2023Activity reportProjectTeamMERGE
RNSR: 202324390R Research center Inria Saclay Centre at Institut Polytechnique de Paris
 In partnership with:Institut Polytechnique de Paris, CNRS
 Team name: Mathematics for Evolution, Reproduction, Growth and Emergence
 In collaboration with:Centre de Mathématiques Appliquées (CMAP)
 Domain:Digital Health, Biology and Earth
 Theme:Computational Biology
Keywords
Computer Science and Digital Science
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.2. Stochastic Modeling
 A6.1.4. Multiscale modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.3. Probabilistic methods
 A6.2.4. Statistical methods
 A6.3.1. Inverse problems
Other Research Topics and Application Domains
 B1. Life sciences
 B1.1. Biology
 B1.1.2. Molecular and cellular biology
 B1.1.6. Evolutionnary biology
 B1.1.8. Mathematical biology
 B2. Health
 B2.2.3. Cancer
 B2.2.6. Neurodegenerative diseases
 B2.3. Epidemiology
 B2.4.2. Drug resistance
 B3. Environment and planet
 B3.6. Ecology
 B3.6.1. Biodiversity
1 Team members, visitors, external collaborators
Research Scientists
 Marie DoumicJauffret [Team leader, Inria, Senior Researcher, HDR]
 Gael Raoul [CNRS, Researcher]
 Milica Tomasevic [CNRS, Researcher]
Faculty Member
 Vincent Bansaye [Ecole Polytechnique, Professor, HDR]
PostDoctoral Fellow
 Laetitia Colombani [Ecole Polytechnique, from Sep 2023]
PhD Students
 Sirine Boucenna [Ecole Polytechnique]
 Claire Ecotière [Ecole Polytechnique, until Oct 2023]
 Ana Fernandez Baranda [Ecole Polytechnique]
 Guillaume Garnier [Inria]
 Madeleine Kubasch [Ecole Polytechnique]
 Maxime Ligonnière [Ecole Polytechnique]
 Ignacio Madrid Canales [Ecole Polytechnique]
 Jules Olayé [Ecole Polytechnique]
 Alexandre Perrin [Ecole Polytechnique, from Oct 2023]
 Anaïs Rat [Inria, until Mar 2023]
Administrative Assistant
 Anna Dib [INRIA]
2 Overall objectives
The wide domain of population dynamics has had many developments in recent years, in probability with the study of stochastic integrodifferential equations 35 as well as in PDE analysis 110, 109. The two approaches are combined more and more frequently, for model analysis 51, 34 as well as for estimation problems 19. In biology, many new questions have appeared, and the very recent development, over the last decade, of the socalled "single cell" or microfluidic methods 124, 75, 89, 39 make these models all the more topical as they can now be quantitatively compared with the data microscopically as well as macroscopically. Many essential medical and social applications are closely related to our research, e.g. cancer treatment (see Section 4.1), biotechnologies (Section 4.3), antibiotic resistance (Section 4.1), species extinction (Section 4.4). Our main theoretical guideline, which can have applications in other fields (SPDE, propagation of uncertainty, PDE analysis...), is to reconcile PDE approaches with stochastic ones, in situations where the two types of dynamics play a fundamental role at different scales. Our main application guideline is to study problems directly inspired by our biologist collaborators' questions, so that even our most theoretical work could have an impact also in biology or medicine.
The applications drive our mathematical research, including the most theoretical ones. Many of our models have several possible applications so that the interests of MERGE members converge, since for instance we are interested in modelling mutations both for bacteria and for leukemic cells; emergence of survivors for senescent yeasts as well as for bacteria under antibiotic treatments; evolutionary questions for bacterial populations as well as tree populations submitted to the climate change. Moreover, most of our mathematical models have even wider applications than in biology  among many other possible examples, fragmentation processes occur in mineral crushing in the mining industry, cell division models are close to models for the TCPIP protocol. The main application domain, shared by all team members, concerns unicellular organism populations.
Our research program is organised along three main axes. First, the study of "models through scales", i.e. the links between various stochastic or PDE models through convergence analysis of individualbased models towards mesoscopic or macroscopic ones, is essential for our models to have a solid foundation. The second axis is their mathematical analysis, which allows one to qualitatively compare them to biological systems and use them as predictive and exploration tools, whereas the third one develops methods for their quantitative comparison to data. For each research axis, we outline what we consider to be the major current research issues of the field, and then use a few non exhaustive examples of work in progress to give a concrete description of our work programme in the short and medium term.
To make the links between our research program and the applications more obvious, we have specified the main research axes concerned for each application.
3 Research program
Our research program is entirely devoted to the modelling and study of interacting populations. In many cases, we will also develop methods for quantitative modeldata comparison through estimation methods and inverse problems.
The first research axis, "Models through scales", is devoted to mathematical problems which appear in order to obtain rigorous links between microscopic, mesoscopic and macroscopic models. These questions are closely related to the modelling work, which we have not detailed in a specific section, as it is carried out through exchanges with our medical doctors and biologists collaborators and is a direct continuation of the application questions outlined above. The second axis gathers qualitative analysis problems for the structured population models that we wrote during such modelling work, or inspired by our interdisciplinary discussions. The third axis, "Modeldata comparison", goes back to the data, through inverse problems theoretical and numerical solution.
3.1 Axis 1: Models through scales
Permament members: Vincent Bansaye, Marie Doumic, Sylvie Méléard, Gaël Raoul, Milica Tomašević
When we describe noninteracting populations which undergo mutations, growth, movement, division and death, the stochastic branching process modelling each individual behaviour may be translated to a structured population equation or system in a rather direct way, by the use of random measures 19 or from the expectation of the empirical measure linked to the branching tree and socalled manytoone formulae 57. This is no more true once interaction between the cells or with the environment is considered: in such cases, meanfield limits have to be derived 78, by making the number of individuals in the population tend to infinity. Making such limits rigorous, and relating the asymptotic models to specific parameter regimes, is a very active research field not only for structured populations but also in physics. One faces several fundamental questions: how to describe and quantify the emergence of an initially very small number of individuals, inside multispecies interacting populations which, depending on available resources and space, will finally succeed to become dominant? How to keep track of microscopic fluctuation at the macroscopic level? How to perform a macroscopic limit when each individual interacts only with its closest neighbours rather than with the overall population? Finally, how stochasticity and heterogeneity between individuals impact macroscopic behaviours? These issues drive our work. Let us now detail some more specific problems we want to study.
From stochastic processes to constrained HamiltonJacobi (HJ) equations.
Permanent members: Sylvie Méléard, Gaël Raoul
Most models, for instance for the "normal" bacterial division cycle 70, 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 74 (see also the application section 4.1). The individualbased population process is given for any $K$ by the jump pointmeasure Markov process ${\left({\mu}_{t}^{K}\right)}_{t}$ on a trait subset of ${\mathbb{R}}^{d}$ 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 integrodifferential equation with non local non linearities due to both competition and transfer. Uniqueness of its solution is obvious but its longtime 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 63, successfully developed in 41, 107, 40, and in many extensions so far. Concentrations in such equations are too fast for realistic evolution 108. Indeed the evolutionary dynamics strongly depend on the positivity of the density although it is exponentially small for some traits. Different papers 108, 84, 100 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 50 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 52. This approach was very fruitful and allowed to quantify the complete scheme from individuals to macroscopic behaviors as suggested in 97 and 62. 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 125, 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 HamiltonJacobi equation and to understand the role of the tradeoff between demography and transfer.
