RESEARCH CENTRE

Inria Saclay Centre
at Institut Polytechnique de
Paris

IN PARTNERSHIP WITH:

Institut Polytechnique de Paris, CNRS

2023
ACTIVITY REPORT

Project-Team

MERGE

Mathematics for Evolution, Reproduction,
Growth and Emergence

IN COLLABORATION WITH: Centre de Mathématiques Appliquées (CMAP)

DOMAIN

Digital Health, Biology and Earth

THEME

Computational Biology

Contents

Project-Team MERGE

1 Team members, visitors, external collaborators

2 Overall objectives

1

2

2

3 Research program

3
3
3.1 Axis 1: Models through scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2 Axis 2: Qualitative analysis of structured populations . . . . . . . . . . . . . . . . . . . . . . .
8
3.3 Axis 3: Model-data comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Software development and dissemination . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.5 CellDiv: a platform for biologists to estimate cell division rates . . . . . . . . . . . . . . . . . 10

4 Application domains

10
4.1 Bacterial growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Cancer and aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Fragmentation, aggregation, filamentation phenomena . . . . . . . . . . . . . . . . . . . . . 12
4.4 Evolutionary epidemiology and ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Social and environmental responsibility

6 Highlights of the year

7 New software, platforms, open data

13

13

14

8 New results

14
8.1 Axis 1: Models through scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.1 From the distributions of times of interactions to preys and predators dynamical

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.2 Scaling limit of bisexual Galton-Watson process . . . . . . . . . . . . . . . . . . . . . . 14
8.1.3 From individual-based evolutionary models to Hamilton-Jacobi equations
. . . . . 14
8.1.4 Multispecies cross diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.1.5 Particle approximation of the doubly parabolic Keller-Segel equation . . . . . . . . . 15
8.2 Axis 2: qualitative analysis of structured populations . . . . . . . . . . . . . . . . . . . . . . . 15
8.2.1 Claire Ecotiere’s PhD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2.2 Evolutionary dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.2.3 A growth-fragmentation-isolation process on random recursive trees and contact

tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8.2.4 Propagation of chaos for stochastic particle systems with singular mean-field inter-

action of Lq − Lp type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.2.5 Blow-up for a stochastic model of chemotaxis driven by conservative noise on R2 . . 17
8.2.6 Reducing exit-times of diffusions with repulsive interactions . . . . . . . . . . . . . . 17
8.3 Axis 3: Model-data comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.1 Telomere shortening, a unifying model
. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.2 Calibrating division rates in population dynamics . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . 18
8.3.3 Fragmentation estimation through short-time dynamics

9 Partnerships and cooperations

18
9.1 International initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9.2 International research visitors
9.2.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9.2.2 Visits to international teams
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.3.1 Horizon Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.4 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.3 European initiatives

10 Dissemination

19
10.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.1.2 Journal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1.3 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1.4 Leadership within the scientific community . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1.5 Scientific expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1.6 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.2.1 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.2.2 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.3 Popularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.3.1 Interventions for undergraduate and highschool students . . . . . . . . . . . . . . . . 21
10.3.2 General audience talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

11 Scientific production

21
11.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.2 Publications of the year
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
11.3 Cited publications

Project MERGE

1

Project-Team MERGE

Creation of the Project-Team: 2023 March 01

Keywords

Computer sciences and digital sciences

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

2

Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Marie Doumic-Jauffret [Team leader, Inria, Senior Researcher, HDR]

• Gael Raoul [CNRS, Researcher]

• Milica Tomasevic [CNRS, Researcher]

Faculty Member

• Vincent Bansaye [Ecole Polytechnique, Professor, HDR]

Post-Doctoral 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 integro-differential equations [36] as well as in PDE analysis [108, 107]. The two
approaches are combined more and more frequently, for model analysis [51, 35] 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 so-called "single cell" or micro-fluidic 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.

Project MERGE

3

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 TCP-IP 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 individual-based
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 model-data 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, "Model-data 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 Tomaševi´c

When we describe non-interacting 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 so-called many-to-
one formulae [57]. This is no more true once interaction between the cells or with the environment is
considered: in such cases, mean-field 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 multi-species 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.

4

Inria Annual Report 2023

From stochastic processes to constrained Hamilton-Jacobi (HJ) equations.
Permanent members: Sylvie Méléard, Gaël Raoul

Most models, for instance for the "normal" bacterial division cycle [68], 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 individual-based population process is given for any K by the jump
point-measure Markov process (µK
t )t on a trait subset of Rd weighted by 1/K . An individual with trait x
gives birth to a new individual with rate b(x). With probability 1 − pK , the new individual carries the trait
x and with probability pK , it carries a mutant trait z chosen according to the distribution m(x, z)d z. An
individual with trait x in the population µK dies with death rate d (x) +C 〈µK (t ), 1〉. Further an individual
with trait x chooses a partner with trait y at rate hK (x, y, µK ) = τ(x, y)/K 〈µK , 1〉 and after transfer, the
couple (x, y) becomes (x, x). Then for any “good” test function ϕ, we have

〈µK

t , ϕ〉 =〈µK
0 , ϕ〉 +
(cid:90)
(cid:90) t

+

0

R

R
0
τ(x, y)
〈µK
s , 1〉

(cid:90) t

(cid:90)

(cid:110)(cid:179)

b(x) − d (x) −C 〈µK

s , 1〉

(cid:180)

ϕ(x) + pK b(x)

(cid:90)

R

(cid:161)ϕ(z) − ϕ(x)(cid:162)m(x, z)d z

(cid:111)

µK

s (d x)d s

(cid:161)ϕ(x) − ϕ(y)(cid:162)µK

s (d x)µK

s (d y)d s + M K ,ϕ

(t ),

(1)

where M K ,ϕ
is a square integrable martingale whose quadratic variation can be easily made explicit. By
letting K go to infinity, and pK go to p, we can derive an integro-differential equation with non local non
linearities due to both competition and transfer. Uniqueness of its solution is obvious but its long-time
behaviour is unknown, as well as the existence of stationary solutions. Formally, applying a limiting
procedure for small mutations and time rescaling usually leads to a HJ type equation with constraints,
in the formalism introduced in [63], successfully developed in [41, 109, 40], and in many extensions so
far. Concentrations in such equations are too fast for realistic evolution [110]. Indeed the evolutionary
dynamics strongly depend on the positivity of the density although it is exponentially small for some traits.
Different papers [110, 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 Hamilton-Jacobi equation

and to understand the role of the trade-off between demography and transfer.

Space-and-trait structured models

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

Project MERGE

5

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 individual-
based model, where each cell repulses and maybe attracts its neighbours, but how do these local interac-
tion 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 [70], Marie Doumic, Sophie Hecht and Diane Peurichard proposed a purely-repulsive
individual-based model of rod-shaped 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 2D-cells dividing into equally-
sized daughters, with an interaction force φ, 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

∂

(cid:161)g (r )u(cid:162) + ∇ · (G[u]u) + β(r )u
∂t u(t , x, r ) + ∂
∂r
(cid:112)

(cid:112)

= 2

2β(

2r ) (cid:82) 1
0

κ(θ)u(t , x + αr (cos(2πθ), sin(2πθ)) ,

(cid:112)

2r )d θ,

t ≥ 0, x ∈ Ω, r > 0,

(2)

u(0, x) = u0(x),

g (0)u(t , x, 0) = 0,

G[u] = − (cid:82)

Ω×(0,∞)

′

φ(| x−x

r +r ′ |2)(x − x

′

)u(d x

′

, d r

′

).

