From stochastic processes to constrained Hamilton-Jacobi (HJ) equations.

Permanent members: Sylvie Méléard, Gaël Raoul

Most models, for instance for the "normal" bacterial division cycle 82, consider asexual populations with clonal reproduction and vertical inheritance. We want to consider here a more general model with a transfer term, justified by biological considerations in the case of bacterial transfer 86 (see also the application section 4.1). The individual-based population process is given for any $K$ by the jump point-measure Markov process ${\left({\mu }_t^K\right)}_t$ on a trait subset of ${ℝ}^d$ weighted by $1/K$. An individual with trait $x$ gives birth to a new individual with rate $b\left(x\right)$. With probability $1-p_K$, the new individual carries the trait $x$ and with probability $p_K$, it carries a mutant trait $z$ chosen according to the distribution $m(x,z)dz$. An individual with trait $x$ in the population ${\mu }^K$ dies with death rate $d\left(x\right)+C\langle {\mu }^K\left(t\right),1\rangle$. Further an individual with trait $x$ chooses a partner with trait $y$ at rate $h_K(x,y,{\mu }^K)=\tau (x,y)/K\langle {\mu }^K,1\rangle$ and after transfer, the couple $(x,y)$ becomes $(x,x)$. Then for any “good” test function $\phi$, we have

where $M^{K,\phi }$ is a square integrable martingale whose quadratic variation can be easily made explicit. By letting $K$ go to infinity, and $p_K$ go to $p$, we can derive an integro-differential equation with non local non linearities due to both competition and transfer. Uniqueness of its solution is obvious but its long-time behaviour is unknown, as well as the existence of stationary solutions. Formally, applying a limiting procedure for small mutations and time rescaling usually leads to a HJ type equation with constraints, in the formalism introduced in 74, successfully developed in 52, 117, 51, and in many extensions so far. Concentrations in such equations are too fast for realistic evolution 118. Indeed the evolutionary dynamics strongly depend on the positivity of the density although it is exponentially small for some traits. Different papers 118, 94, 110 proposed to slow down the concentration speed by the addition of artificial terms. With Nicolas Champagnat, Sylvie Méléard introduced another point of view. The rare mutation assumption introduced in 61 allowed to obtain a time scale separation between demography and mutation. Under this assumption, they were able to characterize rigorously a general evolutionary jump process describing the successive evolutionary population states 63. This approach was very fruitful and allowed to quantify the complete scheme from individuals to macroscopic behaviors as suggested in 107 and 73. Nevertheless, the assumptions imposed very small mutation rates considered too slow to explain evolution (especially for microorganisms), but also too slow to capture the concentration effects of the HJ equations. At this point one may recall the usual critics by some biologists 136, of unrealistic evolutionary time scale, at least for certain species.

The first task in this study is to integrate fast mutation time scales and to show how stochastic models based on logarithmic scales can capture small populations in large approximations and explain deterministic concentration phenomena. In particular we aim to obtain new singular and constrained HJ equations taking into account the local population extinctions. We hope that these new scales will provide an intermediate approach consistent with biological observations.

The second task is to characterize the different asymptotic behaviors in the Hamilton-Jacobi equation and to understand the role of the trade-off between demography and transfer.

Space-and-trait structured models

Permanent members: Vincent Bansaye, Marie Doumic, Gaël Raoul

Effects of spatial heterogeneity on structured population dynamics need to be studied for many applications, in ecology as well as in microbiology. Here again, relating macroscopic to individual-based models is of key importance for a correct interpretation of macroscopically observed phenomena such as morphogenesis or front propagation. Let us develop two examples: size-and-space structured models and phenotypic trait-and-space structured models.

Non-markovian interactions: from local interaction model to the parabolic-parabolic Keller-Segel system

Permanent members: Milica Tomašević

An important mathematical challenge is to derive mean-field limits for non-Markovian interaction, i.e., when the past also needs to be taken into account. Such models appear for instance in neuroscience 64, 65 and chemotaxis. To model chemotaxis, the parabolic-parabolic Keller-Segel model has been stated phenomenologically, but to interpret it we need to introduce interaction memory, which provides tremendous analysis difficulties since particles are now non-markovian both in time and space. New methods have been proposed by Milica Tomašević, with a stochastic representation of the mild formulation of the equation and a particle approximation 125, 129, 127. The equations obtained have been little studied before Milica Tomašević's Ph.D thesis, so that many questions remain open. Concerning the convergence of the particle systems towards the Keller-Segel model, an important problem is the obtention of explicit convergence rates, when the number of particles tends to infinity, for the propagation of chaos of the particle system in 1D. A possible way is to extend techniques developed by Jabin and Wang 95 for the quantitative study of the mean-field boundaries of particle systems in non-regular Markovian interaction. The aim is to control the relative entropy between the joint law of the particles and the law of $N$ independent copies of the Keller-Segel system. By exploiting the results on the Keller-Segel nonlinearity in 1 dimension and on the Sobolev type estimation on the densities of the system (chapter 4 in 128), a regularization of the interaction kernel of the particles allows to obtain a first convergence rate for the marginal laws in time of an arbitrary particle, explicit but suboptimal (due to the regularization procedure). To obtain the optimal convergence rate, we think to develop an essentially probabilistic approach suggested by the recent works of Veretennikov 133 and Lacker 101 as well as by the partial Girsanov transformations introduced in 96.

Diffusion-growth-fragmentation processes and equations

Permanent members: Marie Doumic, Sylvie Méléard

To model the growth of a bacterial population in a chemostat, a new model of growth and fragmentation, coupled to a differential equation for the resource, was proposed by Josué Tchouanti in his thesis 126. Using a combination of probabilistic and analytical methods, he proved the existence, uniqueness and regularity of solutions, as well as the convergence in large populations of the individual-based model. This model also present similiarities with the proliferation of parasites in dividing cells studied by Vincent Bansaye 49, 48.

One of the very interesting novelties of this model is to consider the growth not as a purely deterministic process, leading to a transport term in the size structured equation $\frac{\partial }{\partial x}\left(\tau \left(x\right)u(t,x)\right)$, but to take into account the intrinsic stochasticity in growth, so that a diffusion-type term $\frac{{\partial }^2}{\partial x^2}\left(D\left(x\right)u(t,x)\right)$ is added, which degenerates at the boundary $x=0.$ We thus want to study further this equation, its long-time dynamics with and without interaction (i.e. in the linear case as well as with nonlinear couplings), how it differs from the much more studied growth-fragmentation equation, and which model seems more relevant in which applicative case. We also want to adapt the model to metabolic heterogeneity cases, i.e. when we model the capacity for bacteria to feed on two distinct nutrients, which leads to distinguish two populations competing for two resources.

Ergodicity analysis and exponential convergence for multi-dimensional growth-fragmentation processes and equations

Permanent members: Vincent Bansaye, Sylvie Méléard, Milica Tomašević

Based on data from the Edinburgh's lab of Meriem El Karoui, Ignacio Madrid Canales introduced during his thesis an adder growth-fragmentation stochastic process modelling the growth of bacteria. He studied its long time behaviour and proved that conveniently renormalized, the associated semigroup converges exponentially to a well defined measure. The aim is now to generalize this result in higher dimensions, motivated by the different growth strategies that bacteria can have under stress. Mathematically the question is also largely open.

Understanding the links between genealogical and population behaviours

Permanent members: Vincent Bansaye, Marie Doumic and Sylvie Méléard

Microfluidic experiments allow one to follow a genealogical lineage of cells, whereas most previous experiments as well as "natural" conditions consist in observing a full population dynamics. The natural question then comes to relate the two models, and to understand how certain phenomena may be observed in one setting but not in the other - for instance, how a few individuals may finally invade the whole population; how survivor cells may emerge from a senescent population; or yet, how to find the "signature" of a phenomenon that happened in the past from the observation of a population at a fixed time.