Spaceandtrait 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 individualbased models is of key importance for a correct interpretation of macroscopically observed phenomena such as morphogenesis or front propagation. Let us develop two examples: sizeandspace structured models and phenotypic traitandspace structured models.
Microcolony morphogenesis.
A bacterial microcolony may form out of one single cell, growing and dividing in a petri dish without movement except due to the growth. We can describe it by an individualbased model, where each cell repulses and maybe attracts its neighbours, but how do these local interaction forces influence the overall shape of the microcolony? When and how do specific patterns emerge? Do the bacteria only repulse each other or is attraction possible? Which mesoscopic or macroscopic description would be valid? These are some of the questions we want to address.
As a first step, in 69, Marie Doumic, Sophie Hecht and Diane Peurichard proposed a purelyrepulsive individualbased model of rodshaped bacteria, where growth, division and repulsion were sufficient to explain the main characteristics of microcolonies observed.
A first research direction consists in deriving rigorously a kinetic model, including both a spatial structure and a structuring trait such as size. For a model with spherical 2Dcells dividing into equallysized daughters, with an interaction force $\varphi ,$ an example of limit model satisfied by $u(t,x,r)$ the density of cells at time $t,$ position $x$ and radius $r$ is as follows
This should generalize the model proposed in 102. However, the main drawback is that to prove rigorously this limit, departing from a stochastic differential equation of the same kind as (3.1.0.1) when the number of cells $K$ tends to infinity, one needs to assume a nonlocal interaction kernel $G,$ so that at the limit each cell interacts with infinitely many others. This is false for many applications and in particular for morphogenesis. We thus want to derive, from (1), a macroscopic model where the nonlocal interaction kernel $G$ boils down to a local one, cells interacting only with the ones at the same macroscopic position $x$ 49. However, even for simpler cases  for instance forgetting with the growth and division terms  many difficulties appear, since existing methods 105, 104, based on energy inequalities and compactness embeddings 88, 93, cannot apply due to the lack of compactness in the size variable.
Another research direction, for not isotropic cells but rather rodshaped bacteria like E. coli, is to include a direction for each individual. In this spirit, nematic liquid crystal models 32, 92 have been proposed to describe a variety of biological active fluids, e.g. cellular monolayers 66, 123, 126; though, how they may be derived from individualbased models such as the hardrod model of 126, 69 or the models of 55, 123 remains unclear. We aim at deriving, formally and then  on simplified versions of the model  rigorously, a continuous model of liquid crystal type.This could then be a step towards the reverse question: how to estimate the microscopic interaction function from a macroscopic picture of the colony at a given time, see Section 3.3.
Space and phenotype species models.
Sexual reproductions imply the recombination of DNA during reproductions. The models describing the effect of recombinations on traitstructured species can be divided into two classes: the ones describing the dynamics of a small number of loci (typically less than 3), and the ones considering an infinite number of loci. In the latter case, the main model used is the socalled infinitesimal model, that was developed by Fisher in 1919 76. This model is reminiscent of collision models from statistical physics, which provides an interesting perspective to study the dynamics of these models, in particular when this phenotypic structure is coupled to a distribution of the species in space.
Our first goal will be to generalize the derivation of mascroscopic limits 101, 43 to situations where a finite (but large) number of loci are present, and/or where the reproduction is partially asexual. We would like to study the spatial dynamics of such species compare to asexual species on one side, and to the infinitesimal model case on the other side. From a ecological stand point, this would help us understand the impact of recombination on species' range.
Our second goal will be to use these macroscopic limits to build travelling waves for the structured population models. We would then take advantage of the diffusion operator that represents the effect of the spatial dispersion of individuals. The main roadblock here will be to develop a good framework for the macroscopic travelling waves 98. This is difficult because the macroscopic equations (describing the population by its size and mean phenotypic trait in each location) involves a socalled gene flow term, that we do not fully apprehend yet. This difficulty is directly related to ecological questions: gene flow is an important effect of sexual reproduction on a species' evolutionary dynamics.
The last objective on this topic would be to develop a software able to simulate the dynamics of a species' range. Based on the travelling wave analysis we have developed 27, we believe we could use recently developed fastmarching algorithms 99 to propose a description of the effect of climate change on a given species.
From local interaction models to crossdiffusion equations.
Many interactions of species and cells are local, which means that they occur when individuals are close enough, at a distance negligible for the macroscopic scale. Going from the individual level to macroscopic models raises several mathematical challenges linked in particular to the control of the non linearity in the motion component 77. This issue is linked to the control of the limiting PDE (stability, nonexplosion, invariant distribution, entropic structure) and the distance involved in the convergence of the stochastic process. Vincent Bansaye, Ayman Moussa and Felipe Munoz have developed duality estimates to prove stability of the limit and get a strong convergence of the stochastic model seen as a random perturbation 36.
Nonmarkovian interactions: from local interaction model to the parabolicparabolic KellerSegel system
Permanent members: Milica Tomašević
An important mathematical challenge is to derive meanfield limits for nonMarkovian interaction, i.e., when the past also needs to be taken into account. Such models appear for instance in neuroscience 53, 54 and chemotaxis. To model chemotaxis, the parabolicparabolic KellerSegel model has been stated phenomenologically, but to interpret it we need to introduce interaction memory, which provides tremendous analysis difficulties since particles are now nonmarkovian 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 114, 118, 116. 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 KellerSegel 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 85 for the quantitative study of the meanfield boundaries of particle systems in nonregular 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 KellerSegel system. By exploiting the results on the KellerSegel nonlinearity in 1 dimension and on the Sobolev type estimation on the densities of the system (chapter 4 in 117), 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 122 and Lacker 91 as well as by the partial Girsanov transformations introduced in 86.
3.2 Axis 2: Qualitative analysis of structured populations
Diffusiongrowthfragmentation 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 115. 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 individualbased model. This model also present similiarities with the proliferation of parasites in dividing cells studied by Vincent Bansaye 38, 37.
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 diffusiontype 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 longtime 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 growthfragmentation 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 multidimensional growthfragmentation 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 growthfragmentation 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.
Differential influence of the initial condition
Time to extinction in the case of genealogical data differs drastically from time to extinction for a dividing whole population, so that observing the first occurs at a much faster timescale than the second. Relating the two in simple models like the GaltonWatson tree is straightforward, but much more involved in more complex cases 35, especially if rare mutation events occur (see Section 3.1). With a view towards telomere shortening models and the interpretation of experiments carried out by Teresa Teixeira's lab, we want to assess rigorously the relations between these two observation cases in increasingly complex models. Reversing time in population models which are in a stationary regime has been well developed during the past decades, using coalescent and duality theories, in particular in the case of fixed population size. Understanding the genealogical structure in transitory regime (such as growth), keeping track of the initial conditions (in particular in finite window size of experiments or cancer treatment or epidemics) or capturing the effect of variations of the populations raise new and fundamental mathematical issues. For that purpose, we aim at developing spinal approaches, which consist in a forward construction distinguishing an individual bound to be the sample at a future time.
Time reversed trajectories.