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 (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 (2), 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 rod-shaped 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 individual-based models such as the hard-rod model of [126, 70] 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 trait-structured 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
so-called 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

6

Inria Annual Report 2023

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 so-called 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 fast-marching algorithms [99] to propose a description of the effect of climate change
on a given species.

From local interaction models to cross-diffusion 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, non-explosion, 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 [37].

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

Permanent members: Milica Tomaševi´c

An important mathematical challenge is to derive mean-field limits for non-Markovian interaction,
i.e., when the past also needs to be taken into account. Such models appear for instance in neuros-
cience [54, 53] and chemotaxis. To model chemotaxis, the parabolic-parabolic Keller-Segel model has
been stated phenomenologically, but to interpret it we need to introduce interaction memory, which
provides tremendous analysis difficulties since particles are now non-markovian both in time and space.
New methods have been proposed by Milica Tomaševi´c, 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 Tomaševi´c’s Ph.D thesis, so that many questions remain open. Con-
cerning the convergence of the particle systems towards the Keller-Segel model, an important problem
is the obtention of explicit convergence rates, when the number of particles tends to infinity, for the
propagation of chaos of the particle system in 1D. A possible way is to extend techniques developed
by Jabin and Wang [85] for the quantitative study of the mean-field boundaries of particle systems in
non-regular Markovian interaction. The aim is to control the relative entropy between the joint law of
the particles and the law of N independent copies of the Keller-Segel system. By exploiting the results
on the Keller-Segel nonlinearity in 1 dimension and on the Sobolev type estimation on the densities of
the system (chapter 4 in [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

Diffusion-growth-fragmentation processes and equations

Permanent members: Marie Doumic, Sylvie Méléard

To model the growth of a bacterial population in a chemostat, a new model of growth and frag-
mentation, 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,

Project MERGE

7

uniqueness and regularity of solutions, as well as the convergence in large populations of the individual-
based model. This model also present similiarities with the proliferation of parasites in dividing cells
studied by Vincent Bansaye [38, 34].

inistic process, leading to a transport term in the size structured equation

One of the very interesting novelties of this model is to consider the growth not as a purely determ-
∂x (τ(x)u(t , x)), but to take
∂2
into account the intrinsic stochasticity in growth, so that a diffusion-type term
∂x2 (D(x)u(t , x)) is added,
which degenerates at the boundary x = 0. We thus want to study further this equation, its long-time
dynamics with and without interaction (i.e. in the linear case as well as with nonlinear couplings), how
it differs from the much more studied growth-fragmentation equation, and which model seems more
relevant in which applicative case. We also want to adapt the model to metabolic heterogeneity cases, i.e.
when we model the capacity for bacteria to feed on two distinct nutrients, which leads to distinguish two
populations competing for two resources.

∂

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

Permanent members: Vincent Bansaye, Sylvie Méléard, Milica Tomaševi´c

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

Understanding the links between genealogical and population behaviours

Permanent members: Vincent Bansaye, Marie Doumic and Sylvie Méléard

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

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 Galton-Watson
tree is straightforward, but much more involved in more complex cases [36], 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

8

Inria Annual Report 2023

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 so-called 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 non-interacting 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 super-process 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 → ∞)
and the spinal techniques developed for branching processes (cf [82, 81], [94] and references therein). We
have seen that the stochastic process (1) is close to the solution of an integro-differential equation. There-
fore, we can construct for large K a coupling between the stochastic process and a non-homogeneous
structured branching process where the interaction terms have been replaced by their deterministic
approximation. We should obtain some non-homogeneous biased Markov process by giving its asso-
ciated 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 non-homogeneous 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: Model-data comparison

Permanent members: Marie Doumic, Sylvie Méléard, Milica Tomaševi´c

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, 69]. 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 non-interacting 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 cell-to-cell interaction (or yet by antibiotic resistance or by competition), and reciprocally,
how distributed growth and division rates may have an influence on the morphogenesis of the bacterial
microcolony. In this task, we aim to provide answers based on more realistic individual-based models.
We plan the following steps:

Project MERGE

9

• Develop parametric and non-parametric inference of the interaction function from single individual
tracking. A similar study has been carried out by Laetitia Della Maestra and Marc Hoffmann [61] for
Mc-Kean-Vlasov equation; we would like to add a size structure and a non-constant number of
individuals. We will first assume that the growth and division rates do not depend on the interaction
between cells, so that prior to this step we have used the methods already developed to infer these
functional parameters. We may also build upon biophysical studies such as in [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 so-called Distribution of Fitness Effects (DFE) which characterizes
the accumulation of mutations in bacteria [111]. As shown in [67], these are so-called severely ill-posed
inverse problems, for which we aim at developing new approaches, two in particular: rely on short-
time instead of long-term behaviour, adapt statistical methods developed for decoumpounding Poisson
processes and deconvolution [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
transport-diffusion equation; this new problem is closer to the experimental system but gives rise to
a severely ill-posed inverse problem, for which we want to find an observation inequality thanks to
Carleman estimates [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 the quantities they are supposed
to approximate.

3.4 Software development and dissemination

Permanent members: Marie Doumic, Milica Tomaševi´c

10

Inria Annual Report 2023

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 best-fit 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 best-fit 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 so-called "incremental model", where the structuring variable which
triggers division is the size increase of the bacteria since birth, to experimental data. This opens up
new questions to refine and analyse this model, test its validity in extreme growth conditions such as
antibiotic treatments, and understand its links with intracellular mechanisms. Main research axis: 3, and
the CellDiv platform.

Antibiotic response and resistance emergence To address the emergence of antibiotic-resistant strains
of bacteria, it is essential to understand quantitatively the response of bacteria to antibiotic treatments.
Under the action of an antibiotic that causes damage to cellular DNA, bacteria change their growth
strategy and do not respond homogeneously to this stress. Of particular importance is the so-called
SOS response: in response to DNA damage induced by antibiotic treatments, the cell cycle is arrested
and DNA repair and mutagenesis are induced (cf. [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.

Project MERGE

11

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 incid-
ence and the effects of mutations on fitness. Mutation dynamics has been for the first time analyzed
directly by Lydia Robert and co-authors [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 so-called "distribution of fitness effects" (DFE) of mutations. How to assess better the
shape of the DFE, and apply the method not only to deleterious or neutral but also to possibly beneficial
mutations, is one of our goals. Main research axis: 3, short term (Guillaume Garnier’s ongoing Ph.D).

