized for the invited speakers, more details coming soon.

Abstracts.

Site frequency spectrum of a rescued population under rare resistant mutations

Céline Bonnet, 10:30 — 11:15

The aim of this talk is to study the impact of resistance acquisition on the distribution of neutral mutations in a cell population under therapeutic pressure. The cell population is modeled by a bi-type branching process. Initially, the cells all carry type 0, associated with a negative growth rate. Mutations towards type 1 are assumed to be rare and random, and lead to the survival of cells under treatment, i.e. type 1 is associated with a positive growth rate, and thus models the acquisition of a resistance. Cells also carry neutral mutations, acquired at birth and accumulated by inheritance, that do not affect their type. We describe the expectation of the "Site Frequency Spectrum" (SFS), which is an index of neutral mutation distribution in a population, under the asymptotic of rare events of resistance acquisition and of large initial population. Precisely, we give asymptotically-equivalent expressions of the expected number of neutral mutations shared by both a small and a large number of cells. To identify the influence of relatives on the SFS, our work also lead us to study in detail subcritical binary Galton–Watson trees, where each leaf is marked with a small probability. As a by-product of this study, we thus provide the law of the generation of a randomly chosen leaf in such a Galton–Watson tree conditioned on the number of marks. Back to program ⤴️

Dust solutions and the nested Kingman coalescent

Florin Boenkost, 11:15 — 12:00

The nested Kingman Coalescent \((\mathcal{K}^n_t, t\geq 0)\) is a model for the genealogy of lineages belonging to different species, which can be thought of as (gene-)trees within a (species-)tree. We assume that the species tree is given as a Kingman tree and once species coalesce the inner gene lineages are allowed to coalesce.

Here, we focus on the empirical measure \(g^n_{t}= \frac{1}{s^n_t} \sum_{i=1}^{s_t^n} \delta_{\Pi_t^n (i)}\) of block sizes in the nested coalescent, where \(\Pi_t^n (i)\) denotes the number of gene lineages in species \(i\). As the number of species tends to infinity we show convergence towards a solution \(u(t,x)\) of the Smoluchowski coagulation-transport equation \[ \partial_t \, u = \partial_x \left( \frac{x^2}{2} u\right) + \frac{1}{t} (u \star u - u), \] where \(u \star u\) denotes the convolution. In particular, if there are fewer lineages per species than species itself, \(g_t^n\) converges to a dust solution \(u(t,x)\), meaning that \(u(t,x) \to \delta_0(dx)\) as \(t\to 0.\) Dust solutions of this equation appeared in the work of [Lambert and Schertzer 2020] and a priori there are infinitely many. We show that the solution is unique up to a scaling parameter which encodes the initial number of lineages per species.

This is accomplished through a probabilistic representation of the solution, which in turn can be extended to provide a stochastic representation of all dust solutions in terms of a specific martingale. Lastly, we discuss an application of this result and provide a short glimpse onto other species trees (in particular \(\Lambda\)-coalescents). This talk is based on joint work in progress with Emmanuel Schertzer. Back to program ⤴️

On multi-type Cannings models and their multi-type limiting coalescents

Martin Möhle, 13:30 — 14:15

A multi-type neutral Cannings population model with mutation and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type coalescent with mutation allowing for simultaneous multiple collisions of ancestral lineages. The limiting coalescent shares the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions into reproductive and mutational parts. Back to program ⤴️

Epidemic dynamics on large multi-level contact networks including households and workplaces

Madeleine Kubasch, 14:15 — 15:00

The spread of an epidemic is naturally influenced by the way individuals encounter one another. In particular, human contacts are structured by different social contexts in which they occur, such as at home, in school or at the office. In addition, control measures like teleworking and school closures act by reducing specific contacts, further motivating the study of models which distinguish different types contacts.

Here, we study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space specifies the remaining infectious period length for each infected. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. We start by establishing the convergence to the unique deterministic solution of a measure-valued equation. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics.

Finally, through numerical explorations, we show that the large population limit of the household-workplace model can approximate well the epidemic, even if some assumptions on the contact network are relaxed. Back to program ⤴️

The multi-type bisexual Galton–Watson process

Nicolás Zalduendo, 15:45 — 16:30

The bisexual Galton–Watson process is an extension of the classical Galton–Watson process, but taking into account the mating of females and males, which form couples that can accomplish reproduction. Properties such as extinction conditions and asymptotic behaviour have been studied in the past years, but multi-type versions have only been treated in some particular cases. In this work we deal with a general multi-dimensional version of the model, where we consider different types of females and males, which mate according to a "mating function". We consider that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. One of the main difficulties in the study of this process is the absence of a linear operator that is the key to understand its behavior in its asexual counterpart, but in our case it turns out to be only concave. To overcome this issue, we use a concave Perron–Frobenius theory which ensures the existence of eigen-elements for some concave operators. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the convergence of the process in the long-time through the identification of a supermartingale. Back to program ⤴️

Some growing, living networks and branching processes

Vincent Bansaye, 16:30 — 17:15

We will study fragmentation growth models to describe the expansion of biological networks, particularly in the growth of filamentous fungi or morphogenesis. The models combine linear growth of fragments, several types of fragmentation, and the possibility of fusion when two fragments meet. We will first examine the branching process without fusion and establish the long-term behavior of the structured population of fragments. This will lead us to general results for describing the renormalized empirical measure, that is, to an infinite-dimensional version of the Kesten–Stigum theorem, relaxing the uniformity of the semigroup estimates from the work of Asmussen and Hering. We will then focus on a highly simplified spatial model incorporating fusion events, and we will explore how to study it in detail using a new structured branching process. This talk is linked to a previous work with M. Tomasevic and A. Veber and two works in progress respectively with T. Berah and B. Cloez, and G. Raoul and M. Tomasevic. Back to program ⤴️