A natural question is to get information on individuals from observation on the whole population at a given time. More precisely, given a finite sample of living individuals, we aim first, to find their genealogical and trait's history and second, to find the explicit time reversed path from a sampled individual to its ancestor. A particularly interesting case is the one when the initial density of the whole stochastic process is close to a Dirac measure. This question motivated an abundant literature in population genetics with the socalled Kingman coalescent (see 90, 44 and references therein), or lookdown processes 65, 73 in a context of fixed and small population size, almost neutrality and individuals independence. Genealogy of branching processes models have also been introduced, allowing demographic structure but no interactions (cf. 94). Our framework is different: we focus on bacteria or cells which form large populations and for which assumptions of neutrality, extrinsic control of population size or noninteracting individuals are violated. Developing methods which relax such hypotheses is a contemporary challenge, which could be used in different contexts (see below how this point of view can be of particular relevance to study the individuals responsible of the population survival in case of environmental changes). Inspired by Perkins 106, Sylvie Méléard and Viet Chi Tran constructed in 96 a nonlinear historical superprocess with values in a paths measure space, capturing the history of a large population. It is a heavy object which might not be tractable for our goal.
Our purpose is to introduce more tractable tools, exploiting the large population assumption ($K\to \infty $) and the spinal techniques developed for branching processes (cf 82, 81, 94 and references therein). We have seen that the stochastic process (3.1.0.1) is close to the solution of an integrodifferential equation. Therefore, we can construct for large $K$ a coupling between the stochastic process and a nonhomogeneous structured branching process where the interaction terms have been replaced by their deterministic approximation. We should obtain some nonhomogeneous biased Markov process by giving its associated infinitesimal generator. The next step would consist in finding the time reversed trajectory of a sample individual. This will be done using time reversal theory for nonhomogeneous Markov processes, see 56, 103. This program has already been developed in the Gaussian case 48 and lead to a precise quantitative description of the reverse trajectories explaining the genetic or phenotypic characteristics of a living individual.
3.3 Axis 3: Modeldata comparison
Permanent members: Marie Doumic, Sylvie Méléard, Milica Tomašević
Comparing models to data, either qualitatively or quantitatively, is an essential step for all the previously seen tasks, especially the asymptotic studies through scales. It is often done in a purely informal way, by recursive discussions with our biologists collaborators and qualitative comparison, see Section 4 and for examples of models we design in such interdisciplinary work 30, 45, 46, 68. It may also be carried out with the use of theoretical analysis as in Axis 2, or by sensitivity analysis on the parameters (as for instance in 33, 79, 120), or by relatively standard data analysis tools , as has been done for instance in 42, 47, 95 by various members of the team; our added value then lies in the biological conclusions and models conception rather than in methodological novelties. In other cases however, no standard method is available, or yet, we are led by experimentalists to formulate new inverse problem questions, see for instance 19 for a review of the estimation of the division rate in structured population equations, or yet 29, 71 for the study of inverse problems formulated with biologists.
In this section, we thus explain some of the methodological developments that will be carried out in MERGE in this field of ("deterministic" or "statistical") inverse problems. The underlying question, throughout the section, is to estimate growth and division functional parameters of the individuals. Though we work with external collaborators who are experts in statistics, our team would greatly benefit from the recruitment of a statistician, in order to stay at the cutting edge of new methods like bayesian approaches or machine learning.
Estimate growth, division, interaction features in structured populations
The estimation of the division rate in noninteracting populations has been developed in a series of papers over the last decade 19. The question we want to address now is whether growth and division rates are modified by celltocell 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 individualbased models. We plan the following steps:
 Develop parametric and nonparametric inference of the interaction function from single individual tracking. A similar study has been carried out by Laetitia Della Maestra and Marc Hoffmann 61 for McKeanVlasov equation; we would like to add a size structure and a nonconstant 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 127.
 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 64.
Estimate mutation or fragmentation kernel density
The question of estimating the fragmentation kernel in polymer breakage experiments 67 surprisingly rejoins the question of estimating the socalled Distribution of Fitness Effects (DFE) which characterizes the accumulation of mutations in bacteria 111. As shown in 67, these are socalled severely illposed inverse problems, for which we aim at developing new approaches, two in particular: rely on shorttime instead of longterm behaviour, adapt statistical methods developed for decoumpounding Poisson processes and deconvolution 72.
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 29, we address the question of estimating it from the next order approximation, namely a transportdiffusion equation; this new problem is closer to the experimental system but gives rise to a severely illposed inverse problem, for which we want to find an observation inequality thanks to Carleman estimates 59, 58.
Calibrating the mycelial network model
The model developed in 119 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 64 for the exponential growth rates compare with the one obtained in 119 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.
3.4 Software development and dissemination
Permanent members: Marie Doumic, Milica Tomašević
3.5 CellDiv: a platform for biologists to estimate cell division rates
The CellDiv plateform has been developed by Cédric Doucet and Adeline Fermanian and is already available at . It allows a biologist to upload experimental data of dividing cells, either along a genealogical lineage (microfluidic experiments) or inside an exponentially growing culture (petri dish case), and to get the bestfit estimation of the division rate, according to estimation methods combining statistics and PDE analysis 19.
This plateform will be maintained and completed, to accept other types of data (for instance cells dividing into unequal daughters or with heterogeneous growth rates), estimate the division in more general structured population models, and add statistical tests to select the bestfit model. To date, only Marie Doumic being involved in this project, we need to hire an engineer to continue to develop it  a short term goal being to write a proposal for the help of an engineer from SED.
4 Application domains
Unicellular organisms population models are a transversal application of our work, in various aspects and with different biologists collaborators that we detail below. There are many fascinating issues raised by the understanding of their growth and evolutionary mechanisms, which have prominent societal and health impact  cancer treatment, prevention of antibiotic resistance, aging diseases, control and evolution of epidemics, population viability analysis.
4.1 Bacterial growth
Permanent members: Vincent Bansaye, Marie Doumic, Sylvie Méléard, Gaël Raoul
Biologists collaborators: Meriem El Karoui (Ecole polytechnique and University of Edinburgh), Lydia Robert (INRAe), Charles Baroud (Institut Pasteur and Ecole polytechnique)
Possible new collaborations (first contacts made): Nicolas Desprat (ENS Paris), Claude Loverdo (Sorbonne University)
Bacteria are ubiquitous unicellular organisms, present in most parts of earth, and among the first living beings in evolution. Most animals carry millions of bacteria one human possesses as many bacteria as one's own cells. They are vital, for instance the ones of the gut for facilitating digestion, and very useful in industry (biofilms, sewage treatment, cheese production...) as well as potentially pathogenic, causing infectious diseases, increasingly more difficult to treat due to their high capacity of developing resistance to antibiotics. Here are some of the questions we want to tackle concerning bacterial growth.
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 28, 113 stated the excellent adequacy of the socalled "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 antibioticresistant 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 socalled SOS response: in response to DNA damage induced by antibiotic treatments, the cell cycle is arrested and DNA repair and mutagenesis are induced (cf. 31). 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 coauthors 111, 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 socalled "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 celltocell 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 longterm research, included in the ERC project SINGER.
4.2 Cancer and aging
MERGE members involved: Vincent Bansaye, Marie Doumic, Sylvie Méléard
Medical doctors and biologists collaborators: Stéphane Giraudier and Raphael tzykson (St Louis hospital), Teresa Teixeira (IBPC), Zhou Xu (Sorbonne University), Michael Rera (CRI)
Cell division dynamics combine several fundamental processes that are involved in aging and cancers, such as replication and mutation, differentiation and proliferation, quiescence. The main research axis concerned by these applications is axis 2, together with an important modelling work performed through interdisciplinary discussions with MD and biologists.
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 longterm goal, in a longlasting collaboration with Teresa Teixeira and Zhou Xu, and has short and midterms objectives, through Anaïs Rat's finishing Ph.D and Jules Olayé forthcoming Ph.D (cosupervised 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 2phases model of ageing has been proposed by Hervé Tricoire and Michael Rera 60, 121, 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.