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

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 hos-
pital), 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,

12

Inria Annual Report 2023

which has developed microfluidic as well as population experiments to follow senescence triggering
in yeast cells. Main research axis: 1 and 2. This application is both a long-term goal, in a long-lasting
collaboration with Teresa Teixeira and Zhou Xu, and has short and mid-terms objectives, through Anaïs
Rat’s finishing Ph.D and Jules Olayé forthcoming Ph.D (co-supervised by Milica Tomaševi´c and Marie
Doumic).

Ageing in drosophyla Ageing’s sensitivity to natural selection has long been discussed because of its
apparent negative effect on an individual’s fitness. In the recent years, a new 2-phases model of ageing
has been proposed by Hervé Tricoire and Michael Rera [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 Tomaševi´c

Biologist collaborators: Human Rezaei (INRAe), Florence Chapeland-Leclerc and Eric Herbert (LIED,
Univ. Paris Diderot), Sascha Martens (Vienna University), Wei-Feng 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 fila-
mentous 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.

Project MERGE

13

Emergence of bacterial resistance in heterogeneous environments

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

Dynamics of species submitted to climate change

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

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 (metapop-
ulations) 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 macro-
scopic 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 project-team brings together mathematicians with complementary competences and in-
terests, in order to integrate at a high level different areas of mathematical analysis (multiscale stochastic
processes, partial or integro-differential equations) and microbiology, ecology, cancer medicine. If suc-
cessful, 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 prob-
lems 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. Suc-
cessful 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.

14

Inria Annual Report 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 non-exponential. An age
structure allows to describe these interactions and get a Markovian setting. The process is characterized by
a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional
setting and get the convergence of the slow-fast 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 Galton-Watson process

Participants:

Vincent Bansaye, Maria-Emilia Caballero,
Sylvie Méléard.

Jaime San Martin,

Bisexual Galton-Watson 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 Galton-
Watson processes is lost. In [5], we prove tightness for conveniently rescaled bisexual Galton-Watson
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 mo-
tions. 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 individual-based evolutionary models to Hamilton-Jacobi 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 sub-populations may strongly influence the dynamics. Our main

Project MERGE

15

result quantifies the asymptotic dynamics of sub-population 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 Hamilton-Jacobi equation. Such Hamilton-
Jacobi equations have already been derived from parabolic integro-differential 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 long-range 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 self-generated 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 cross-diffusion system of quadratic type.

8.1.5 Particle approximation of the doubly parabolic Keller-Segel equation

Participants: Nicolas Fournier, Milica Tomaševi´c.

In [3], we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel
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 time-marginals solves the parabolic-parabolic Keller-Segel 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 two-by-two path-dependant interaction can be controlled by a two-by-two
Coulomb interaction, as in the parabolic-elliptic 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 environ-
mental 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.

16

Inria Annual Report 2023

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

The article [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 eco-evolutionary feedbacks affect
abrupt ecological transitions in shallow lakes. We explore the eco-evolutionary dynamics of submerged
and floating macrophytes in a shallow lake ecosystem under asymmetric competition for nutrients
and light. We show how rapid trait evolution can result in complex dynamics including evolutionary
oscillations, extensive diversification and evolutionary suicide. Overall, this study shows that evolution
can have strong effects in the ecological dynamics of bistable ecosystems.

8.2.3 A growth-fragmentation-isolation 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 mean-field interaction of

Lq − Lp type

Participants: Milica Tomaševi´c.

In this work [14], we prove the well-posedness and propagation of chaos for a stochastic particle
system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable
p
x space. Contrary to the large deviation principle approach recently proposed in [83], the main
L

− L

q
t

Project MERGE

17

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 Blow-up for a stochastic model of chemotaxis driven by conservative noise on R2

Participants:

Avi Mayorcas, Milica Tomaševi´c.

In [13], we establish criteria on the chemotactic sensitivity χ for the non-existence of global weak
solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous,
conservative noise on R2. We show that if χ is sufficiently large then blow-up occurs with probability 1. In
this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity.
However, for χ in an intermediate regime, determined by the variance of the initial data and the spatial
correlation of the noise, we show that blow-up occurs with positive probability.

8.2.6 Reducing exit-times of diffusions with repulsive interactions

Participants:

Paul-Eric Chaudru de Reynal, Pierre Monmarché, Milica Tomaševi´c,
Julian Tugaut.

In this work [9], we prove a Kramers’ type law for the low-temperature behavior of the exittimes
from a metastable state for a class of self-interacting nonlinear diffusion processes. Contrary to previous
works, the interaction is not assumed to be convex, which means that this result covers cases where
the exit-time for the interacting process is smaller than the exit-time for the associated non-interacting
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 non-interacting (linear) Markov process where
the interacting law is replaced by a constant Dirac mass at the fixed point of the deterministic zero-
temperature process.

8.3 Axis 3: Model-data 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 population-scale 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
non-terminal arrests drive cells into a totally different regime, which may promote genome instability
and senescence escape. Our results offer a comprehensive framework for investigating the implications
of telomere length on human diseases.

18

Inria Annual Report 2023

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 short-time dynamics

Participants: Marie Doumic, Miguel Escobedo, Magali Tournus.

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

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 multi-drug resistance of Klebsiella pneumoniae.

- with Vo Hoang Hung, Saigon University, Ho Chi Minh city, Vietnam. They work on the effect of an

age structure on the propagation of population in the context of climate change.

9.1 International initiatives

Milica Tomaševi´c participates in two international projects, in the France-Japan program Sakura (headed
by K. Fujie in Japan and Julian Tugaut in France) and in a franco-brasilian ANR project, ANR PRCI FAPESP
SDAIM : Stochastic and Deterministic Analysis of Irregular Models, 2023-2027, 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).

Project MERGE

19

9.3 European initiatives

9.3.1 Horizon Europe

The ERC Advanced Grant SINGER (Stochastic dynamics of sINgle cells; Growth, Emergence and Resist-
ance) 2022-2027, headed by Sylvie Méléard, involves all members of MERGE and several of our collab-
orators (Sepideh Mirrahimi in Toulouse, Nicolas Champagnat in BIGS, Viet-Chi 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 post-doctoral 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 project-team 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, co-supervised by Milica Tomaševi´c 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 parti-
cipates in the ANR DEEV (159 k€), 2020–2023 on Integro-Differential Equations from EVolutionary
biology, headed by Sepideh Mirrahimi. Milica Tomaševi´c participates to the ANR project METAN-
OLIN (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 Multi-échelle,
headed by Eric Herbert.

• Sylvie Méléard is the P.I. of a newly funded Aviesan-Inserm ITMO Cancer project (261 k€), 2022–2026
on Mathématiques pour une meilleure compréhension des néoplasmes myélo-prolifératifs et leurs
thérapeutiques.

10 Dissemination

Participants:

Vincent Bansaye, Marie Doumic, Sylvie Méléard, Gaël Raoul, Milica To-
maševi´c.

10.1 Promoting scientific activities

10.1.1 Scientific events: organisation

Principle organisation of events Vincent Bansaye, Sylvie Méléard and Milica Tomaševi´c co-organise
the summer school of the MMB chaire, headed by Sylvie Méléard, which takes place every June in Aussois.