Estimate growth, division, interaction features in structured populations

The estimation of the division rate in non-interacting populations has been developed in a series of papers over the last decade 81. The question we want to address now is whether growth and division rates are modified by cell-to-cell interaction (or yet by antibiotic resistance or by competition), and reciprocally, how distributed growth and division rates may have an influence on the morphogenesis of the bacterial microcolony. In this task, we aim to provide answers based on more realistic individual-based models. We plan the following steps:

• Develop parametric and non-parametric inference of the interaction function from single individual tracking. A similar study has been carried out by Laetitia Della Maestra and Marc Hoffmann 72 for Mc-Kean-Vlasov equation; we would like to add a size structure and a non-constant number of individuals. We will first assume that the growth and division rates do not depend on the interaction between cells, so that prior to this step we have used the methods already developed to infer these functional parameters. We may also build upon biophysical studies such as in 138.
• Develop statistical hypothesis testing to accept or reject the assumption made in the previous step that division and growth are not influenced by the interaction inferred. Reciprocally, test whether different division or growth rates would give rise to different morphogenesis.
• Generalise the methods and adapt them to new problems, in particular the mycelial networks 75.

Estimate mutation or fragmentation kernel density

The question of estimating the fragmentation kernel in polymer breakage experiments 78 surprisingly rejoins the question of estimating the so-called Distribution of Fitness Effects (DFE) which characterizes the accumulation of mutations in bacteria 122. As shown in 78, these are so-called severely ill-posed inverse problems, for which we aim at developing new approaches, two in particular: rely on short-time instead of long-term behaviour, adapt statistical methods developed for decoumpounding Poisson processes and deconvolution 84.

State estimation and observation inequalities for depolymerisation models

In depolymerisation experiments, prior to parameter estimation, we began to address the question of state estimation, i.e. how to infer the initial condition out of measurements of moments time dynamics. Whereas it is relatively straightforward if we approximate the discrete system by a backward transport equation 39, we address the question of estimating it from the next order approximation, namely a transport-diffusion equation; this new problem is closer to the experimental system but gives rise to a severely ill-posed inverse problem, for which we want to find an observation inequality thanks to Carleman estimates 70, 69.

Calibrating the mycelial network model

The model developed in 130 paves the way to new parametric calibration methods that we wish to confront with the real observations made by mycological colleagues of the LIED laboratory (Paris Diderot University), as well as with their empirical results.

The parametric calibration based on the solutions of the spectral problem can lead to new simple descriptors that characterize the growth of the fungus.

The first objective is to see how values obtained in 75 for the exponential growth rates compare with the one obtained in 130 as a solution of the spectral problem related to the corresponding growth and fragmentation equation. For the latter, there is an interpretation through the main characteristics of the network (ratio between the number of external nodes and the total length of the network at a sufficiently large time $t$).

Then, we could test how these descriptors change in different growth environments. This will allow us to quantify the impact of various forms of stress (nutrient depletion, pH, ...).

From a theoretical point of view, we would have to justify this empirical approach and demonstrate a "many to one" formula to be able to correctly sample our model. It should also be proved that the estimators thus constructed are consistent and converge, when $t\to \infty$, to the quantities they are supposed to approximate.

The bacterial cell cycle

Coordination between cell growth and division is often carried out by ‘size control’ mechanisms, where the cell size has to reach a certain threshold to trigger some event of the cell cycle, such as DNA replication or cell division. Concerning bacteria, recent articles 38, 124 stated the excellent adequacy of the so-called "incremental model", where the structuring variable which triggers division is the size increase of the bacteria since birth, to experimental data. This opens up new questions to refine and analyse this model, test its validity in extreme growth conditions such as antibiotic treatments, and understand its links with intracellular mechanisms. Main research axis: 3, and the CellDiv platform.

Antibiotic response and resistance emergence

To address the emergence of antibiotic-resistant strains of bacteria, it is essential to understand quantitatively the response of bacteria to antibiotic treatments. Under the action of an antibiotic that causes damage to cellular DNA, bacteria change their growth strategy and do not respond homogeneously to this stress. Of particular importance is the so-called SOS response: in response to DNA damage induced by antibiotic treatments, the cell cycle is arrested and DNA repair and mutagenesis are induced (cf. 41). Cells with high SOS response will grow for an abnormal duration, producing long filaments that are impervious to antibiotics. Understanding the distribution of sizes in the population of bacteria will allow a better quanitfication of antibiotics effects. On this subject, we work with Meriem El Karoui who carries out microfluidic experiments in Edinburg university. Main research axis: 2.

Microcolony morphogenesis

When bacterial microcolonies grow, they can aggregate to one another and form a biofilm. How do they interact? How do their growth and division characteristics translate into the shape of the colony? Inside the gut, it has been proved that the immune response acts not by killing bacteria but by making them aggregate after division; how do these aggregates form and break is another question tackled by Claude Loverdo at Lab. Jean Perrin (Sorbonne). Main research axis: 1 (short term in collaboration with Diane Peurichard and Sophie Hecht).

Bacterial growth in a chemostat; the gut as a chemostat

A chemostat is a specific experiment, where the number of bacteria is let constant by a permanent influx and outflux. The functional mechanism of the very gut could be modeled as a chemostat. Main research axis: 2 (mid to long term / only first contacts made).

Mutations

The pace of evolution and possible trajectories depend on the dynamics of mutation incidence and the effects of mutations on fitness. Mutation dynamics has been for the first time analyzed directly by Lydia Robert and co-authors 122, using two different microfluidic experiments which led them to the conclusion of a Poissonnian appearance of bacterial mutations, and to a first parametric estimation of the so-called "distribution of fitness effects" (DFE) of mutations. How to assess better the shape of the DFE, and apply the method not only to deleterious or neutral but also to possibly beneficial mutations, is one of our goals. Main research axis: 3, short term (Guillaume Garnier's ongoing Ph.D).

Horizontal gene transfer

Microorganisms such as bacteria tend to exhibit a relatively large “evolution speed”. They have also the particularity to exchange genes by direct cell-to-cell contact. We are particularly interested in plasmids horizontal gene transfer (HGT): plasmids carry pathogens or genes coding for antibiotics resistances, and plasmid exchange is considered by biologists as the primary reason for antibiotics resistance. Main research axis: 1, both short and long-term research, included in the ERC project SINGER.

Leukemic mutations and hematopoeisis

Hematopoiesis is the process of producing blood cells from stem cells and progenitors. These highly regulated mechanisms keep at equilibrium the number of blood cells such as red blood cells, white blood cells and platelets (mature cells). We want to understand the emergence of leukaemia or resistance to chemotherapy through the mechanisms of erythropoiesis (production of red blood cells) and leukopoiesis (white blood cell formation). This application also rejoins the application 3.1.

Senescence by telomere shortening

Telomeres cap the ends of linear chromosomes, and help maintain genome integrity by preventing the ends being recognized and processed as accidental chromosomal breaks. When telomeres fall below a critical length, cells enter replicative senescence. However, the exact structure(s) of the short or dysfunctional telomeres either triggering permanent replicative senescence or promoting genome instability remains to be defined; this is the main focus of Teresa Teixeira's lab at IBPC, which has developed microfluidic as well as population experiments to follow senescence triggering in yeast cells. Main research axis: 1 and 2. This application is both a long-term goal, in a long-lasting collaboration with Teresa Teixeira and Zhou Xu, and has short and mid-terms objectives, through Anaïs Rat's finishing Ph.D and Jules Olayé forthcoming Ph.D (co-supervised by Milica Tomašević and Marie Doumic).

Ageing in drosophyla

Ageing’s sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual’s fitness. In the recent years, a new 2-phases model of ageing has been proposed by Hervé Tricoire and Michael Rera 71, 132, describing the ageing process not as being continuous but as made of at least 2 consecutive phases separated by a dramatic transition. It was first observed in drosophila, and then shown to be evolutionary conserved; this raises the question of an active selection of the underlying mechanisms throughout evolution. Main research axis: 2 and 3.