4.3 Fragmentation, aggregation, filamentation phenomena
Permanent members: Vincent Bansaye, Marie Doumic, Milica Tomašević
Biologist collaborators: Human Rezaei (INRAe), Florence ChapelandLeclerc and Eric Herbert (LIED, Univ. Paris Diderot), Sascha Martens (Vienna University), WeiFeng Xue (Univ. of Kent)
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.
4.4 Evolutionary epidemiology and ecology
Permanent members: Vincent Bansaye, Gaël Raoul
Biologists collaborators: Sylvain Billiard, Nicolas Lœuille (Institute of Ecology and Environmental Sciences, Paris), François Massol (Center for Infection and Immunity of Lille), Ophélie Ronce (ISEM, Montpellier), François Deslandes (INRAe), Sylvain Gandon (CEFE Montpellier), Elisabeta Vergu (INRAe)
In ecology, the influence of a spatially heterogeneous environment and of different contact structures is at the heart of current problems (biological invasions, epidemiology, etc.), as well as the interaction between different species. The questions we look at concern how a species can invade the range of another one, leading to its extinction; how an epidemics spreading is influenced by contact structures; resilience and tipping points in ecosystems. Applications are as varied as the links between light and plankton species evolution in shallow water lakes, the replacement of red squirrels by grey squirrels, or the current pandemics. Main research axis: 1.
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 multiresistant 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.
Contacts structured by graphs
In the context of spatial ecology and epidemiology, the contacts between individuals leading to predation or transmission of a desease are often modeled by graph. It may represent the connected sites (metapopulations) or the nature of the contacts (multilevel contact structure) between individuals. The description of the population dynamics is important for prediction : stability, explosion, coexistence... The macroscopic approximation when the population and the graph are large is a key question for model reduction and analysis of these models. The mathematical challenges raised are linked to homogeneisation and spatial random graphs, multiscale modelling and local interactions. Collaborations with Sylvain Billiard (Lille, biologist) and Elisebeta Vergu (INRAe, epidemiologist) and Michele Salvi (Roma, mathematician) and Ayman Moussa (Paris Sorbonne, mathematician). Main research axis: 1.
5 Social and environmental responsibility
The MERGE projectteam brings together mathematicians with complementary competences and interests, in order to integrate at a high level different areas of mathematical analysis (multiscale stochastic processes, partial or integrodifferential equations) and microbiology, ecology, cancer medicine. If successful, this research can have fundamental impacts in these fields. General mathematical frameworks unifying different biological questions from single cell to ecological problems not only can improve modelling and simulations but also create a considerable synergy in all these scientific communities. It will also create collaborations between mathematicians (the links between models through scales, taking into account varying environment, interaction between cells...) which could have potential applications in other domains, beyond biology and ecology. In Mathematics, this research tackles fundamental problems from the representation of stochastic microscopic effects in large approximations to macroscopic representations. Successful results would open a new area of research at the interface of probability and analysis, tracking the rare but fundamental effects.
In Biology, this research addresses fundamental questions of growth, mutation and resistance. Successful results will offer interesting opportunities for medical innovations based on evolutionary or adaptive strategies.
6 Highlights of the year
Sylvie Méléard received the « Médaillon Lecture » of the IMS at the conference INFORMS, NANCY 2023. Together with 16 speakers, including 7 Fields Medal, she has given an invited talk at the Conference "Panorama of Mathematics", Hausdorff Center, Bonn 2023.
7 New software, platforms, open data
When simulations have been carried out, the source codes have been systematically published, in a "reproducible research" perspective.
For the simulations done in the submitted article 25, the codes and software are publicly available on Anais Rat's github project "telomeres".
8 New results
8.1 Axis 1: Models through scales
We refer to 3.1 for a presentation of the research program in this direction.
8.1.1 From the distributions of times of interactions to preys and predators dynamical systems
Participants: Vincent Bansaye, Bertrand Cloez.
In 6, we consider a stochastic individual based model where each predator searches during a random time and then manipulates its prey or rests. The time distributions may be nonexponential. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measurevalued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slowfast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.
8.1.2 Scaling limit of bisexual GaltonWatson process
Participants: Vincent Bansaye, MariaEmilia Caballero, Jaime San Martin, Sylvie Méléard.
Bisexual GaltonWatson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of GaltonWatson processes is lost. In 5, we prove tightness for conveniently rescaled bisexual GaltonWatson processes, based on recent techniques developed by Bansaye, Caballero and Méléard. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.
8.1.3 From individualbased evolutionary models to HamiltonJacobi equations
Participants: Nicolas Champagnat, Sylvie Méléard, Sepideh Mirrahimi, Viet Chi Tran.
In 2, we consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small subpopulations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a HamiltonJacobi equation. Such HamiltonJacobi equations have already been derived from parabolic integrodifferential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts. We have thus provided a first answer to a question that has been open for a long time, and we continue to progress in order to generalise these results.
8.1.4 Multispecies cross diffusion
Participants: Marie Doumic, Sophie Hecht, Benoit Perthame, Diane Peurichard.
Systems describing the longrange interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems are quadratic, written under the form of transport equations with a nonlocal selfgenerated drift. In the article 23, to be published in the Journal of Differential Equations, 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 our 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 crossdiffusion system of quadratic type.
8.1.5 Particle approximation of the doubly parabolic KellerSegel equation
Participants: Nicolas Fournier, Milica Tomašević.
In 3, we study a stochastic system of $N$ particles associated with the parabolicparabolic KellerSegel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any , we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as , solves some nonlinear martingale problem, which in particular implies that its family of timemarginals solves the parabolicparabolic KellerSegel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the twobytwo pathdependant interaction can be controlled by a twobytwo Coulomb interaction, as in the parabolicelliptic case.
8.2 Axis 2: qualitative analysis of structured populations
We refer to 3.2 for a presentation of the research program in this direction.
8.2.1 Claire Ecotiere's PhD
Participants: Claire Ecotière, Sylvie Méléard, Régis Ferrière.
Claire Ecotiere's thesis focuses on the stochastic modeling of human behavior in the face of environmental change and the study of related mathematical models. She has mainly developed a stochastic model describing the coupled dynamics of the environment and the population, through the proportion of active individuals facing the environment, in a population composed of active and passive individuals. Individuals can switch from one behavior to another through social interactions or their own assessment of environmental degradation. Active behavior contributes less to environmental degradation, but is more costly to adopt than passive behavior. The stochastic model and its behavior in long time are studied as well as its approximation in large population leading to a deterministic system.
8.2.2 Evolutionary dynamics
Participants: Sirine Boucenna, Vasilis Dakos, Quentin Griette, Sylvie Alfaro, Sylvain Gandon, Gaël Raoul.
The article 80 has been accepted 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 longterm 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 112 is in revision for Theoretical Population 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 ecoevolutionary feedbacks affect abrupt ecological transitions in shallow lakes. We explore the ecoevolutionary 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.
8.2.3 A growthfragmentationisolation process on random recursive trees and contact tracing
Participants: Vincent Bansaye, Chenlin Gu, Linglong Yuan.
In 7, we consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalized empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated with the size of clusters, which is inherited from the splitting property of random recursive trees. This work is motivated by the control of epidemics and contact tracing where clusters correspond to trees of infected individuals that can be identified and isolated. We complement this work by providing results on the Malthusian exponent to describe the effect of control policies on epidemics.