Milica Tomaševi´c co-organised the conference in honour of Denis Talay at CIRM, Marseille.
Vincent Bansaye organised an invited session in the SPA Conference "Stochastic models for epidemi-

ology and evolution", Lisboa, July 2023.

20

Inria Annual Report 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 "Mathem-
atics 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 Comptes-Rendus 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 5-7 ; at the workshop "Topics on Neuroscience, Collective
Migration and Parameter Estimation", Oxford, July 3-7 ; at the "Journées EDP", Aussois, June 19-23 ; at
the READINET conference, Orsay, January 3-6.

Sylvie Méléard gave 12 invited talks, including 7 abroad.
Milica Tomaševi´c gave invited talks at the Mean field interactions with singular kernels and their
approximations workshop, IHP Paris 18-22 déc 2023 ; at the Conference Stochastic Flows, thematic
program Order and Randomness in Partial Differential Equations, Institut Mittag-Leffler, nov 2023 ; at the
Second Berlin-Leipzig Workshop on Fluctuating Hydrodynamics, Freie University, Berlin September 19-
20, 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 Polytech-
nique 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 post-doctoral 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.

Project MERGE

21

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 Math-Club, 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] V. Bansaye, C. Gu and L. Yuan. ‘A growth-fragmentation-isolation process on random recursive
trees and contact tracing’. In: The Annals of Applied Probability 33.6B (2023), pp. 5233–5278. DOI:
10.1214/23-AAP1947. URL: https://hal.science/hal-04406491.

[2] N. Champagnat, S. Méléard, S. Mirrahimi and V. Chi Tran. ‘Filling the gap between individual-
based evolutionary models and Hamilton-Jacobi equations’. In: Journal de l’École polytechnique
— Mathématiques 10 (2023), pp. 1247–1275. DOI: 10.5802/jep.244. URL: https://hal.scienc
e/hal-03656608.

[3] N. Fournier and M. Tomaševi´c. ‘Particle approximation of the doubly parabolic Keller-Segel
equation in the plane’. In: Journal of Functional Analysis 285.7 (2023), p. 110064. DOI: 10.1016/j
.jfa.2023.110064. URL: https://hal.science/hal-03855110.

[4] A. Rat, M. Doumic, M. T. Teixeira and Z. Xu. Individual cell fate and population dynamics revealed
by a mathematical model linking telomere length and replicative senescence. 22nd Nov. 2023. DOI:
10.1101/2023.11.22.568287. URL: https://hal.science/hal-04305005.

22

Inria Annual Report 2023

11.2 Publications of the year

International journals

[5] V. Bansaye, M.-E. Caballero, S. Méléard and J. San Martín. ‘Scaling limits of bisexual Galton-
Watson processes’. In: Stochastics: An International Journal of Probability and Stochastic Processes
95.5 (6th Feb. 2023), pp. 749–784. DOI: 10.1080/17442508.2022.2123706. URL: https://hal
.science/hal-02859954.

[6] V. Bansaye and B. Cloez. ‘From the distributions of times of interactions to preys and predators
dynamical systems’. In: Journal of Mathematical Biology 87.1 (12th June 2023), p. 2. DOI: 10.1007
/s00285-023-01925-5. URL: https://hal.science/hal-03183538.

[7] V. Bansaye, C. Gu and L. Yuan. ‘A growth-fragmentation-isolation process on random recursive
trees and contact tracing’. In: The Annals of Applied Probability 33.6B (2023), pp. 5233–5278. DOI:
10.1214/23-AAP1947. URL: https://hal.science/hal-04406491.

[8] N. Champagnat, S. Méléard, S. Mirrahimi and V. Chi Tran. ‘Filling the gap between individual-
based evolutionary models and Hamilton-Jacobi equations’. In: Journal de l’École polytechnique
— Mathématiques 10 (2023), pp. 1247–1275. DOI: 10.5802/jep.244. URL: https://hal.scienc
e/hal-03656608.

[9] P.-E. Chaudru de Raynal, M. H. Duong, P. Monmarché, M. Tomaševi´c and J. Tugaut. ‘Reducing
exit-times of diffusions with repulsive interactions’. In: ESAIM: Probability and Statistics 27 (2023),
pp. 723–748. DOI: 10.1051/ps/2023012. URL: https://hal.science/hal-03762603.

[10] M. Fathi, P. Le Bris, A. Menegaki, P. Monmarché, J. Reygner and M. Tomaševi´c. ‘Recent progress
on limit theorems for large stochastic particle systems’. In: ESAIM: Proceedings and Surveys 75
(19th Dec. 2023), pp. 2–23. DOI: 10.1051/proc/202375002. URL: https://enpc.hal.science
/hal-03711772.

[11] N. Fournier and M. Tomaševi´c. ‘Particle approximation of the doubly parabolic Keller-Segel
equation in the plane’. In: Journal of Functional Analysis 285.7 (2023), p. 110064. DOI: 10.1016/j
.jfa.2023.110064. URL: https://hal.science/hal-03855110.

[12] B. Henry, S. Méléard and V. Chi Tran. ‘Time reversal of spinal processes for linear and non-linear
branching processes near stationarity’. In: Electronic Journal of Probability 28.none (1st Jan. 2023).
DOI: 10.1214/23-EJP911. URL: https://cnrs.hal.science/hal-03525599.

[13] A. Mayorcas and M. Tomaševi´c. ‘Blow-up for a stochastic model of chemotaxis driven by conser-
vative noise on R2’. In: Journal of Evolution Equations 23.57 (Sept. 2023). DOI: 10.1007/s00028-
023-00900-3. URL: https://inria.hal.science/hal-04191089.

[14] M. Tomaševi´c. ‘Propagation of chaos for stochastic particle systems with singular mean-field
interaction of Lq =Lp type’. In: Electronic Communications in Probability 28.none (29th Aug. 2023),
p. 13. DOI: 10.1214/23-ECP539. URL: https://hal.science/hal-03086253.

Conferences without proceedings

[15] M. Kubasch. ‘Large population limit for a multilayer SIR model including households and work-
places’. In: Rencontres Mathématiques de Rouen. Rouen, France, 28th June 2023. URL: https:
//hal.science/hal-04366603.

[16] M. Kubasch, V. Bansaye, F. Deslandes and E. Vergu. ‘A tribute to Elisabeta Vergu: Multi-level contact
models of epidemics’. In: Journées 2023 de l’Action Coordonnée "Modélisation des Maladies
Infectieuses". Paris, France, 24th Oct. 2023. URL: https://hal.science/hal-04366616.

[17] M. Kubasch, V. Bansaye, F. Deslandes and E. Vergu. ‘Approximations déterministes de modèles
stochastiques multi-niveaux en épidémiologie’. In: Colloque Jeunes Probabilistes et Statisticiens.
Saint Pierre d’Oléron, France, 1st Oct. 2023. URL: https://hal.science/hal-04366610.

[18] M. Kubasch, V. Bansaye, F. Deslandes and E. Vergu. ‘Modèles structurés multi-niveaux de dy-
namiques épidémiques’. In: École de Printemps de la chaire MMB. Aussois, France, 11th June
2023. URL: https://hal.science/hal-04366602.