Protein polymerisation: amyloid formation and autophagy

Protein polymerisation occurs in many different situations, from functional situations (actin filaments, autophagy) to toxic ones (amyloid diseases). It involves complex reaction networks, making it a challenge to identify the key mechanisms, for instance which mechanisms lead to the initial formation of polymers during the first reaction steps (nucleation), how and where the polymers break, or yet the aggregates formation, out of (at least) two different proteins, in autophagy. With our biologist collaborators, our aim in these applications is to isolate the most meaningful reactions, study their behaviour(Research axis 2), and compare them - qualitatively and, if possible, quantitatively - with experimental data.

Mycelial network

Filamentous fungi are complex expanding organisms that are omnipresent in nature. They form filamentous structures, growing and branching to create huge networks called mycelia. We aim at modelling, understanding and estimating the main mechanisms of mycelial formation. We have already studied a first model without interactions and we will now study the impact of fusion of filaments on the growth of the network. Main research axes: 2 and 3.

Emergence of bacterial resistance in heterogeneous environments

When an antibiotic treatment is applied to a population, bacteria resistant to the treatment have an opportunity to develop. If several treatments are used, life threatening multi-resistant bacteria can appear. Understand the dynamics of bacterial populations in such heterogeneous environments would provide interesting perspectives to improve treatments and keep antibiotic resistance in control. On this topic, we will collaborate with S. Gandon lab at CEFE, that tackles this problem with a combination of theory and experiments. This also rejoins the application domain 3.1., and the main research axis is axis 2.

Dynamics of species submitted to climate change

The impact of climate change on natural species is a complicated matter. An important research effort has been made on the modification of species' niche in coming years, but this is only a partial clue for the future of species. In collaboration with Ophélie Ronce at ISEM, we will investigate how the local adaptation of species will is shook by global changes. With François Massol in CIIL and Nicolas Loeuille in IEES, we will focus on the impact of interspecies effects: predation, parasitism, cooperation, etc. Main research axis: 1.

7.1.1 telomeres

• Name:
  Simulation of cell populations and lineages during replicative senescence.
• Keywords:
  Stochastic models, Statistical inference, Monte-Carlo methods, Branching system, Population approach, Computational biology
• Functional Description:

  Code associated with "Mathematical model linking telomeres to senescence in Saccharomyces cerevisiae reveals cell lineage versus population dynamics".

  Preprint version of the associated article: https://www.biorxiv.orgcontent/10.1101/2023.11.22.568287v1 See also Chapter 3 of the PhD thesis: https://theses.hal.science/tel-04250492

  The telomeres package contains all the necessary auxiliary code. This is where the mathematical model is encoded, with its default parameters (parameters.py). More generally, it contains all the functions allowing to

  Posttreat the raw data (make_*.py) Simulate the model (simulation.py) Plot the simulated and experimental data, the laws of the model... (plot.py)

  The scripts in this folder are not intended to be modified (unless you find errors, in which case please let me know) or used directly to run simulations.

  The makeFiles folder contains scripts to run to generate the data/processed directory, that contains the posstreated data.

  The main folder contains the scripts that should be run to perform the simulations and plot their results.

• URL:
  https://­zenodo.­org/­records/­14525443
• Contact:
  Anais Rat

8.1.1 Chemotaxis models

Participants: Charles Bertucci, Mathias Rakotomalala, Milica Tomasevic, Guillaume Woessner.

Curvature in chemotaxis: A model for ant trail pattern formation

In 21, we propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, and the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In the article 16, we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ${ℝ}^D$ for $d\ge 3$. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ${ℝ}^d$. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

8.1.2 Limits of large populations with local or nonlocal interaction and heterogeneity

Participants: Vincent Bansaye, Marie Doumic, Sophie Hecht, Marc Hoffmann, Ayman Moussa, Felipe Munoz, Benoit Perthame, Diane Peurichard.

For the study of large populations with local interaction, Vincent Bansaye, together with Felipe Munoz and Ayman Moussa, developed approaches in a discrete space that grows simultaneously with the local size of the population 47. We introduced duality techniques to handle the convergence of the stochastic process to a cross-diffusion and the stability of the limiting PDE. Ayman Moussa and Vincent Bansaye are currently supervising a pre-thesis student, Alexandre Bertoloni, who is extending these resu lts by incorporating births and deaths with local density dependence, along with a reaction term.

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of a series of articles, in a collaboration between Marie Doumic, members of the Inria project-team MUSCLEES and Marc Hoffmann, is to explore models through different scales for a spatial population of interacting, growing and dividing particles. After the modelling and simulation article 80, we studied the rigorous limits through scales of a model including growth, division and interaction.

In 26, we start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, we then obtain entropy estimates, which provide us with a localization limit at the macroscopic level. Finally, we perform a thorough numerical study in order to compare the three modeling scales. An important difficulty of this work is to take into account the continuous size structure, which leads to a lack of compactness for the localisation limit.

In the article 12, we study a multi-species system, which may be seen as a discrete counterpart of the size-structured one 26. At the mesoscopic level, the system is quadratic, written under the form of transport equations with a nonlocal self-generated drift. We establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in this analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

8.1.3 A scenario for an evolutionary selection of ageing

Participants: Tristan Roget, Sylvie Méléard, Michael Rera.

Signs of ageing become apparent only late in life, after organismal development is finalized. Ageing, most notably, decreases an individual’s fitness. As such, it is most commonly perceived as a non-adaptive force of evolution and considered a by-product of natural selection. Building upon the evolutionarily conserved age-related Smurf phenotype, we propose a simple mathematical life-history trait model in which an organism is characterized by two core abilities: reproduction and homeostasis 15. Through the simulation of this model, we observe 1) the convergence of fertility’s end with the onset of senescence, 2) the relative success of ageing populations, as compared to non-ageing populations, and 3) the enhanced evolvability (i.e. the generation of genetic variability) of ageing populations. In addition, we formally demonstrate the mathematical convergence observed in 1). We thus theorize that mechanisms that link the timing of fertility and ageing have been selected and fixed over evolutionary history, which, in turn, explains why ageing populations are more evolvable and therefore more successful. Broadly speaking, our work suggests that ageing is an adaptive force of evolution.

8.1.4 Phenotypic plasticity trade-offs in an age-structured model of bacterial growth under stress

Participants: Meriem El Karoui, Ignacio Madrid, Sylvie Méléard.

Under low concentrations of antibiotics causing DNA damage, Escherichia coli bacteria can trigger stochastically a stress response known as the SOS response. While the expression of this stress response can make individual cells transiently able to overcome antibiotic treatment, it can also delay cell division, thus impacting the whole population's ability to grow and survive. In order to study the trade-offs that emerge from this phenomenon, we propose a bi-type age-structured population model that captures the phenotypic plasticity observed in the stress response 2, 3. Individuals can belong to two types: either a fast-dividing but prone to death “vulnerable" type, or a slow-dividing but “tolerant" type. We study the survival probability of the population issued from a single cell as well as the population growth rate in constant and periodic environments. We show that the sensitivity of these two different notions of fitness with respect to the parameters describing the phenotypic plasticity differs between the stochastic approach (survival probability) and the deterministic approach (population growth rate). Moreover, under a more realistic configuration of periodic stress, our results indicate that optimal population growth can only be achieved through fine-tuning simultaneously both the induction of the stress response and the repair efficiency of the damage caused by the antibiotic.

8.1.5 Dynamics of a kinetic model describing protein transfers in a cell population

Participants: Pierre Magal$†$, Gaël Raoul.

We consider a cell population structured by a positive real number which represents the number of P-glycoproteins carried by the cell. In this article, we introduce a kinetic model to describe the dynamics of the cell population, and consider an asymptotic limit of this equation: if transfers are frequent, the population can be described through a system of two coupled ordinary differential equations. The main idea of this manuscript is to combine Wasserstein distance estimates on the kinetic operator to more classical estimates on the macroscopic quantities.

8.1.6 Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

Participants: Gaël Raoul.