8.2.4 Propagation of chaos for stochastic particle systems with singular meanfield interaction of ${L}^{q}{L}^{p}$ type
Participants: Milica Tomašević.
In this work 14, we prove the wellposedness and propagation of chaos for a stochastic particle system in meanfield interaction under the assumption that the interacting kernel belongs to a suitable ${L}_{t}^{q}{L}_{x}^{p}$ space. Contrary to the large deviation principle approach recently proposed in 83, the main ingredient of the proof here are the Partial Girsanov transformations introduced in 87 and developed in a general setting in this work.
8.2.5 Blowup for a stochastic model of chemotaxis driven by conservative noise on ${\mathbb{R}}^{2}$
Participants: Avi Mayorcas, Milica Tomašević.
In 13, we establish criteria on the chemotactic sensitivity $\chi $ for the nonexistence of global weak solutions (i.e., blowup in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on ${\mathbb{R}}^{2}$. We show that if $\chi $ is sufficiently large then blowup occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for $\chi $ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blowup occurs with positive probability.
8.2.6 Reducing exittimes of diffusions with repulsive interactions
Participants: PaulEric Chaudru de Reynal, Pierre Monmarché, Milica Tomašević, Julian Tugaut.
In this work 9, we prove a Kramers’ type law for the lowtemperature behavior of the exittimes from a metastable state for a class of selfinteracting nonlinear diffusion processes. Contrary to previous works, the interaction is not assumed to be convex, which means that this result covers cases where the exittime for the interacting process is smaller than the exittime for the associated noninteracting process. The technique of the proof is based on the fact that, under an appropriate contraction condition, the interacting process is conveniently coupled with a noninteracting (linear) Markov process where the interacting law is replaced by a constant Dirac mass at the fixed point of the deterministic zerotemperature process.
8.3 Axis 3: Modeldata comparison
We refer to 3.3 for a presentation of the research program in this direction.
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 populationscale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In the submitted article 4, 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 nonterminal 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.
8.3.2 Calibrating division rates in population dynamics
Participants: Marie Doumic, Marc Hoffmann.
Modelling, analysing and inferring triggering mechanisms in population reproduction is fundamental in many biological applications. It is also an active and growing research domain in mathematical biology. In the book chapter 19, we review the main results developed over the last decade for the estimation of the division rate in growing and dividing populations in a steady environment. These methods combine tools borrowed from PDE's and stochastic processes, with a certain view that emerges from mathematical statistics. A focus on the application to the bacterial cell division cycle provides a concrete presentation, and may help the reader to identify major new challenges in the field.
8.3.3 Fragmentation estimation through shorttime dynamics
Participants: Marie Doumic, Miguel Escobedo, Magali Tournus.
Given a phenomenon described by a selfsimilar 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 shorttime observations and initial data close to a Dirac measure should be a welladapted 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.
9 Partnerships and cooperations
Sylvie Méléard has a regular partnership with the CMM (Chile), mainly with Servet Martinez, Joaquin Fontbona and Jaime San Martin.
Gaël Raoul has a regular collaboration 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 multidrug 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.
9.1 International initiatives
Milica Tomašević participates in two international projects, in the FranceJapan program Sakura (headed by K. Fujie in Japan and Julian Tugaut in France) and in a francobrasilian ANR project, ANR PRCI FAPESP SDAIM : Stochastic and Deterministic Analysis of Irregular Models, 20232027, PI : F. Russo (ENSTA).
9.2 International research visitors
9.2.1 Visits of international scientists
Other international visits to the team
Juan Velazquez visited the team for one week in September, and gave a talk at the CMAP.
9.2.2 Visits to international teams
Research stays abroad
Sylvie Méléard spent 2 weeks at Santiago (Chile).
9.3 European initiatives
9.3.1 Horizon Europe
The ERC Advanced Grant SINGER (Stochastic dynamics of sINgle cells; Growth, Emergence and Resistance) 20222027, headed by Sylvie Méléard, involves all members of MERGE and several of our collaborators (Sepideh Mirrahimi in Toulouse, Nicolas Champagnat in BIGS, VietChi Tran in Univ. G. Eiffel). Amount: 2M€.
9.4 National initiatives
 The MMB Chaire, Modélisation Mathématique et Biodiversité, headed by Sylvie Méléard since 2009, has been renewed till 2027. It funds PhD and postdoctoral grants, a yearly summer school and scientific meetings every two month. This has a great role in uniting our community.
 Our research on telomere shortening modelling is structured around several fundings:
 The INCa Projet TheFinalCut, headed by Teresa Teixeira (total: 0,78 M€), 2020–2024
 Following the funding of the PEPR MathVives, a project on telomere shortening modelling, DyLT (approx. 1MEuro), Influence of telomere length dynamics and environmental conditions on biological and clinical aspects of aging, has been accepted. Headed by Nicolas Champagnat (Inria projectteam TOSCA), and Marie Doumic being the head of Axis 2 of the project, it will be a meeting place for mathematicians and biologists in this field and will be an important opportunity for the pooling of forces on this important topic.
 Jules Olayé's Ph.D, cosupervised by Milica Tomašević and Marie Doumic, has been funded by the EDMH.
 We are part of many ANR projects: Marie Doumic participates to the ANR ODISSE (411 k€), 2019–2023 on Synthèse d'observateur pour des systèmes de dimension infinie, headed by Vincent Andrieu, and to a newly funded ANR project ENERGENCE ( 433 k€), 2022–2026, ENERgy driven modelling of tissue architecture emerGENCE and homeorhesis, headed by Diane Peurichard. Gaël Raoul participates in the ANR DEEV (159 k€), 2020–2023 on IntegroDifferential Equations from EVolutionary biology, headed by Sepideh Mirrahimi. Milica Tomašević participates to the ANR project METANOLIN (87 k€), 2019–2023 on Metastability for nonlinear processes, headed by Julian Tugaut, and to the ANR project NEMATIC (367 k€), 2021–2025 on Analyse Modelisation et Simulation Multiéchelle, headed by Eric Herbert.
 Sylvie Méléard is the P.I. of a newly funded AviesanInserm ITMO Cancer project (261 k€), 2022–2026 on Mathématiques pour une meilleure compréhension des néoplasmes myéloprolifératifs et leurs thérapeutiques.
10 Dissemination
Participants: Vincent Bansaye, Marie Doumic, Sylvie Méléard, Gaël Raoul, Milica Tomašević.
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
Principle organisation of events
Vincent Bansaye, Sylvie Méléard and Milica Tomašević coorganise the summer school of the MMB chaire, headed by Sylvie Méléard, which takes place every June in Aussois.
Milica Tomašević coorganised the conference in honour of Denis Talay at CIRM, Marseille.
Vincent Bansaye organised an invited session in the SPA Conference "Stochastic models for epidemiology and evolution", Lisboa, July 2023.
Member of scientific committees
Sylvie Méléard has been a member of the scientific committee of ICIAM 2023 and of the conference in honour of Denis Talay at CIRM, Marseille.
10.1.2 Journal
Member of the editorial boards
Vincent Bansaye has been Guest Editor for the special issue "Mathematics and biology" of the journal Maths in Action (SMAI).
Marie Doumic is editor in chief of ESAIM Proceedings and Surveys, and associate editor of the Journal of Mathematical Biology, Kinetic and Related Models and le Bulletin des sciences mathématiques.
Sylvie Méléard is associate editor of the ComptesRendus de l'Académie des Sciences, of Stochastic Processes and their Applications, and of the Lecture Notes on mathematical Modelling in the Life Sciences.