Project MERGE

23

Scientific book chapters

[19] M. Doumic and M. Hoffmann. ‘Individual and population approaches for calibrating division rates
in population dynamics: Application to the bacterial cell cycle’. In: Modeling and Simulation for
Collective Dynamics. Vol. 40. Lecture Notes Series, Institute for Mathematical Sciences, National
University of Singapore. WORLD SCIENTIFIC, 7th Feb. 2023, pp. 1–81. DOI: 10.1142/978981126
6140_0001. URL: https://hal.science/hal-03328781.

Reports & preprints

[20] V. Bansaye, P.-Y. Boëlle, S. Cauchemez, B. Cazelles, P. Crepey, L. Cristancho-Fajardo, J.-S. Dhersin,
S. Dumitrescu, P. Ezanno, A. Flahault, J.-L. Golmard, P. Hoscheit, H. Mermoz Kouye, M. Kubasch,
C. Laredo, A. Mallet, L. Opatowski, C. Prieur and A. Véber. Hommage à Elisabeta Vergu. 29th Nov.
2023. URL: https://hal.science/hal-04230307.

[21] V. Bansaye, F. Deslandes, M. Kubasch and E. Vergu. The epidemiological footprint of contact
structures in models with two levels of mixing. 7th Mar. 2023. URL: https://hal.science/hal-
04012906.

[22] V. Bansaye, X. Erny and S. Méléard. Sharp approximation and hitting times for stochastic invasion

processes. 11th Dec. 2023. URL: https://hal.science/hal-03915479.

[23] M. Doumic, S. Hecht, B. Perthame and D. Peurichard. Multispecies cross-diffusions: from a nonlocal
mean-field to a porous medium system without self-diffusion. 26th May 2023. URL: https://hal
.science/hal-04108050.

[24] M. Kubasch. Large population limit for a multilayer SIR model including households and work-

places. 25th May 2023. URL: https://hal.science/hal-04106800.

[25] A. Rat, M. Doumic, M. T. Teixeira and Z. Xu. Individual cell fate and population dynamics revealed
by a mathematical model linking telomere length and replicative senescence. 22nd Nov. 2023. DOI:
10.1101/2023.11.22.568287. URL: https://hal.science/hal-04305005.

[26] M. Tomaševi´c and G. Woessner. On a multi-dimensional McKean-Vlasov SDE with memorial and
singular interaction associated to the parabolic-parabolic Keller-Segel model. 22nd Sept. 2022. URL:
https://inria.hal.science/hal-04191117.

11.3 Cited publications

[27] M. Alfaro, J. Coville and G. Raoul. ‘Travelling waves in a nonlocal reaction-diffusion equation as a
model for a population structured by a space variable and a phenotypic trait’. In: Communications
in Partial Differential Equations 38.12 (2013), pp. 2126–2154.

[28] A. Amir. ‘Cell size regulation in bacteria’. In: Physical Review Letters 112.20 (2014), p. 208102.

[29] A. Armiento, M. Doumic, P. Moireau and H. Rezaei. ‘Estimation from Moments Measurements for
Amyloid Depolymerisation’. In: Journal of Theoretical Biology 397 (Mar. 2016), pp. 68–88. DOI: 10
.1016/j.jtbi.2016.02.037. URL: https://hal.archives-ouvertes.fr/hal-01248255.

[30] A. Armiento, P. Moireau, D. Martin, N. Lepejova, M. Doumic and H. Rezaei. ‘The mechanism of
monomer transfer between two structurally distinct PrP oligomers’. In: PLOS ONE 12.7 (July 2017).
DOI: 10.1371/journal.pone.0180538. URL: https://hal.archives-ouvertes.fr/hal-0
1574346.

[31] Z. Baharoglu and D. Mazel. ‘SOS, the formidable strategy of bacteria against aggressions’. In: FEMS
Microbiology Reviews 38.6 (Nov. 2014), pp. 1126–1145. DOI: 10.1111/1574-6976.12077.

[32]

J. M. Ball and A. Majumdar. ‘Nematic liquid crystals: from Maier-Saupe to a continuum theory’.
In: Molecular crystals and liquid crystals 525.1 (2010), pp. 1–11.

[33] H. T. Banks, M. Doumic, C. Kruse, S. Prigent and H. Rezaei. ‘Information content in data sets for a

nucleated-polymerization model’. In: Journal of biological dynamics 9.1 (2015), pp. 172–197.

[34] V. Bansaye. ‘Proliferating parasites in dividing cells: Kimmel’s branching model revisited’. In: The

Annals of Applied Probability 18.3 (2008), pp. 967–996.

24

Inria Annual Report 2023

[35] V. Bansaye, B. Cloez and P. Gabriel. ‘Ergodic behavior of non-conservative semigroups via general-

ized Doeblin conditions’. In: Acta Applicandae Mathematicae (2019), pp. 1–44.

[36] V. Bansaye and S. Méléard. Stochastic Models for Structured Populations. Vol. 16. Springer, 2015.

[37] V. Bansaye, A. Moussa and F. Muñoz-Hernández. ‘Stability of a cross-diffusion system and ap-
proximation by repulsive random walks: a duality approach’. In: arXiv preprint arXiv:2109.07146
(2021).

[38] V. Bansaye and V. C. Tran. ‘Branching Feller diffusion for cell division with parasite infection’. In:

ALEA: Latin American Journal of Probability and Mathematical Statistics 8 (2011), pp. 95–127.

[39] A. Barizien, M. Suryateja Jammalamadaka, G. Amselem and C. N. Baroud. ‘Growing from a few
cells: combined effects of initial stochasticity and cell-to-cell variability’. In: Journal of the Royal
Society Interface 16.153 (2019), p. 20180935.

[40] G. Barles, S. Mirrahimi and B. Perthame. ‘Concentration in Lotka-Volterra parabolic or integral
equations: a general convergence result’. In: Methods and Applications of Analysis 16.3 (2009),
pp. 321–340.

[41] G. Barles and B. Perthame. ‘Concentrations and constrained Hamilton-Jacobi equations arising in

adaptive dynamics’. In: Contemporary Mathematics 439 (2007), pp. 57–68.

[42] M. Barthe, J. Tchouanti, P. H. Gomes, C. Bideaux, D. Lestrade, C. Graham, J.-P. Steyer, S. Méléard,
J. Harmand and N. Gorret. ‘Availability of the Molecular Switch XylR Controls Phenotypic Hetero-
geneity and Lag Duration during Escherichia coli Adaptation from Glucose to Xylose’. In: Mbio
11.6 (2020), e02938–20.

[43] N. Barton. ‘Adaptation at the edge of a species’ range’. In: Integrating ecology and evolution in a

spatial context. 2001.

[44] N. Berestycki. ‘Recent progress in coalescent theory’. In: ENSAIOS MATEMÁTICOS 16 (2009),

pp. 1–193.