We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We combine a contraction argument based on Wasserstein estimates on the phenotypic variable with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology.

8.2.1 Long-time behaviours

Participants: Pierre Collet, Claire Ecotière, Jules Olayé, Sylvie Méléard, Milica Tomasevic.

Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel.

In the article 4, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified.

Long-time behaviour of a degenerate stochastic system modeling the response of a population to its environmental perception.

In 23, accepted for publication in Electronic Communications in Probability, we study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two differentbehaviors, responding more or less intensively to their environmental perception. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. This allows us to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibria of the dynamical system having the smallest diffusion coefficient.

8.2.2 Time reversal and ancestral lineages

Participants: Vincent Bansaye, Pierre Collet, Jaime San Martin, Sylvie Méléard.

Ancestral lineage for interacting populations.

In 8, we consider Markov jump processes describing structured populations with interactions via density dependance. We propose a Markov construction with a distinguished individual which allows to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of type space of individuals to include the state of the population. The jump rates outside the spine are also modified. We apply this approach to some issues concerning evolution of populations and competition. For single type populations, we derive the diagram phase of a growth fragmentation model with competition and the growth of the size of birth and death processes with multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

Branching diffusion processes and spectral properties of Feynman Kac semigroup

The article 24 is motivated by the study of the long time behavior of linear functionals of birth and death diffusion processes as well as the time reversal of the spinal process by means of spectral properties of the associated Feynman-Kac semigroup. We generalize for this non Markovian semigroup the theory of quasi-stationary distribution (q.s.d.) and $Q$-process. We consider a situation where the underlying diffusion process doesn't come down rapidly from infinity but the compactness properties follow from the divergence of the death rate at infinity. We establish the complete spectral decomposition for the Feynman-Kac semigroup. An interesting consequence is the identification of the law of the reversal time spinal process issued from q.s.d. with the $Q$-process of the Feynman-Kac semigroup.

8.2.3 Evolutionary dynamics - stochastic and deterministic mutation models

Participants: Matthieu Alfaro, Vincent Bansaye, Sirine Boucenna, Vasilis Dakos, Marie Doumic, Xavier Erny, Quentin Griette, Anouar Jeddi, Sylvain Gandon, Sylvie Méléard, Sepideh Mirrahimi, Gaël Raoul, Anaïs Rat, Magali Tournus.

Sharp approximation and hitting times for stochastic invasion processes We are interested in the invasion phase for stochastic processes with interactions. A single mutant with positive fitness arrives in a large resident population at equilibrium. By a now classical approach, the first stage of the invasion is well approximated by a branching process. The macroscopic phase, when the mutant population is of the same order as the resident population, is described by the limiting dynamical system. We capture the intermediate mesoscopic phase for the invasive population and obtain sharp approximations. It allows us to describe the fluctuations of the hitting times of thresholds, which inherit a large variance from the first stage. We apply our results to two models which are first motivations. In particular, we quantify the hitting times of critical values in cancer emergence and epidemics 7.

Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation

In the preprint 32, Anouar Jeddi (Ph.D student supervised by Sylvie Méléard and Sepideh Mirrahimi) derives a Hamilton-Jacobi equation with obstacle from a discrete linear integro-differential model in population dynamics, with exponentially decaying mutation kernel. The fact that the kernel has exponential decay leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models as in Barles-Mirrahimi-Perthame. We consider a population composed of individuals characterized by a quantitative trait, subject to selection and mutation. In the regime of large population, small mutations and large time, we prove that the WKB transformation of the density converges to the unique viscosity solution of a Hamilton-Jacobi equation with obstacle.

The article 13 has been published in Evolution Letters. In this article, we have analysed the evolutionary dynamics of pathogens spreading in a heterogeneous host population where selection varies periodically in space. We study both the transient dynamics taking place at the front of the epidemic and the long-term evolution far behind the front. In particular, we identify the conditions where a generalist pathogen carrying multiple adaptations can outrace a coalition of specialist pathogens. We also show that finite host populations promote the spread of generalist pathogens because demographic stochasticity enhances the extinction of locally maladapted pathogens.

The article 123 is submitted to the Journal of Theoretical Biology. Shallow lakes ecosystems may experience abrupt shifts (ie tipping points) from one state to a contrasting degraded alternative state as a result of gradual environmental changes. It is crucial to elucidate how eco-evolutionary feedbacks affect abrupt ecological transitions in shallow lakes. We explore the eco-evolutionary dynamics of submerged and floating macrophytes in a shallow lake ecosystem under asymmetric competition for nutrients and light. We show how rapid trait evolution can result in complex dynamics including evolutionary oscillations, extensive diversification and evolutionary suicide. Overall, this study shows that evolution can have strong effects in the ecological dynamics of bistable ecosystems.

Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection.

We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady-states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady-states.

Is there an advantage of displaying heterogeneity in a population where the individuals grow and divide by fission? This is a wide-ranging question, for which a universal answer cannot be easily provided. In 28, we aim at providing a quantitative answer in the specific context of growth rate heterogeneity by comparing the fitness of homogeneous versus heterogeneous populations. We focus on a size-structured population, where an individual's growth rate is chosen at its birth through heredity and/or random mutations. We use the long-term behaviour to define the Malthus parameter of such a population, and compare it to the ones of averaged homogeneous populations. We obtain analytical formulae in two paradigmatic cases: first, constant rates for growth and division, second, linear growth rates and uniform fragmentation. Surprisingly, these two cases happen to display similar analytical formulae linking effective and individual fitness. They allow us to investigate quantitatively the crossed influence of heredity and heterogeneity, and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity.

8.2.4 The epidemiological footprint of contact structures in models with two levels of mixing

Participants: Vincent Bansaye, François Deslandes, Madeleine Kubasch, Elisabeta Vergu$†$.

In human contact networks, individuals often belong to several contact structures, such as a home and a workplace or school. This social organization is relevant to study in an epidemic context, as it is the subject of control measures such as telecommuting or school closures. However, the influence of these structures on the epidemic is not yet well understood. We are therefore investigating a model with two levels of mixing, namely a uniformly mixing global level, and a local level divided into two layers of contacts within households and workplaces, respectively. We are seeking to develop reduced models that closely approximate these epidemic dynamics, while being more manageable for numerical and/or theoretical analysis. In 1, a simulation study compares several telecommuting strategies, and shows that there are more effective strategies than the most naive strategy of allowing only a number of workers in the workplace proportional to the size of the workplace. Next, we highlight two indicators of the epidemic impact of the location size distribution, namely the variance and the exponential growth rate observed at the start of an epidemic. In addition, we calibrate a uniformly mixing epidemic model to provide a good approximation of the home-work model, as shown by the numerical exploration. This reduced model thus provides an easily parameterizable and numerically satisfactory approximation. Finally, in 100, a sensitivity study enables us to understand the impact of model parameters on the performance of this reduced model, by quantifying the impact of epidemic parameters on its ability to predict key features of the epidemic.

8.2.5

Participants: Marie Doumic, Klemens Fellner, Mathieu Mezache, Juan Velazquez.

To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-Döring system has been proposed in (Doumic, Fellner, Mezache, Rezaei, J. of Theor. Biol., 2019). When all reaction rates are constant, the equations are the following:

where $u$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.

8.2.6 Ongoing PhD theses

Participants: Ana Fernandez Baranda, Mateo Deangelo Bravo, Jules Olayé, Alexandre Perrin.

Following 19, the second Ana Fernandez Baranda's work is to model hematopoiesis (formation of blood cells) as a continuum process. Until now, the modeling of this phenomenon has assumed a hierarchical structure where, at the moment of differentiation, a cell transitions into a new type of cell. However, recent biological studies suggest that differentiation is instead a continuous process, where the phenotype of cells varies widely and does not allow for a clear separation of cells into distinct categories. Starting from a stochastic model with a finite number, $N$, of compartments, where each compartment corresponds to a stage in the cell's maturation, the idea is to determine the dynamics as the number of compartments approaches infinity, which represents continuous differentiation, as opposed to stepwise differentiation. To study this, we consider a measure that aggregates the compartments of immature cells, from 2 to $N-1$. This results in three coupled processes: one describing the stem cells, another for the immature cells, and a third for the red blood cells.