10.1.3 Invited talks
Vincent Bansaye gave invited talks at the Conference on "Branching Processes and applications", Angers, may 23rd; on the Journée Analyse Appliquée Hauts de France, October 17th; at the Evolution Everevol Conference, Grenoble, December 14th.
Marie Doumic gave an invited talk at the 5th International Symposium on Pathomechanisms of Amyloid Diseases, Bordeaux, September 57 ; at the workshop "Topics on Neuroscience, Collective Migration and Parameter Estimation", Oxford, July 37 ; at the "Journées EDP", Aussois, June 1923 ; at the READINET conference, Orsay, January 36.
Sylvie Méléard gave 12 invited talks, including 7 abroad.
Milica Tomašević gave invited talks at the Mean field interactions with singular kernels and their approximations workshop, IHP Paris 1822 déc 2023 ; at the Conference Stochastic Flows, thematic program Order and Randomness in Partial Differential Equations, Institut MittagLeffler, nov 2023 ; at the Second BerlinLeipzig Workshop on Fluctuating Hydrodynamics, Freie University, Berlin September 1920, 2023; at the Conference "A random walk in the land of stochastic processes and numerical probability", CIRM sept 2023 ; at the Conference Mean Field Models, Rennes, 12 juin  16 juin 2023 ; and 4 seminar talks.
10.1.4 Leadership within the scientific community
Vincent Bansaye has become vice president of the applied mathematics department of Ecole Polytechnique and vice director of the Fondation Mathématiques Jacques Hadamard (FMJH) since September 2023.
Sylvie Méléard leads the Chaire MMB, joint between École Polytechnique, the Muséum national d'Histoire naturelle, the Fondation de l'École Polytechnique and VEOLIA Environnement. It is a major player in bringing together our community of mathematicians, modellers and ecologists, and funds several PhD and postdoctoral grants.
10.1.5 Scientific expertise
Vincent Bansaye is a member of the scientific council of ModCov19.
Marie Doumic has been a member of the CID51 of CNRS (hiring committees for junior and senior research scientists of CNRS), and has been a member for the hiring committee of an associate professor at ENSTA.
10.1.6 Research administration
Vincent Bansaye is a member of the steering committee of the MMB Chaire.
Marie Doumic is a member of the scientific committee of INSMI, of Inria Saclay and of the steering committee of FMJH.
Sylvie Méléard has been a member of the Evaluation Committee of Inria till September 2023, and is a member of the scientific committees of CRM (Montreal), Hausdorff Center (Bonn, Germany) and CMM (Chile), and of the steering committees of FMJH and E4H.
10.2 Teaching  Supervision  Juries
10.2.1 Supervision
Claire Ecotière defended her Ph.D on October 18th, 2023, on the Study and stochastic modelling of human behaviour in the face of environmental change, under the supervision of Sylvie Méléard and Régis Ferrière.
Anaïs Rat defended her Ph.D on May 31st, 2023, on Structured population dynamics: theory, asymptotic and numerical analysis. Application to populations with heterogeneous growth rates and to replicative senescence, under the supervision of Magali Tournus and Marie Doumic.
10.2.2 Juries
Marie Doumic has been a member of the Ph.D thesis committees for Darryl Ondoua (under Yvon Maday's supervision), Léo Meyer (under Magali Ribot and Romain Yvinec's supervision), and for the habilitation thesis of Bertrand Cloez (January 5, 2023).
Sylvie Méléard has been a member for the Marc Yor 2023 prize and the Dargelos 2023 prize committees.
10.3 Popularization
10.3.1 Interventions for undergraduate and highschool students
Marie Doumic gave a talk "Chiche!" at Inria, December 4, 2023, and a talk for high school students at the SMAI/MAM Musée des Arts et Métiers, Paris, March 9.
Sylvie Méléard gave a talk to the MathClub, Université Paris Cité (bachelor students) and a "Maths en Jeans" conference at Potsdam University in March 2023, organised by the Berlin lycée français and Potsdam University.
10.3.2 General audience talks
Sylvie Méléard gave a talk for her Conférence entry as a foreign correspondent to the Chilean Academy of Sciences, January 2023, and a talk at the actuaries meeting, SCOR Foundation, December 2023.
11 Scientific production
11.1 Major publications
 1 articleA growthfragmentationisolation process on random recursive trees and contact tracing.The Annals of Applied Probability336B2023, 52335278HALDOI
 2 articleFilling the gap between individualbased evolutionary models and HamiltonJacobi equations.Journal de l'École polytechnique — Mathématiques102023, 12471275HALDOIback to text
 3 articleParticle approximation of the doubly parabolic KellerSegel equation in the plane.Journal of Functional Analysis28572023, 110064HALDOIback to text
 4 miscIndividual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence.November 2023HALDOIback to text
11.2 Publications of the year
International journals
 5 articleScaling limits of bisexual GaltonWatson processes.Stochastics: An International Journal of Probability and Stochastic Processes955February 2023, 749784HALDOIback to text
 6 articleFrom the distributions of times of interactions to preys and predators dynamical systems.Journal of Mathematical Biology871June 2023, 2HALDOIback to text
 7 articleA growthfragmentationisolation process on random recursive trees and contact tracing.The Annals of Applied Probability336B2023, 52335278HALDOIback to text
 8 articleFilling the gap between individualbased evolutionary models and HamiltonJacobi equations.Journal de l'École polytechnique — Mathématiques102023, 12471275HALDOI
 9 articleReducing exittimes of diffusions with repulsive interactions.ESAIM: Probability and Statistics272023, 723748HALDOIback to text
 10 articleRecent progress on limit theorems for large stochastic particle systems.ESAIM: Proceedings and Surveys75December 2023, 223HALDOI
 11 articleParticle approximation of the doubly parabolic KellerSegel equation in the plane.Journal of Functional Analysis28572023, 110064HALDOI
 12 articleTime reversal of spinal processes for linear and nonlinear branching processes near stationarity.Electronic Journal of Probability28noneJanuary 2023HALDOI

13
articleBlowup for a stochastic model of chemotaxis driven by conservative noise on
${}^{2}$ .Journal of Evolution Equations2357September 2023HALDOIback to text 
14
articlePropagation of chaos for stochastic particle systems with singular meanfield interaction of
${L}^{q}{L}^{p}$ type.Electronic Communications in Probability28noneAugust 2023, 13HALDOIback to text
Conferences without proceedings
 15 inproceedingsA tribute to Elisabeta Vergu: Multilevel contact models of epidemics.Journées 2023 de l'Action Coordonnée "Modélisation des Maladies Infectieuses"Paris, FranceOctober 2023HAL
 16 inproceedingsApproximations déterministes de modèles stochastiques multiniveaux en épidémiologie.Colloque Jeunes Probabilistes et StatisticiensSaint Pierre d'Oléron, FranceOctober 2023HAL
 17 inproceedingsModèles structurés multiniveaux de dynamiques épidémiques.École de Printemps de la chaire MMBAussois, FranceJune 2023HAL
 18 inproceedingsLarge population limit for a multilayer SIR model including households and workplaces.Rencontres Mathématiques de RouenRouen, FranceJune 2023HAL
Scientific book chapters
 19 inbookIndividual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle.40Modeling and Simulation for Collective DynamicsLecture Notes Series, Institute for Mathematical Sciences, National University of SingaporeWORLD SCIENTIFICFebruary 2023, 181HALDOIback to textback to textback to textback to textback to textback to text
Reports & preprints
 20 miscHommage à Elisabeta Vergu.November 2023HAL
 21 miscThe epidemiological footprint of contact structures in models with two levels of mixing.March 2023HAL
 22 miscSharp approximation and hitting times for stochastic invasion processes.December 2023HAL