[45] C. Bonnet, P. Gou, V. Bansaye, C. Lacout, K. Bailly, M.-h. Schlagetter, E. Lauret, S. Meleard and S.
Giraudier. Modeling the behavior of hematopoietic compartments from stem to red cells in murine
steady state and stress hematopoiesis. 2019.

[46] C. Bonnet, P. Gou, S. Girel, V. Bansaye, C. Lacout, K. Bailly, M.-H. Schlagetter, E. Lauret, S. Méléard
and S. Giraudier. ‘Multistage hematopoietic stem cell regulation in the mouse: a combined
biological and mathematical approach’. In: iScience (2021), p. 103399.

[47] T. Bourgeron, Z. Xu, M. Doumic and M. T. Teixeira. ‘The asymmetry of telomere replication
contributes to replicative senescence heterogeneity’. In: Scientific Reports 5 (Oct. 2015), p. 15326.
DOI: 10.1038/srep15326. URL: https://hal.sorbonne-universite.fr/hal-01272075.

[48] V. Calvez, B. Henry, S. Méléard and V. C. Tran. ‘Dynamics of lineages in adaptation to a gradual

environmental change’. In: Annales Henri Lebesgue, to appear (2021).

[49]

J. A. Carrillo, K. Craig and Y. Yao. ‘Aggregation-diffusion equations: dynamics, asymptotics, and
singular limits’. In: Active Particles, Volume 2. Springer, 2019, pp. 65–108.

[50] N. Champagnat. ‘A microscopic interpretation for adaptive dynamics trait substitution sequence

models’. In: Stochastic processes and their applications 116.8 (2006), pp. 1127–1160.

[51] N. Champagnat, R. Ferrière and S. Méléard. ‘Unifying evolutionary dynamics: from individual
stochastic processes to macroscopic models’. In: Theoretical Population Biology 69.3 (2006),
pp. 297–321.

[52] N. Champagnat and S. Méléard. ‘Polymorphic evolution sequence and evolutionary branching’.

In: Probability Theory and Related Fields 151.1-2 (2011), pp. 45–94.

[53]

[54]

J. Chevallier. ‘Mean-field limit of generalized Hawkes processes’. In: Stochastic Processes and their
Applications 127.12 (2017), pp. 3870–3912.

J. Chevallier, M. J. Cacéres, M. Doumic and P. Reynaud-Bouret. ‘Microscopic approach of a time
elapsed neural model’. In: Mathematical Models and Methods in Applied Sciences (Dec. 2015),
http://www.worldscientific.com/doi/10.1142/S021820251550058X. DOI: 10.1142/S0
21820251550058X. URL: http://hal.upmc.fr/hal-01159215.

Project MERGE

25

[55] H. Cho, H. Jönsson, K. Campbell, P. Melke, J. W. Williams, B. Jedynak, A. M. Stevens, A. Groisman
and A. Levchenko. ‘Self-organization in high-density bacterial colonies: efficient crowd control’.
In: PLOS Biology 5.11 (2007), e302.

[56] K. L. Chung and J. B. Walsh. ‘To reverse a Markov process’. In: Acta Mathematica 123 (1969),

pp. 225–251.

[57] B. Cloez. ‘Limit theorems for some branching measure-valued processes’. In: Advances in Applied

Probability 49.2 (2017), pp. 549–580.

[58] P. Cornilleau and S. Guerrero. ‘Controllability and observability of an artificial advection–diffusion

problem’. In: Mathematics of Control, Signals, and Systems 24.3 (2012), pp. 265–294.

[59]

J.-M. Coron and S. Guerrero. ‘Singular optimal control: a linear 1-D parabolic–hyperbolic example’.
In: Asymptotic Analysis 44.3, 4 (2005), pp. 237–257.

[60] E. Dambroise, L. Monnier, L. Ruisheng, H. Aguilaniu, J.-S. Joly, H. Tricoire and M. Rera. ‘Two
phases of aging separated by the Smurf transition as a public path to death’. In: Scientific Reports
6.1 (2016), pp. 1–7.

[61] L. Della Maestra and M. Hoffmann. ‘Nonparametric estimation for interacting particle systems:

McKean–Vlasov models’. In: Probability Theory and Related Fields (2021), pp. 1–63.

[62] U. Dieckmann and R. Law. ‘The dynamical theory of coevolution: a derivation from stochastic

ecological processes’. In: Journal of mathematical biology 34.5 (1996), pp. 579–612.

[63] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame. ‘The dynamics of adaptation: an illumin-
ating example and a Hamilton–Jacobi approach’. In: Theoretical Population Biology 67.4 (2005),
pp. 257–271.

[64]

J. Dikec, A. Olivier, C. Bobée, Y. D’angelo, R. Catellier, P. David, F. Filaine, S. Herbert, C. Lalanne and
H. Lalucque. ‘Hyphal network whole field imaging allows for accurate estimation of anastomosis
rates and branching dynamics of the filamentous fungus Podospora anserina’. In: Scientific Reports
10.1 (2020), pp. 1–16.

[65] P. Donnelly and T. G. Kurtz. ‘Genealogical processes for Fleming-Viot models with selection and

recombination’. In: Annals of Applied Probability (1999), pp. 1091–1148.

[66] A. Doostmohammadi, S. P. Thampi, T. B. Saw, C. T. Lim, B. Ladoux and J. M. Yeomans. ‘Celebrating
Soft Matter’s 10th Anniversary: Cell division: a source of active stress in cellular monolayers’. In:
Soft Matter 11.37 (2015), pp. 7328–7336.

[67] M. Doumic, M. Escobedo and M. Tournus. ‘Estimating the division rate and kernel in the frag-
mentation equation’. In: Ann. Inst. H. Poincaré Anal. Non Linéaire 35.7 (2018), pp. 1847–1884. DOI:
10.1016/j.anihpc.2018.03.004. URL: https://doi.org/10.1016/j.anihpc.2018.03.0
04.

[68] M. Doumic, M. Hoffmann, N. Krell and L. Robert. ‘Statistical estimation of a growth-fragmentation

model observed on a genealogical tree’. In: Bernoulli 21.3 (2015), pp. 1760–1799.

[69] M. Doumic, K. Fellner, M. Mezache and H. Rezaei. ‘A bi-monomeric, nonlinear Becker–Döring-
type system to capture oscillatory aggregation kinetics in prion dynamics’. In: Journal of Theoret-
ical Biology 480 (2019), pp. 241–261.

[70] M. Doumic, S. Hecht and D. Peurichard. ‘A purely mechanical model with asymmetric features
for early morphogenesis of rod-shaped bacteria micro-colony’. In: Mathematical Biosciences and
Engineering 17.6 (2020), pp. 6873–6908.

[71] M. Doumic, A. Olivier and L. Robert. ‘Estimating the division rate from indirect measurements of

single cells.’ In: Discrete and Continuous Dynamical Systems-Series B 25.10 (2020).

[72] C. Duval and J. Kappus. ‘Adaptive procedure for Fourier estimators: application to deconvolution

and decompounding’. In: Electronic Journal of Statistics 13.2 (2019), pp. 3424–3452.