Mateo Deangelo Bravo has begun his Ph.D in September 2024. He models the evolution of a population of bacteria (through a measure-valued process) characterized by real values (possibly representing a genetic trait, the number of plasmids or an adaptability to the environment). This population is subject to the dynamics of births, deaths, competition, mutation and horizontal transfer (such as conjugation, plasmid transfer). The aim is the study of the ancestral lines, i.e. to trace the evolution of genetic traits in a lineage back to the initial time, starting with an individual sampled at a fixed time $T>0$.

After a 6-month pre-doc, supervised by Sarah Kaakai, Marie Doumic and Michael Rera, Luce Breuil has begun her Ph.D in September 2024. She worked on a 2-phase mathematical model of aging, based on the biological discovery by Michaël Rera 121, 132 of two consecutive phases in the aging of drosophila. With non-parametric kernel estimation and parametric estimation, we estimated the rates of transition from each phase to the next and the potential dependence between both phases through a thorough statistical study of in-vitro drosophila data. She also worked on proving the convergence of the hazard rate kernel estimator for a very general class of kernels, much less restrictive than what is usually found in the literature. Finally, she also derived a stochastic model for 2-phase aging in the wild, adding competition and birth, and studied its convergence to a deterministic model in large population.

Maxime Ligonnière, co-supervised by Vincent Bansaye and Marc Peigné, works on the study of multitype Galton-Watson processes in random environment (MGWREs), with infinitely many types. He also studied some associated products of random operators. MGWREs are a class of stochastic, individual based population models, with a structuration according to a type or trait. The offpsring of an individual is random and depends both on the type or trait of this individual and on the state of the environment at the time of the reproduction. The environment evolves through time according to a random stationary process. We are particularly interested in the so called supercritical regime, where the population survives with positive probability. We prove the mathusian growth of the population, as well as the convergence of the distribution of types at large time to a trajectory determined by the environmental sequence. We particularly introduce an example of a process modelling a population with a discrete age structure, with infinitely many age classes. In this context, we provide more tractable criterions which guarantee our various assumptions are met. A prerequisite for this work was to obtain ergodicity results for some products of random operators which act on some infinite dimensional measure spaces. To establish these ergodicity properties, we used a Doeblin-type hypothesis which assumes the existence of a special type produced in the offpsring of individuals of all types. These generalizes previously known results for $d\times d$ matrices with positive entries. Once again we focused particularly on the case of operators modelling a age structured population, called infinite Leslie matrices which are an infinite dimensional extension of the famous Leslie matrices commounly used in demographic studies and matrix population models.

Jules Olayé's Ph.D focuses on the study of mathematical models in relation to the biological phenomenon of telomere degradation, under the supervision of Marie Doumic and Milica Tomašević. His collaborative projects are described in Section 8.3.5, 8.2.1. A second work, supervised by Marie Doumic, involves solving an inverse problem related to biological phenomena. We assume that we are observing the times at which cells enter a cemetery state, called the “senescence state”, and we wish to recover the initial telomere size distribution that existed at the start of the dynamics. This work was completed during 2024, and has resulted in a pre-publication 34. In parallel with these two works, he has been working on a stochastic modeling project supervised by Frédérique Clément. They modeled the phenomenon of neurogenesis with compound Poisson processes, implemented it, and studied its moments in detail 10.

Alexandre Perrin has begun his Ph.D, funded by the ERC Advanced Grant SINGER, in September 2023, and is supervised by Sylvie Méléard, Meriem El Karoui and Marie Doumic. While significant efforts have been made to model bacterial cell division, few models have incorporated DNA replication into the control of this process. To date, models that attempt to capture the coordination between replication and division cycles are based on fundamentally different assumptions, and yet no study provide a rigorous quantitative comparaison. As a result, key questions regarding how replication quantitatively controls cell division remain unresolved. To address this, we have developed a robust mathematical framework to compare models of coordination of replication and division cycles proposed in the literature. Through theoretical analysis, we identified necessary and sufficient conditions for these models to exhibit physiological behaviour in cell cycles, and tested whether these conditions accords with experimental data. Additionally, a comprehensive statistical analysis allowed us to assess the models' ability to reproduce the joint distributions of cell cycles related quantities. This in-depth analysis led to the development of a novel model for the coordination of replication and division cycles, which provides an accurate fit to the data across a wider range of growth conditions than previous models.

Guillaume Garnier's Ph.D, co-supervised by Lydia Robert, Marc Hoffmann and Marie Doumic, is devoted to the study of the effects of mutations on the fitness of the bacteria E. coli. He has developed a non-parametric statistical method based on Fourier estimators that can be used to reconstruct the Distribution of Fitness Effects (DFE) from microfluidic data of "Mother Machine", see 122 and also Sections 8.3.3, 8.3.4, 8.3.5. This work has enabled us to explore various methods and construct a statistical estimator of this density. Extensive analytical work was carried out to formally demonstrate their convergence property, which was illustrated using numerical simulations 30.

In collaboration with Marie Doumic and Miguel Escobedo, Guillaume Garnier is also currently working on an integro-PDE, satisfied in expectation by the empirical measure inferred above. This work is an in-depth theoretical analysis of the long-time evolution of the fitness distribution.

8.3.1 Telomere shortening, a unifying model

Participants: Anaïs Rat, Marie Doumic, Teresa Teixeira, Zhou Xu.

Progressive shortening of telomeres ultimately causes replicative senescence and is linked with aging and tumor suppression. Studying the intricate link between telomere shortening and senescence at the molecular level and its population-scale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In the article 5, accepted for publication in Nature Communications, we developed a mathematical model for telomere shortening and the onset of replicative senescence using data from Saccharomyces cerevisiae without telomerase. Our model tracks individual cell states, their telomere length dynamics, and lifespan over time, revealing selection forces within a population. We discovered that both cell genealogy and global telomere length distribution are key to determine the population proliferation capacity. We also discovered that cell growth defects unrelated to telomeres also affect subsequent proliferation and may act as confounding variables in replicative senescence assays. Overall, while there is a deterministic limit for the shortest telomere length, the stochastic occurrence of non-terminal arrests drive cells into a totally different regime, which may promote genome instability and senescence escape. Our results offer a comprehensive framework for investigating the implications of telomere length on human diseases. Our model has also been used further in another experimental device, where one telomere is cut at a given very short length thanks to CrisPr-Cas9 technique 20. Alongside this modelling and simulation approach, Jules Olayé's Ph.D work focused on several interesting problems raised by this collaboration, see Sections 8.2.1, 8.2.6 and 34, 4.

8.3.2 Classical Myelo-Proliferative Neoplasms emergence and development in patients based on real-life incidence and mathematical modeling

Participants: Ana Fernandez Baranda, Vincent Bansaye, Emelyne Lauret, M Mounier, Sylvie Méléard, Stéphane Giraudier.

Mathematical modeling offers the opportunity to test hypothesis concerning Myeloproliferative emergence and development. In 19, we tested different mathematical models based on a training cohort ($n=264$ patients) (Registre de la côte d'Or) to determine the emergence and evolution times before JAK2V617F classical Myeloproliferative disorders (respectively Polycythemia Vera and Essential Thrombocytemia) are diagnosed. We dissected the time before diagnosis as two main periods: the time from embryonic development for the JAK2V617F mutation to occur, not disappear and enter in proliferation, and a second time corresponding to the expansion of the clonal population until diagnosis. We demonstrate using progressively complexified models that the rate of active mutation occurrence is not constant and doesn't just rely on individual variability, but rather increases with age and takes a median time of $63.1+/-13$ years. A contrario, the expansion time can be considered as constant: 8.8 years once the mutation has emerged. Results were validated in an external cohort (national FIMBANK Cohort, $n=1248$ patients). Analyzing JAK2V617F Essential Thrombocytema versus Polycythemia Vera, we noticed that the first period of time (rate of active homozygous mutation occurrence) for PV takes approximatively 1.5 years more than for ET to develop when the expansion time was quasi-similar. In conclusion, our multi-step approach and the ultimate time-dependent model of MPN emergence and development demonstrates that the emergence of a JAK2V617F mutation should be linked to an aging mechanism, and indicates a 8-9 years period of time to develop a full MPN.