 23 miscMultispecies crossdiffusions: from a nonlocal meanfield to a porous medium system without selfdiffusion.May 2023HALback to text
 24 miscLarge population limit for a multilayer SIR model including households and workplaces.May 2023HAL
 25 miscIndividual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence.November 2023HALDOIback to text
 26 miscOn a multidimensional McKeanVlasov SDE with memorial and singular interaction associated to the parabolicparabolic KellerSegel model.September 2022HAL
11.3 Cited publications
 27 articleTravelling waves in a nonlocal reactiondiffusion equation as a model for a population structured by a space variable and a phenotypic trait.Communications in Partial Differential Equations38122013, 21262154back to text
 28 articleCell size regulation in bacteria.Physical Review Letters112202014, 208102back to text
 29 articleEstimation from Moments Measurements for Amyloid Depolymerisation.Journal of Theoretical Biology397March 2016, 68  88HALDOIback to textback to text
 30 articleThe mechanism of monomer transfer between two structurally distinct PrP oligomers.PLOS ONE127July 2017HALDOIback to text
 31 articleSOS, the formidable strategy of bacteria against aggressions.FEMS Microbiology Reviews386November 2014, 11261145DOIback to text
 32 articleNematic liquid crystals: from MaierSaupe to a continuum theory.Molecular crystals and liquid crystals52512010, 111back to text
 33 articleInformation content in data sets for a nucleatedpolymerization model.Journal of biological dynamics912015, 172197back to text
 34 articleErgodic behavior of nonconservative semigroups via generalized Doeblin conditions.Acta Applicandae Mathematicae2019, 144back to text
 35 bookStochastic Models for Structured Populations.16Springer2015back to textback to text
 36 articleStability of a crossdiffusion system and approximation by repulsive random walks: a duality approach.arXiv preprint arXiv:2109.071462021back to text
 37 articleProliferating parasites in dividing cells: Kimmel’s branching model revisited.The Annals of Applied Probability1832008, 967996back to text
 38 articleBranching Feller diffusion for cell division with parasite infection.ALEA: Latin American Journal of Probability and Mathematical Statistics82011, 95127back to text
 39 articleGrowing from a few cells: combined effects of initial stochasticity and celltocell variability.Journal of the Royal Society Interface161532019, 20180935back to text
 40 articleConcentration in LotkaVolterra parabolic or integral equations: a general convergence result.Methods and Applications of Analysis1632009, 321340back to text
 41 articleConcentrations and constrained HamiltonJacobi equations arising in adaptive dynamics.Contemporary Mathematics4392007, 5768back to text
 42 articleAvailability of the Molecular Switch XylR Controls Phenotypic Heterogeneity and Lag Duration during Escherichia coli Adaptation from Glucose to Xylose.Mbio1162020, e0293820back to text
 43 incollectionAdaptation at the edge of a species' range.Integrating ecology and evolution in a spatial context2001back to text
 44 articleRecent progress in coalescent theory.ENSAIOS MATEMÁTICOS162009, 1193back to text
 45 miscModeling the behavior of hematopoietic compartments from stem to red cells in murine steady state and stress hematopoiesis.2019back to text
 46 articleMultistage hematopoietic stem cell regulation in the mouse: a combined biological and mathematical approach.iScience2021, 103399back to text
 47 articleThe asymmetry of telomere replication contributes to replicative senescence heterogeneity.Scientific Reports5October 2015, 15326HALDOIback to text
 48 articleDynamics of lineages in adaptation to a gradual environmental change.Annales Henri Lebesgue, to appear2021back to text
 49 incollectionAggregationdiffusion equations: dynamics, asymptotics, and singular limits.Active Particles, Volume 2Springer2019, 65108back to text
 50 articleA microscopic interpretation for adaptive dynamics trait substitution sequence models.Stochastic processes and their applications11682006, 11271160back to text
 51 articleUnifying evolutionary dynamics: from individual stochastic processes to macroscopic models.Theoretical Population Biology6932006, 297321back to text
 52 articlePolymorphic evolution sequence and evolutionary branching.Probability Theory and Related Fields151122011, 4594back to text
 53 articleMicroscopic approach of a time elapsed neural model.Mathematical Models and Methods in Applied SciencesDecember 2015, http://www.worldscientific.com/doi/10.1142/S021820251550058XHALDOIback to text
 54 articleMeanfield limit of generalized Hawkes processes.Stochastic Processes and their Applications127122017, 38703912back to text
 55 articleSelforganization in highdensity bacterial colonies: efficient crowd control.PLOS Biology5112007, e302back to text
 56 articleTo reverse a Markov process.Acta Mathematica1231969, 225251back to text
 57 articleLimit theorems for some branching measurevalued processes.Advances in Applied Probability4922017, 549580back to text
 58 articleControllability and observability of an artificial advectiondiffusion problem.Mathematics of Control, Signals, and Systems2432012, 265294back to text
 59 articleSingular optimal control: a linear 1D parabolichyperbolic example.Asymptotic Analysis443, 42005, 237257back to text
 60 articleTwo phases of aging separated by the Smurf transition as a public path to death.Scientific Reports612016, 17back to text
 61 articleNonparametric estimation for interacting particle systems: McKeanVlasov models.Probability Theory and Related Fields2021, 163back to text
 62 articleThe dynamical theory of coevolution: a derivation from stochastic ecological processes.Journal of mathematical biology3451996, 579612back to text
 63 articleThe dynamics of adaptation: an illuminating example and a HamiltonJacobi approach.Theoretical Population Biology6742005, 257271back to text
 64 articleHyphal network whole field imaging allows for accurate estimation of anastomosis rates and branching dynamics of the filamentous fungus \it Podospora anserina.Scientific Reports1012020, 116back to textback to text
 65 articleGenealogical processes for FlemingViot models with selection and recombination.Annals of Applied Probability1999, 10911148back to text
 66 articleCelebrating Soft Matter's 10th Anniversary: Cell division: a source of active stress in cellular monolayers.Soft Matter11372015, 73287336back to text
 67 articleEstimating the division rate and kernel in the fragmentation equation.Ann. Inst. H. Poincaré Anal. Non Linéaire3572018, 18471884URL: https://doi.org/10.1016/j.anihpc.2018.03.004DOIback to textback to text
 68 articleA bimonomeric, nonlinear BeckerDöringtype system to capture oscillatory aggregation kinetics in prion dynamics.Journal of Theoretical Biology4802019, 241261back to text
 69 articleA purely mechanical model with asymmetric features for early morphogenesis of rodshaped bacteria microcolony.Mathematical Biosciences and Engineering1762020, 68736908back to textback to text
 70 articleStatistical estimation of a growthfragmentation model observed on a genealogical tree.Bernoulli2132015, 17601799back to text
 71 articleEstimating the division rate from indirect measurements of single cells..Discrete and Continuous Dynamical SystemsSeries B25102020back to text
 72 articleAdaptive procedure for Fourier estimators: application to deconvolution and decompounding.Electronic Journal of Statistics1322019, 34243452back to text
 73 articleGenealogical constructions of population models.The Annals of Probability4742019, 18271910back to text
 74 articleSystematic detection of horizontal gene transfer across genera among multidrugresistant bacteria in a single hospital.Elife92020, e53886back to text