[73] A. M. Etheridge and T. G. Kurtz. ‘Genealogical constructions of population models’. In: The Annals

of Probability 47.4 (2019), pp. 1827–1910.

26

Inria Annual Report 2023

[74] D. R. Evans, M. P. Griffith, A. J. Sundermann, K. A. Shutt, M. I. Saul, M. M. Mustapha, J. W. Marsh,
V. S. Cooper, L. H. Harrison and D. Van Tyne. ‘Systematic detection of horizontal gene transfer
across genera among multidrug-resistant bacteria in a single hospital’. In: Elife 9 (2020), e53886.

[75] A. Fievet, A. Ducret, T. Mignot, O. Valette, L. Robert, R. Pardoux, A. R. Dolla and C. Aubert. ‘Single-
Cell Analysis of Growth and Cell Division of the Anaerobe Desulfovibrio vulgaris Hildenborough’.
In: Frontiers in Microbiology 6 (2015), p. 1378.

[76] R. A. Fisher. ‘XV.—The correlation between relatives on the supposition of Mendelian inheritance.’
In: Earth and Environmental Science Transactions of the Royal Society of Edinburgh 52.2 (1919),
pp. 399–433.

[77]

J. Fontbona and S. Méléard. ‘Non local Lotka-Volterra system with cross-diffusion in an hetero-
geneous medium’. In: Journal of Mathematical Biology 70.4 (2015), pp. 829–854.

[78] N. Fournier and S. Méléard. ‘A microscopic probabilistic description of a locally regulated popula-
tion and macroscopic approximations’. In: The Annals of Applied Probability 14.4 (2004), pp. 1880–
1919.

[79] C. Graham, J. Harmand, S. Méléard and J. Tchouanti. ‘Bacterial metabolic heterogeneity: from
stochastic to deterministic models’. In: Mathematical Biosciences and Engineering 17.5 (2020),
pp. 5120–5133.

[80] Q. Griette, M. Alfaro, G. Raoul and S. Gandon. ‘Evolution and spread of multidrug resistant
pathogens in a spatially heterogeneous environment’. working paper or preprint. Dec. 2022. DOI:
10.1101/2022.07.16.500289. URL: https://hal.science/hal-03880206.

[81] S. C. Harris, S. G. Johnston and M. I. Roberts. ‘The coalescent structure of continuous-time

Galton–Watson trees’. In: The Annals of Applied Probability 30.3 (2020), pp. 1368–1414.

[82] S. C. Harris and M. I. Roberts. ‘The many-to-few lemma and multiple spines’. In: Annales de

l’Institut Henri Poincaré, Probabilités et Statistiques 53.1 (2017), pp. 226–242.

[83]

J. Hoeksema, T. Holding, M. Maurelli and O. Tse. ‘Large deviations for singularly interacting
diffusions’. In: arXiv preprint arXiv:2002.01295 (2020).

[84] P.-E. Jabin. ‘Small populations corrections for selection-mutation models’. In: Networks & Hetero-

geneous Media 7.4 (2012), p. 805.

[85] P.-E. Jabin and Z. Wang. ‘Quantitative estimates of propagation of chaos for stochastic systems
kernels’. In: Inventiones mathematicae 214.1 (2018), pp. 523–591.

with W

−1,∞

[86]

[87]

J.-F. Jabir, D. Talay and M. Tomaševi´c. ‘Mean-field limit of a particle approximation of the one-
dimensional parabolic-parabolic Keller-Segel model without smoothing’. In: Electronic Commu-
nications in Probability 23 (2018), pp. 1–14.

J. Jabir, D. Talay and M. Tomasevic. ‘Mean-field limit of a particle approximation of the one-
dimensional parabolic-parabolic keller-segel model without smoothing’. In: Electronic Commu-
nications in Probability 23 (2018), pp. 1–14.

[88] A. Jüngel, S. Portisch and A. Zurek. ‘Nonlocal cross-diffusion systems for multi-species populations

and networks’. In: arXiv preprint arXiv:2104.06292 (2021).

[89] A. N. Kapanidis, A. Lepore and M. El Karoui. ‘Rediscovering bacteria through single-molecule

imaging in living cells’. In: Biophysical Journal 115.2 (2018), pp. 190–202.

[90]

J. F. C. Kingman. ‘The coalescent’. In: Stochastic processes and their applications 13.3 (1982),
pp. 235–248.

[91] D. Lacker. ‘On a strong form of propagation of chaos for McKean-Vlasov equations’. In: Electronic

Communications in Probability 23 (2018), pp. 1–11.

[92] F. Lin and C. Wang. ‘Recent developments of analysis for hydrodynamic flow of nematic liquid

crystals’. In: Phil. Trans. R. Soc. A 372.2029 (2014), p. 20130361.

[93] P.-L. Lions and S. Mas-Gallic. ‘Une méthode particulaire déterministe pour des équations dif-
fusives non linéaires’. In: Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 332.4
(2001), pp. 369–376.

Project MERGE

27

[94] A. Marguet. ‘Uniform sampling in a structured branching population’. In: Bernoulli 25.4A (2019),

pp. 2649–2695.

[95] H. Martin, M. Doumic, M. T. Teixeira and Z. Xu. ‘Telomere shortening causes distinct cell division

regimes during replicative senescence in Saccharomyces cerevisiae’. In: bioRxiv (2021).

[96] S. Méléard and V. C. Tran. ‘Nonlinear historical superprocess approximations for population

models with past dependence’. In: Electronic Journal of Probability 17 (2012), pp. 1–32.

[97]

[98]

[99]

[100]

[101]

J. Metz, S. A. Geritz, G. Meszéna, F. Jacobs and J. Van Heerwaarden. ‘Adaptive dynamics: a Geo-
metrical Study of the Consequences of Nearly Faithful Reproduction’. In: Stochastic and Spatial
Structures of Dynamical Systems. North-Holland, 1996, pp. 183–231.

J. R. Miller. ‘Invasion waves and pinning in the Kirkpatrick–Barton model of evolutionary range
dynamics’. In: Journal of mathematical biology 78.1 (2019), pp. 257–292.

J.-M. Mirebeau and J. Portegies. ‘Hamiltonian fast marching: A numerical solver for anisotropic
and non-holonomic eikonal PDEs’. In: Image Processing On Line 9 (2019), pp. 47–93.

S. Mirrahimi, G. Barles, B. Perthame and P. E. Souganidis. ‘A Singular Hamilton–Jacobi Equation
Modeling the Tail Problem’. In: SIAM Journal on Mathematical Analysis 44.6 (2012), pp. 4297–4319.

S. Mirrahimi and G. Raoul. ‘Dynamics of sexual populations structured by a space variable and a
phenotypical trait’. In: Theoretical Population Biology 84 (2013), pp. 87–103.

[102] S. Motsch and D. Peurichard. ‘From short-range repulsion to Hele-Shaw problem in a model of

tumor growth’. In: Journal of mathematical biology 76.1-2 (2018), pp. 205–234.

[103] M. Nagasawa. ‘Time reversions of Markov processes’. In: Nagoya Mathematical Journal 24 (1964),

pp. 177–204.