8.3.3 Quantitative effects of the stress response to DNA damage in the cell size control of Escherichia coli

Participants: Ignacio Madrid Canales, James Broughton, Sylvie Méléard, Meriem El Karoui.

In Escherichia coli the response to DNA damage shows strong cell-to-cell-heterogenity. This results in a random delay in cell division and asymmetrical binary fission of single cells, which can compromise the size homeostasis of the population. To quantify the effect of the heterogeneous response to genotoxic stress (called SOS response in E. coli) on the growth of the bacterial population, we propose a flexible time-continuous parametric model of individual-based population dynamics 22. We construct a stochastic model based on the "adder" size-control mechanism, extended to incorporate the dynamics of the SOS response and its effect on cell division. The model is fitted to individual lineage data obtained in a 'mother machine' microfluidic device. We show that the heterogeneity of the SOS response can bias the observed division rate. In particular, we show that the adder division rate is decreased by SOS induction and that this perturbative effect is stronger in fast-growing conditions.

8.3.4 A unifying mathematical approach for the coordination of DNA replication and cell division in E. coli

Participants: Alexandre Perrin, Marie Doumic, Meriem El Karoui, Sylvie Méléard.

While significant efforts have been made to model bacterial cell division, few models have incorporated DNA replication into the control of this process. To date, models that attempt to capture the coordination between replication and division cycles are based on fundamentally different assumptions, and yet no study provide a rigorous quantitative comparaison. As a result, key questions regarding how replication quantitatively controls cell division remain unresolved. To address this, we have developed a robust mathematical framework to compare models of coordination of replication and division cycles proposed in the literature. Through theoretical analysis, we identified necessary and sufficient conditions for these models to exhibit physiological behaviour in cell cycles, and tested whether these conditions accords with experimental data. Additionally, a comprehensive statistical analysis allowed us to assess the models' ability to reproduce the joint distributions of cell cycles related quantities. This in-depth analysis led to the development of a novel model for the coordination of replication and division cycles, which provides an accurate fit to the data across a wider range of growth conditions than previous models.

8.3.5 Exploitation of microfluidic data

Participants: Vincent Bansaye, Charles Baroud, Jules Olayé, Guillaume Garnier.

In 35, we describe fluctuations in population sizes of Bellman-Harris processes to estimate lifetime distributions from temporal population size tracking. This involves determining two fluctuation regimes, leveraging recent results for Crump-Mode-Jagers processes, and applying these findings to microfluidic data.

8.3.6 Asymptotic inverse problems for fragmentation and depolymerisation models

Participants: Marie Doumic, Miguel Escobedo, Philippe Moireau, Magali Tournus.

Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution ? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a ”short” time observation is. Our analysis is complemented by a numerical investigation 11.

In another study, we focused on depolymerisation reactions, which constitute frequent experiments, for instance in biochemistry for the study of amyloid fibrils. The quantities experimentally observed are related to the time dynamics of a quantity averaged over all polymer sizes, such as the total polymerised mass or the mean size of particles. The question analysed here is to link this measurement to the initial size distribution. To do so, we first derive, from the initial reaction system two asymptotic models: at first order, a backward transport equation, and at second order, an advection-diffusion/Fokker-Planck equation complemented with a mixed boundary condition at x = 0. We estimate their distance to the original system solution. We then turn to the inverse problem, i.e., how to estimate the initial size distribution from the time measurement of an average quantity, given by a moment of the solution. This question has been already studied for the first order asymptotic model, and we analyse here the second order asymptotic. Thanks to Carleman inequalities and to log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularization. We then develop a Kalman-based observer approach, and implement it on simulated observations. Despite its severely ill-posed character, the second order approach appears numerically more accurate than the first-order one 27.

9.1.1 Visits of international scientists

9.1.2 Visits to international teams

Gaël Raoul has a regular collaboration and long-term visits every year with research groups in Vietnam:

• with Marc Choisy and Pham Thanh Duy, Oxford Clinical Research Unit, Ho Chi Minh city, Vietnam. They work on the multi-drug resistance of Klebsiella pneumoniae.
• with Vo Hoang Hung, Saigon University, Ho Chi Minh city, Vietnam. They work on the effect of an age structure on the propagation of population in the context of climate change.

9.2.1 Other european programs/initiatives

M. Tomasevic is the PI in the project IEA - International Emerging Actions of CNRS with University of Bath, Title: Chemotaxis in random environments: from microscopic to macroscopic viewpoints. Funding 4K 2025, 4K 2024.

10.1.1 Scientific events: organisation

The team organised, on April 3rd, a kick-off meeting for the launching of the team (50 participants, and a series of four "twin talks" with a biologist and a mathematician speaking together on a given interdisciplinary project).

Sylvie Méléard organised, as every year, the summer school of the MMB Chaire in Aussois, 10-14 June, 2024 (70 participants, including many members of MERGE - Vincent Bansaye, Luce Breuil, Marie Doumic, Alexandre Perrin, Ana Fernandez Baranda, Anouar Jeddi, Milica Tomasevic - and many collaborators of our team).

Sylvie Méléard also organised a retreat of her ERC SINGER, on December 2-4, to which many members of MERGE participated.

We organise local seminars, called "MERGE aperitif", where our students or invited students give talks. Luce Breuil, Jules Olayé, Alexandre Perrin, Ana Baranda, as well as Claudia Fonte Sanchez and Manuel Esser gave talks in 2024.

M. Tomasevic co-organized two research schools at CIRM in 2024 : Stochastic and Deterministic Analysis for Irregular Models (8-12 jan) and Collective behavior and Pattern formation (1-5 july). The second one obtained the MJC funding of Inria.

M. Tomasevic co-organized the Annual research school of the Chaire MMB at Aussois (juin 2024). She also co-organized an invited session on the Journée MAS entitled Mean field limits and control (Poitiers, Aug 2024) and the Conference 50 ans CMAP at Ecole polytechnique (sept 2024).

M. Tomasevic co-organizes the probability seminar of CMAP.

M. Tomasevic is co-responsable of the thematic group MABIOME of SMAI and a member of the Scientific council of the RT Maths bio santé.

Marie Doumic organised a mini-symposium at the ECMTB Conference in Toledo, Spain, on Telomere dynamics modelling, where Jules Olayé, Anaïs Rat and Marek Kimmel gave talks.

Sylvie Méléard co-organised the conference A lifelong journey in stochastic analysis: from branching processes to statistical mechanics, IHP Paris 2024.

10.1.2 Scientific events and invited talks

Vincent Bansaye participated in:

• Meeting « Probability, Ecology and Evolution » in Besancon, 10/12/2024, talk on branching processes and networks
• Conference « Ecological networks, complex systems, stability » , 28/10/2024, talk on Stochastic functional responses in ecology
• Conference « Non local branching Markov process », in CIRM , 20th September 2024, talk on Branching processes for growing, living network;
• Meeting Inov3PT, Paris, 9 janvier 2024, talk on invasive stochastic process and applications

Marie Doumic gave

• a talk in the TeloEMBO meeting, April 6-10 2024, Rome
• a 3-hour minicourse at the Mittag-Leffler Institute for the "Kinetic Theory arising from mathematical biology" conference, Stockholm, Sweden, July 1-5 2024
• several seminars: Toulouse (September 30th), Brest (September 9th), Versailles (December 16th), CMAP.