 75 articleSingleCell Analysis of Growth and Cell Division of the Anaerobe \it Desulfovibrio vulgaris H\it ildenborough.Frontiers in Microbiology62015, 1378back to text
 76 articleXV.—The correlation between relatives on the supposition of Mendelian inheritance..Earth and Environmental Science Transactions of the Royal Society of Edinburgh5221919, 399433back to text
 77 articleNon local LotkaVolterra system with crossdiffusion in an heterogeneous medium.Journal of Mathematical Biology7042015, 829854back to text
 78 articleA microscopic probabilistic description of a locally regulated population and macroscopic approximations.The Annals of Applied Probability1442004, 18801919back to text
 79 articleBacterial metabolic heterogeneity: from stochastic to deterministic models.Mathematical Biosciences and Engineering1752020, 51205133back to text
 80 unpublishedEvolution and spread of multidrug resistant pathogens in a spatially heterogeneous environment.December 2022, working paper or preprintHALDOIback to text
 81 articleThe coalescent structure of continuoustime GaltonWatson trees.The Annals of Applied Probability3032020, 13681414back to text
 82 articleThe manytofew lemma and multiple spines.Annales de l'Institut Henri Poincaré, Probabilités et Statistiques5312017, 226242back to text
 83 articleLarge deviations for singularly interacting diffusions.arXiv preprint arXiv:2002.012952020back to text
 84 articleSmall populations corrections for selectionmutation models.Networks & Heterogeneous Media742012, 805back to text

85
articleQuantitative estimates of propagation of chaos for stochastic systems with W
${}^{1,}$ kernels.Inventiones mathematicae21412018, 523591back to text  86 articleMeanfield limit of a particle approximation of the onedimensional parabolicparabolic KellerSegel model without smoothing.Electronic Communications in Probability232018, 114back to text
 87 articleMeanfield limit of a particle approximation of the onedimensional parabolicparabolic kellersegel model without smoothing.Electronic Communications in Probability232018, 114back to text
 88 articleNonlocal crossdiffusion systems for multispecies populations and networks.arXiv preprint arXiv:2104.062922021back to text
 89 articleRediscovering bacteria through singlemolecule imaging in living cells.Biophysical Journal11522018, 190202back to text
 90 articleThe coalescent.Stochastic processes and their applications1331982, 235248back to text
 91 articleOn a strong form of propagation of chaos for McKeanVlasov equations.Electronic Communications in Probability232018, 111back to text
 92 articleRecent developments of analysis for hydrodynamic flow of nematic liquid crystals.Phil. Trans. R. Soc. A37220292014, 20130361back to text
 93 articleUne méthode particulaire déterministe pour des équations diffusives non linéaires.Comptes Rendus de l'Académie des SciencesSeries IMathematics33242001, 369376back to text
 94 articleUniform sampling in a structured branching population.Bernoulli254A2019, 26492695back to textback to text
 95 articleTelomere shortening causes distinct cell division regimes during replicative senescence in Saccharomyces cerevisiae.bioRxiv2021back to text
 96 articleNonlinear historical superprocess approximations for population models with past dependence.Electronic Journal of Probability172012, 132back to text
 97 incollectionAdaptive dynamics: a Geometrical Study of the Consequences of Nearly Faithful Reproduction.Stochastic and Spatial Structures of Dynamical SystemsNorthHolland1996, 183231back to text
 98 articleInvasion waves and pinning in the KirkpatrickBarton model of evolutionary range dynamics.Journal of mathematical biology7812019, 257292back to text
 99 articleHamiltonian fast marching: A numerical solver for anisotropic and nonholonomic eikonal PDEs.Image Processing On Line92019, 4793back to text
 100 articleA Singular HamiltonJacobi Equation Modeling the Tail Problem.SIAM Journal on Mathematical Analysis4462012, 42974319back to text
 101 articleDynamics of sexual populations structured by a space variable and a phenotypical trait.Theoretical Population Biology842013, 87103back to text
 102 articleFrom shortrange repulsion to HeleShaw problem in a model of tumor growth.Journal of mathematical biology76122018, 205234back to text
 103 articleTime reversions of Markov processes.Nagoya Mathematical Journal241964, 177204back to text
 104 articleLarge systems of interacting particles and the porous medium equation.Journal of Differential Equations8821990, 294346back to text
 105 articleOn the derivation of reactiondiffusion equations as limit dynamics of systems of moderately interacting stochastic processes.Probability Theory and Related Fields8241989, 565586back to text
 106 bookOn the martingale problem for interactive measurevalued branching diffusions.549American Mathematical Soc.1995back to text
 107 articleDirac concentrations in LotkaVolterra parabolic PDEs.Indiana University Mathematics Journal2008, 32753301back to text
 108 articleSurvival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations.Mathematical Medicine and Biology: a journal of the IMA2732010, 195210back to textback to text
 109 bookParabolic equations in biology.Lecture Notes on Mathematical Modelling in the Life SciencesSpringer, Cham2015, xii+199URL: http://dx.doi.org/10.1007/9783319195001DOIback to text
 110 bookTransport equations in biology.Frontiers in MathematicsBaselBirkhäuser Verlag2007, x+198back to text
 111 articleMutation dynamics and fitness effects followed in single cells.Science35963812018, 12831286back to textback to text
 112 unpublishedEvolutionary outcomes arising from bistability in ecosystem dynamics.2023back to text
 113 articleCellSize Control and Homeostasis in Bacteria.Current Biology11679172015back to text
 114 articleA new McKeanVlasov stochastic interpretation of the parabolicparabolic KellerSegel model: The onedimensional case.Bernoulli2622020, 13231353back to text
 115 unpublishedWell posedness and stochastic derivation of a diffusiongrowthfragmentation equation in a chemostat.2021, preprintback to text
 116 articleA new McKeanVlasov stochastic interpretation of the parabolicparabolic KellerSegel model: The twodimensional case.The Annals of Applied Probability3112021, 432459back to text
 117 phdthesisOn a probabilistic interpretation of the KellerSegel parabolicparabolic equations.Université Côte d'AzurNovember 2018HALback to text

118
unpublishedPropagation of chaos for stochastic particle systems with singular meanfield interaction of
${L}^{q}{L}^{p}$ type.2020, preprintback to text  119 unpublishedErgodic behaviour of a multitype growthfragmentation process modelling the mycelial network of a filamentous fungus.2020, preprintback to textback to text
 120 articleInsights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation.PLOS Computational Biology17909 2021, 121URL: https://doi.org/10.1371/journal.pcbi.1008964DOIback to text
 121 articleA new, discontinuous 2 phases of aging model: Lessons from Drosophila melanogaster.PLOS One10112015, e0141920back to text
 122 articleOn meanfield (GI/GI/1) queueing model: existence and uniqueness.Queueing Systems9432020, 243255back to text
 123 articleBiomechanical ordering of dense cell populations.Proceedings of the National Academy of Sciences105402008, 1534615351back to textback to text
 124 articleRobust Growth of \it Escherichia coli.Current Biology20122010, 1099103back to text
 125 article20 questions on adaptive dynamics.Journal of evolutionary biology1852005, 11391154back to text
 126 articleGeometry and mechanics of microdomains in growing bacterial colonies.Physical Review X832018, 031065back to textback to text
 127 articleCollective motion and density fluctuations in bacterial colonies.Proceedings of the National Academy of Sciences107312010, 1362613630back to text