[104] K. Oelschläger. ‘Large systems of interacting particles and the porous medium equation’. In:

Journal of Differential Equations 88.2 (1990), pp. 294–346.

[105] K. Oelschläger. ‘On the derivation of reaction-diffusion equations as limit dynamics of systems of
moderately interacting stochastic processes’. In: Probability Theory and Related Fields 82.4 (1989),
pp. 565–586.

[106] E. A. Perkins. On the martingale problem for interactive measure-valued branching diffusions.

Vol. 549. American Mathematical Soc., 1995.

[107] B. Perthame. Parabolic equations in biology. Lecture Notes on Mathematical Modelling in the
Life Sciences. Springer, Cham, 2015, pp. xii+199. DOI: 10.1007/978- 3- 319- 19500- 1. URL:
http://dx.doi.org/10.1007/978-3-319-19500-1.

[108] B. Perthame. Transport equations in biology. Frontiers in Mathematics. Basel: Birkhäuser Verlag,

2007, pp. x+198.

[109] B. Perthame and G. Barles. ‘Dirac concentrations in Lotka-Volterra parabolic PDEs’. In: Indiana

University Mathematics Journal (2008), pp. 3275–3301.

[110] B. Perthame and M. Gauduchon. ‘Survival thresholds and mortality rates in adaptive dynamics:
conciliating deterministic and stochastic simulations’. In: Mathematical Medicine and Biology: a
journal of the IMA 27.3 (2010), pp. 195–210.

[111] L. Robert, J. Ollion, J. Robert, X. Song, I. Matic and M. Elez. ‘Mutation dynamics and fitness effects

followed in single cells’. In: Science 359.6381 (2018), pp. 1283–1286.

[112] G. R. S. Boucenna and V. Dakos. ‘Evolutionary outcomes arising from bistability in ecosystem

dynamics’. 2023.

[113]

S. Taheri-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. Vergassola and S. Jun.
‘Cell-Size Control and Homeostasis in Bacteria’. In: Current Biology 11679.1-7 (2015).

[114] D. Talay and M. Tomaševi´c. ‘A new McKean–Vlasov stochastic interpretation of the parabolic–
parabolic Keller–Segel model: The one-dimensional case’. In: Bernoulli 26.2 (2020), pp. 1323–
1353.

28

Inria Annual Report 2023

[115]

J. Tchouanti. ‘Well posedness and stochastic derivation of a diffusion-growth-fragmentation
equation in a chemostat’. preprint. 2021.

[116] M. Tomasevic. ‘A new McKean–Vlasov stochastic interpretation of the parabolic-parabolic Keller–
Segel model: The two-dimensional case’. In: The Annals of Applied Probability 31.1 (2021), pp. 432–
459.

[117] M. Tomasevic. ‘On a probabilistic interpretation of the Keller-Segel parabolic-parabolic equations’.
Theses. Université Côte d’Azur, Nov. 2018. URL: https://tel.archives-ouvertes.fr/tel-0
1932777.

[118] M. Tomasevic. ‘Propagation of chaos for stochastic particle systems with singular mean-field

interaction of Lq − Lp type’. preprint. 2020.

[119] M. Tomasevic, A. Véber and V. Bansaye. ‘Ergodic behaviour of a multi-type growth-fragmentation

process modelling the mycelial network of a filamentous fungus’. preprint. 2020.

[120] M. Tournus and Xue. ‘Insights into the dynamic trajectories of protein filament division revealed
by numerical investigation into the mathematical model of pure fragmentation’. In: PLOS Com-
putational Biology 17.9 (Sept. 2021), pp. 1–21. DOI: 10.1371/journal.pcbi.1008964. URL:
https://doi.org/10.1371/journal.pcbi.1008964.

[121] H. Tricoire and M. Rera. ‘A new, discontinuous 2 phases of aging model: Lessons from Drosophila

melanogaster’. In: PLOS One 10.11 (2015), e0141920.

[122] A. Y. Veretennikov. ‘On mean-field (GI/GI/1) queueing model: existence and uniqueness’. In:

Queueing Systems 94.3 (2020), pp. 243–255.

[123] D. Volfson, S. Cookson, J. Hasty and L. S. Tsimring. ‘Biomechanical ordering of dense cell popula-
tions’. In: Proceedings of the National Academy of Sciences 105.40 (2008), pp. 15346–15351.

[124] P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright and S. Jun. ‘Robust Growth of

Escherichia coli’. In: Current Biology 20.12 (2010), pp. 1099–103.

[125] D. Waxman and S. Gavrilets. ‘20 questions on adaptive dynamics’. In: Journal of evolutionary

biology 18.5 (2005), pp. 1139–1154.

[126] Z. You, D. J. Pearce, A. Sengupta and L. Giomi. ‘Geometry and mechanics of microdomains in

growing bacterial colonies’. In: Physical Review X 8.3 (2018), p. 031065.

[127] H.-P. Zhang, A. Be’er, E.-L. Florin and H. L. Swinney. ‘Collective motion and density fluctuations in
bacterial colonies’. In: Proceedings of the National Academy of Sciences 107.31 (2010), pp. 13626–
13630.
<scratchpad>
- Main topic: Research Centre MERGE (Inria Saclay Centre)
- Key points:
  - The MERGE project focuses on modelling and studying interacting populations, with applications in biology, ecology, and medicine.
  - The research program is organized into three axes: Models through scales, Qualitative analysis of structured populations, and Model-data comparison.
  - The team has collaborated with biologists, mathematicians, and ecologists to develop new models and methods for studying population dynamics.
  - The project has explored various applications, including bacterial growth, cancer, aging, and evolutionary epidemiology.
  - The team has developed new software and platforms, such as CellDiv, to facilitate data analysis and model simulation.
- Structure: The document is divided into sections on Project MERGE, Research Program, Application Domains, and Dissemination.
- Potential artifacts to ignore: The document contains links to external web pages, HTML tags, and unnecessary formatting.
</scratchpad>

<final_summary>
The MERGE project is a research centre that focuses on modelling and studying interacting populations, with applications in biology, ecology, and medicine. The project is organized into three axes: Models through scales, Qualitative analysis of structured populations, and Model-data comparison. The team has collaborated with biologists, mathematicians, and ecologists to develop new models and methods for studying population dynamics, with applications in bacterial growth, cancer, aging, and evolutionary epidemiology. The project has also developed new software and platforms, such as CellDiv, to facilitate data analysis and model simulation. The MERGE project aims to provide a comprehensive understanding of population dynamics and its applications in various fields.
The MERGE project is a research centre that focuses on modelling and studying interacting populations, with applications in biology, ecology, and medicine. The project is organized into three axes: Models through scales, Qualitative analysis of structured populations, and Model-data comparison. The team has collaborated with biologists, mathematicians, and ecologists to develop new models and methods for studying population dynamics, with applications in bacterial growth, cancer, aging, and evolutionary epidemiology. The project has also developed new software and platforms, such as CellDiv, to facilitate data analysis and model simulation. The MERGE project aims to provide a comprehensive understanding of population dynamics and its applications in various fields.