Madeleine Kubasch gave talks in

• Dec 2024 — « Besançon meeting on Probability, Ecology and Evolution », Université Bourgogne Franche-Comté, Besançon.
• August 2024 — 11th Bernoulli-IMS World Congress in Probability and Statistics, Organized Contributed Paper Session « Stochastic Epidemic Models », Bochum (Germany).
• May 2024 — Workshop « Spatial Epidemic Models (Including Graphs and Graphons) », Rice Global Paris Center, Paris.

Maxime Ligonnière gave talks in

• Séminaire de Probabilités de l'Institut Fourier, Grenoble, November 2023
• Workshop of the ANR RAWABRANCH, Angers, March 2024
• Séminaire de probabilités LMBA, Brest, April 2024
• Séminaire de Probabilités, LPSM, Sorbonne université, Paris, May 2024
• Workshop Produits de matrices et branchement, ESAIP, Angers, June 2024
• Workshop on branching processes and products of random matrices, LMBA, UBS, Vannes, July 2024

Sylvie Méléard gave talks in:

• Valparaiso 2024, Biostochastic
• HKIAS International Conference on Mathematical Analysis and its Applications, City University of Hong Kong, 2024
• 30 ans du Laboratoire Manceau de Mathématiques : Probabilités - Statistique - Risque, 2024
• Non-local branching processes, CIRM 2024
• Seminaries: Inria Dyogene Seminar, February 2024 ; Rennes 2024, Séminaire MoVi.

Jules Olayé

• gave talks at the kick-off meeting of MERGE and of the PEPR Maths VivES DyLT project
• gave a talk at the 13th ECMTB (European Conference on Mathematical and Theoretical Biology), Toledo, Spain
• presented posters at the "Mathematical Biology: Collective Behavior and Pattern Formation", CIRM, Marseille, and at the 50th anniversary of the CMAP

Gaël Raoul gave talks in

• several seminars: Chaire MMB seminar (May 6th 2024), Seoul National University (October 18th 2024), online seminar of the National Center for Theoretical Sciences in Taiwan (0ctober 25th 2024).
• the ReaDiNet conférence in Jeju, Korea (October 23th 2024).

Milica Tomasevic gave talks at the following conferences and seminars:

• Conference PDE and Probability in interaction: functional inequalities, optimal transport and particle systems, jan 2024 CIRM
• French Japanese Conference on Probability & Interactions, Mar 2024, IHES
• Non-local operators, probability and singularities, May 2024, online seminar
• Frontiers in Interacting Particle Systems, Aggregation-Diffusion Equations and Collective Behavior, june 2024 CIRM
• Colloquium de laboratoire de Maths, Créteil, déc 2024.

10.1.3 Journal

Marie Doumic is editor in Chief of ESAIM Proceedings and Surveys.

Marie Doumic and Gaël Raoul are editors of the Journal of Mathematical Biology.

Marie Doumic is associate editor for Kinetic and Related Models and the Bulletin des Sciences Mathématiques.

Sylvie Méléard is associate editor for the Comptes-Rendus de l'Académie des Sciences (CRAS).

10.1.4 Research administration

Vincent Bansaye is vice-president of the Applied Math Department of Ecole Polytechnique and of Fondation Mathématique Jacques Hadamard (FMJH).

Marie Doumic, Sylvie Méléard and Vincent Bansaye are members of the steering committee of the MMB chair.

Marie Doumic is a member of the board of the ESMTB (European Society for Mathematical and Theoretical Biology).

Sylvie Méléard is a member of the Scientific Advertisory Board of HIM (Hausdorff Research Institute for Mathematics, Bonn, Germany), CMM (Center for Mathematical Modeling, Santiago, Chili) and of CRM (Centre de recherches mathématiques, Montréal, Canada).

10.2.1 Teaching

Vincent Bansaye gave 3 courses.

Sylvie Méléard is the head of the master "Mathematics for living sciences" and gave a course on Stochastic Processes in the master.

M. Tomasevic is a part time professor at DMAP, Ecole polytechnique with 64h of service. In addition, she gives an M2 course (24h) at M2 in probability at Jussieu.

Luce Breuil gave probability tutorials for Bachelor 2 students of Ecole Polytechnique as well as linear algebra classes for the Master X-HEC.

10.2.2 Supervision

Several students are co-supervised by two permanent members of MERGE:

• Alexandre Perrin is co-supervised by Marie Doumic and Sylvie Méléard
• Ana Fernandez Baranda is co-supervised by Vincent Bansaye and Sylvie Méléard
• Jules Olayé is co-supervised by Marie Doumic and Milica Tomasevic
• Nadia Belmabrouk's post-doctoral studies are in collaboration with Vincent Bansaye and Sylvie Méléard.

Vincent Bansaye also supervised Maxence Baccara (with Sylvain Billiard and Jean-René Chazottes) and Alexandre Bertolino (with Ayman Moussa).

Marie Doumic also co-supervised Guillaume Garnier (with Marc Hoffmann and the biologist Lydia Robert, funded by Inria-INRAE grants), Luce Breuil (with Sarah Kaakai and the biologist Michael Rera, funding AMX) and Viviana Gavilanes (with the biologist Zhou Xu, funded by MITI of CNRS).

Sylvie Méléard also supervised Ignacio Madrid Canales (Co direction with the biologist M. El Karoui), Defense in February 2024 and co-Supervised ($50\%$) Anouar Jeddi (with Nicolas Champagnat) and Mateo Deangelo Bravo (with Viet Chi Tran).

Gaël Raoul co-advises the PhD thesis of Le Tuyet Nhi Pham with Giovanni Conforti, University of Padova, Italy. The PhD thesis is devoted to the development of new entropic optimal transport methods inspired by biological problems. Sirine Boucenna defended her Ph.D on September 26th, 2024, on the eco-evolutive dynamics of ecosystems under stress, under the supervision of Vasilis Dakos and Gaël Raoul.

Mililca Tomasevic also co-supervises N. Cazacu's Ph.D since 2024 with A. Richard (CentraleSupelec) and the postdoctoral project of Hadamard Lecturer T. Cavalazzi since 2023 with A. Richard. M. Tomasevic co-supervised two internships of level M2.

10.2.3 Juries

Vincent Bansaye participated in 4 juries (3 Ph.D, 1 Habilitation thesis). Sylvie Méléard participated in 5 juries (3 PhD thesis: Ignacio Madrid Canales, Laurent Freoa, Louis-Pierre Chaintron) and 2 habilitation thesis (Boris Nectoux and Clément Foucart).

M. Tomasevic was a vice president of the recruting jury of an assistant professor position at Ecole polytechnique.

Marie Doumic participated to the selection committee of a full professor at INSA Toulouse (May 2024) and to the admission jury of DR Inria. She reviewed the PhD thesis of Nathan Quiblier and Nga Nguyen, chaired Lucie Laurence's defence, and was a member of the habilitation thesis committee of Mélanie Prague.

10.3.1 Specific official responsibilities in science outreach structures

Sylvie Méléard is now member of the Organisation Committee of the SMF-BNF cycle "un texte, une aventure mathématique" : The lecturer chooses a mathematical text dating back several decades, or even much longer, which has particularly influenced him. Based on this text, its author and its history, the lecturer will show how an ancient mathematical problem leads to current questions and ongoing mathematical research. Combining history and mathematics, the conferences enable a wide audience to discover contemporary mathematics.

10.3.2 Productions (articles, videos, podcasts, serious games, ...)

For Sylvie Méléard's Irène Joliot-Curie price, there have been several articles: ,.

For Madeleine Kubasch's L'Oréal-Unesco price, there also have been several articles , , , , .

Marie Doumic has been interviewed for "l'oreille mathématique" , a podcast produced by IHP.

For the kick-off day of MERGE, on April 3rd, Marie Doumic, Sylvie Méléard and Gaël Raoul have been interviewed for the Ecole Polytechnique website and Inria website .

Guillaume Garnier participated to "ma thèse en 180 secondes" .

