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A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric spaces and give some applications of coalescent point processes to the phylogeny of bird species.

However, these models of diversification assume that species are exchangeable particles, and this always leads to the same (Yule) tree shape in distribution. Here, we propose a non-exchangeable, individual-based, point mutation model of diversification, where interspecific pairwise competition is only felt from the part of individuals belonging to younger species. As the initial (meta)population size grows to infinity, the properly rescaled dynamics of species lineages converge to a one-parameter family of coalescent trees interpolating between the caterpillar tree and the Kingman coalescent.

Keywords: ultrametric tree, inference, phylogenetic tree, phylogeny, birth-death process, population dynamics, evolution
A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric ...

60J80 ; 60J85 ; 92D15 ; 92D25 ; 54E45 ; 54E70

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The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other properties of the point process. In particular, such freezing occurs for the extremal process in branching random walks and in certain versions of the (discrete) two dimensional GFF.
Joint work with Eliran Subag
The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other ...

60G55 ; 60J65 ; 60J80

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Multi angle  Branching for PDEs
Warin, Xavier (Auteur de la Conférence) | CIRM (Editeur )

Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities polynomial in u and Du with convergence results. At last we give some numerical schemes to solve the semi-linear case and even the full non linear case but currently without convergence results. Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities ...

60H15 ; 35R60 ; 60J80

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Multi angle  Processus de Pólya à valeur mesure
Mailler, Cécile (Auteur de la Conférence) | CIRM (Editeur )

Une urne de Pólya est un processus stochastique décrivant la composition d'une urne contenant des boules de différentes couleurs. L'ensemble des couleurs est usuellement un ensemble fini {1, ..., d}. A chaque instant n, une boule est tirée uniformément au hasard dans l'urne (notons c sa couleur), remise dans l'urne accompagnée de R(c,i) boules de couleur i pour toute couleur i.
Je présente dans cet exposé une généralisation de ce modèle à un ensemble infini, et même potentiellement indénombrable de couleurs. Dans ce nouveau modèle, la composition de l'urne est une mesure (potentiellement non-atomique) sur un espace Polonais.
Ceci est un travail en collaboration avec Jean-François Marckert.
Une urne de Pólya est un processus stochastique décrivant la composition d'une urne contenant des boules de différentes couleurs. L'ensemble des couleurs est usuellement un ensemble fini {1, ..., d}. A chaque instant n, une boule est tirée uniformément au hasard dans l'urne (notons c sa couleur), remise dans l'urne accompagnée de R(c,i) boules de couleur i pour toute couleur i.
Je présente dans cet exposé une généralisation de ce modèle à un ...

60J80

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The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is the probability that the rightmost particle at time t in a branching Brownian motion (BBM) is to the right of $x$.
One of the beauty of this equation is that for initial conditions that decrease sufficiently fast, a front develops, i.e. there exists a centring term $m(t)$ and an asymptotic shape $\omega(x)$ such that

$\lim_{t \to \infty} h(m(t) + x,t) = \omega(x) \in (0, 1).$

Since the original paper of Kolmogorov et al., the position of the front $m(t)$ has been studied intensely, in particular by Bramson. In this talk, I will present some recent results concerning a prediction of Ebert and van Saarloos about the vanishing corrections of this position.
Based on a joint work with E. Brunet.
The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is ...

60J80 ; 35K57

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Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the transfer of plasmids in bacteria. The transfer rates are either density-dependent (DD) or frequency-dependent (FD) or of Michaelis-Menten form (MM). Our model allows eco-evolutionary feedbacks. In the first part we present a two-traits (alleles or kinds of plasmids, etc.) model with horizontal transfer without mutation and study a large population limit. It’s a ODEs system. We show that the phase diagrams are different in the (DD), (FD) and (MM) cases. We interpret the results for the impact of horizontal transfer on the maintenance of polymorphism and the invasion or elimination of pathogens strains. We also propose a diffusive approximation of adaptation with transfer. In a second part, we study the impact of the horizontal transfer on the evolution. We explain why it can drastically affect the evolutionary outcomes. Joint work with S. Billiard,P. Collet, R. Ferrière, C.V. Tran. Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the ...

60J75 ; 60J80 ; 92D25 ; 92D15

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