10.3.3 Participation in Live events

Sylvie Méléard gave a conference at the Institut Montaigne with title "Mathématiques et biodiversité".

Madeleine Kubasch gave a large audience presentation in the framework of « Entretiens de l’Excellence » at École polytechnique, presenting her work and career for 700 middle-schoolers . She also participated to the Fête de la Science and presented her research work at the Cité des Sciences et de l'Industrie, in the framework of the price "Young Talents France L’Oréal-Unesco For Women and Science".

Marie Doumic participated to middle school's career mornings.

10.3.4 Others science outreach relevant activities

Vincent Bansaye participated in the setting up and took part in the math club at Gérard Philippe secondary school, Massy.

International journals

• 6 articleV.Vincent Bansaye, F.François Deslandes, M.Madeleine Kubasch and E.Elisabeta Vergu. The epidemiological footprint of contact structures in models with two levels of mixing.Journal of Mathematical BiologySeptember 2024HALDOI
• 7 articleV.Vincent Bansaye, X.Xavier Erny and S.Sylvie Méléard. Sharp approximation and hitting times for stochastic invasion processes.Stochastic Processes and their Applications178December 2024, 104458HALDOIback to text
• 8 articleV.Vincent Bansaye. Spine for interacting populations and sampling.Bernoulli302May 2024HALDOIback to text
• 9 articleV.Vincent Calvez, A.Adil El Abdouni, M.Maxime Estavoyer, I.Ignacio Madrid, J.Julien Olivier and M.Magali Tournus. Regime switching on the propagation speed of travelling waves of some size-structured Myxobacteria population models.ESAIM: Proceedings and Surveys77November 2024, 195-212HALDOI
• 10 articleF.Frédérique Clément and J.Jules Olayé. A stochastic model for neural progenitor dynamics in the mouse cerebral cortex.Mathematical Biosciences372June 2024, 109185HALDOIback to text
• 11 articleM.Marie Doumic, M.Miguel Escobedo and M.Magali Tournus. An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation.Annales Henri Lebesgue7September 2024, 621-671HALDOIback to text
• 12 articleM.Marie Doumic, S.Sophie Hecht, B.Benoît Perthame and D.Diane Peurichard. Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion.Journal of Differential Equations389April 2024, 228-256In press. HALDOIback to text
• 13 articleQ.Quentin Griette, M.Matthieu Alfaro, G.Gaël Raoul and S.Sylvain Gandon. Evolution and spread of multiadapted pathogens in a spatially heterogeneous environment.Evolution Letters83June 2024, 427-436HALDOIback to text
• 14 articleA.Anaïs Rat, V.Veronica Martinez Fernandez, M.Marie Doumic, M. T.Maria Teresa Teixeira and Z.Zhou Xu. Individual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence.Nature Communications161January 2025, 1024HALDOI
• 15 articleT.Tristan Roget, C.Claire Macmurray, P.Pierre Jolivet, S.Sylvie Meleard and M.Michael Rera. A scenario for an evolutionary selection of ageing.eLife13November 2024, RP92914HALDOIback to text
• 16 articleM.Milica Tomašević and G.Guillaume Woessner. On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model.Stochastic Analysis and Applications424August 2024, 767-796HALDOIback to text

International peer-reviewed conferences

• 17 inproceedingsM.Marie Doumic. Moments approaches for asymptotic inverse problems of depolymerisation and fragmentation systems.Journées Equations aux dérivées partiellesJournées Equations aux Dérivées PartiellesAussois, France2024HAL

Reports & preprints

• 18 miscV.Vincent Bansaye, P.-Y.Pierre-Yves Boëlle, S.Simon Cauchemez, B.Bernard Cazelles, P.Pascal Crepey, L.Lina Cristancho-Fajardo, J.-S.Jean-Stéphane Dhersin, S.Sorin Dumitrescu, P.Pauline Ezanno, A.Antoine Flahault, J.-L.Jean-Louis Golmard, P.Patrick Hoscheit, H.Henri Mermoz Kouye, M.Madeleine Kubasch, C.Catherine Laredo, A.Alain Mallet, L.Lulla Opatowski, C.Clémentine Prieur and A.Amandine Véber. Hommage à Elisabeta Vergu.2024HAL
• 19 miscA. F.Ana Fernández Baranda, V.Vincent Bansaye, E.Evelyne Lauret, M.Morgane Mounier, V.Valérie Ugo, S.Sylvie Méléard and S.Stéphane Giraudier. Classical Myelo-Proliferative Neoplasms emergence and development based on real life incidence and mathematical modeling.2024HALDOIback to textback to text
• 20 miscP.Prisca Berardi, V.Veronica Martinez Fernandez, A.Anaïs Rat, F.Fernando Rosas Bringas, P.Pascale Jolivet, R.Rachel Langston, S.Stefano Mattarocci, A.Alexandre Maes, T.Théo Aspert, B.Bechara Zeinoun, K.Karine Casier, H.Hinke Kazemier, G.Gilles Charvin, M.Marie Doumic, M.Michael Chang and M. T.Maria Teresa Teixeira. The shortest telomere in telomerase-negative cells triggers replicative senescence at a critical threshold length and also fuels genomic instability.January 2025HALDOIback to text
• 21 miscC.Charles Bertucci, M.Matthias Rakotomalala and M.Milica Tomasevic. Curvature in chemotaxis: A model for ant trail pattern formation.September 2024HALback to text
• 22 miscI. M.Ignacio Madrid Canales, J.James Broughton, S.Sylvie Méléard and M.Meriem El Karoui. Quantitative effects of the stress response to DNA damage in the cell size control of Escherichia coli.September 2024HALback to text
• 23 miscP.Pierre Collet, C.Claire Ecotière and S.Sylvie Méléard. Long time behavior of a degenerate stochastic system modeling the response of a population face to environmental impacts.May 2024HALback to text
• 24 miscP.Pierre Collet, S.Sylvie Méléard and J.Jaime San MARTIN. Branching diffusion processes and spectral properties of Feynman-Kac semigroup.April 2024HALback to text
• 25 miscM.Marie Doumic, K.Klemens Fellner, M.Mathieu Mezache and J. J.Juan J L Velázquez. Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system.July 2024HAL
• 26 miscM.Marie Doumic, S.Sophie Hecht, M.Marc Hoffmann and D.Diane Peurichard. Scaling limits for a population model with growth, division and cross-diffusion.October 2024HALback to textback to text
• 27 miscM.Marie Doumic and P.Philippe Moireau. Asymptotic approaches in inverse problems for depolymerization estimation.September 2024HALback to text
• 28 miscM.Marie Doumic, A.Anaïs Rat and M.Magali Tournus. Comparison Between Effective and Individual Fitness in a Heterogeneous Population.December 2024HALback to text
• 29 miscM.Meriem El Karoui, I.Ignacio Madrid and S.Sylvie Méléard. Phenotypic plasticity trade-offs in an age-structured model of bacterial growth under stress.March 2024HAL
• 30 miscG.Guillaume Garnier. Decompounding with unknown noise through several independents channels.May 2024HALback to text
• 31 miscG.Gaoyue Guo and M.Milica Tomasevic. Mean-field limit of particle systems with absorption.July 2024HAL
• 32 miscA.Anouar Jeddi. Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation.2024HALDOIback to text
• 33 miscM.Maxime Ligonnière. A Kesten Stigum theorem for Galton-Watson processes with infinitely many types in a random environment.November 2024HAL
• 34 miscJ.Jules Olayé. An inverse problem in cell dynamics: Recovering an initial distribution of telomere lengths from measurements of senescence times.January 2025HALback to textback to text
• 35 miscJ.Jules Olayé, H.Hala Bouzidi, A.Andrey Aristov, S. G.Salomé Gutiérrez Ramos, C.Charles Baroud and V.Vincent Bansaye. Estimation of the lifetime distribution from fluctuations in Bellman-Harris processes.February 2024HALback to text
• 36 miscJ.Jules Olayé and M.Milica Tomasevic. Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel.August 2024HAL