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We consider a spectrally positive L��vy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process \cite{GaJa98,Lam10}. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf_{[0,t]} X$ and we let $��$ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$��$ and the post-$��$ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of L��vy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

40 pages, 2 figures; We define a quasistationary monotone labelling of Bienaymé--Galton--Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i;i\ge 1)$, where $A_i$ is the coalescence time between individuals $i$ and $i+1$. There is a Markov process of point measures $(B_i;i\ge 1)$ keeping track of more ancestral relationships, such that $A_i$ is also the first point mass of $B_i$. We further give an inhomogeneous spine decomposition ending at the first surviving particle of generation $h$ in a plane BGW tree conditioned to survive $h$ generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\RR_+$ associated with the limiting continuous-state branching process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting continuous-state population by considering only points with coalescence times greater than $\vareps$. The limiting coalescent point process is the sequence of depths greater than $\vareps$ of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. The question of characterizing this point process in the presence of multiple repeats as $\vareps\to 0$ remains open.

Responsable : Amaury Lambert L’équipe SMILE (Stochastic Models for the Inference of Life Evolution; cf. http://www.proba.jussieu.fr/~smile/) regroupe des mathématiciens, des biologistes et des bio-informaticiens. Les recherches de notre équipe portent principalement sur la modélisation probabiliste de l’évolution. Durant l’année 2011-2012, nous avons notamment fait des contributions importantes, dans le sens croissant des échelles de temps et d’espace, à la génétique des populations hors d’éq...

Ce livre est issu d'un séminaire interdisciplinaire, organisé en mai 2012 à la Fondation Hugot du Collège de France, autour de l'idée de simplexité - propriété fondamentale du vivant de pouvoir inventer des solutions simples aux problèmes que la complexité du réel pose pour sa survie - avancée par Alain Berthoz en 2009 (La Simplexité, éditions Odile Jacob). Ce séminaire réunissait des philosophes, des mathématiciens, des physiciens et des biologistes, des chercheurs en sciences cognitives, en robotique, en didactique, ainsi que des architectes. Le présent ouvrage ne se contente pas de recueillir les articles issus des communications à ce séminaire. Il a été conçu comme un volume "compagnon" du livre d'Alain Berthoz : il vise à replacer au sein du débat théorique entre spécialistes de la modélisation des systèmes complexes un texte plutôt orienté en apparence vers le public général. Les phénomènes qualifiés de simplexes ne sont-ils que l'énumération rhapsodique de dispositifs ou de mécanismes dépourvus d'unité intégrative? Ou manifestent-ils, au contraire, l'exigence d'une nouvelle théorie de la simplexité du vivant, qui puisse ête confrontée aux théories existantes des systèmes complexes et, éventuellement, offrir une alternative à ces théories ?

We seek to construct genetics of branching populations, based upon continuous-state branching processes, also called CB-processes.We get stationary versions of branching trees, by modifying them in two fashions: introducing competition-type interactions between individuals, to regulate population size (`logistic branching process'); applying various conditionings, in the sens of Doob's h-process: conditioning the associated Lévy process to remain in a finite interval, conditioning CB-processes to non-extinction (Q-process), but also CB-processes with interactions, in their diffusion form.We study the fixation probability of a mutant. Using diffusion theory, we propose a unifying framework to compare two classical models and the logistic branching model. Then, we endow each individual with a quantitative trait subject to mutations, and by a micro--macro approach, we follow the evolution of the resident trait (`canonical diffusion of adaptive dynamics').We study the genealogy of CB-processes with immigration, among which lies the aforementioned $Q$-process. We construct branching trees (whose width process is not Markovian), called splitting trees, on which can directly be seen both types of genealogies associated with CB-processes, that were discovered by J.-F. Le Gall and collaborators. We also provide a proof of the Lamperti representation of CB-processes as time-changed Lévy processes.We give a retrospective description of the genealogical structure of CB-processes, and then of splitting trees, as does coalescence theory in modern population genetics.Collaborations in various fields of population biology are also displayed : classical population genetics, ecology of invasions, conservation biology., Nous cherchons à construire une génétique des populations branchantes, basée notamment sur les processus de branchement à espace d'états continu, dits CB-processus. Nous modifions d'abord les arbres branchants afin d'en obtenir des versions stationnaires, de deux façons : en introduisant des interactions de type compétitif entre les individus, de manière à réguler la taille de la population ("processus de branchement logistique"); en appliquant divers conditionnements, au sens des h-processus de Doob : conditionnement du processus de Lévy associé à rester dans un intervalle, conditionnement à la non-extinction des CB-processus (Q-processus), mais aussi des CB-processus avec interactions, sous leur forme diffusion.Nous étudions la probabilité de fixation d'un mutant, question qui conditionne l'évolution de la diversité. En utilisant la théorie des diffusions, nous proposons un cadre unifié permettant de comparer deux modèles classiques et le modèle de branchement logistique. Puis nous munissons les individus d'un trait quantitatif soumis à des mutations, et nous suivons, par une approche micro--macro, l'évolution du trait résident ("diffusion canonique de la dynamique adaptative"). Nous étudions la généalogie associée aux CB-processus avec immigration, dont un cas particulier est le Q-processus cité plus haut. Nous construisons des arbres branchants (dont la largeur n'est pas markovienne), dits \emph{arbres de ramification}, sur lesquels se voient directement les deux types de généalogies associées aux CB-processus, qui ont été découvertes par J.-F. Le Gall et ses collaborateurs. Nous donnons également une démonstration de la représentation de Lamperti des CB-processus comme processus de Lévy changés de temps.Nous décrivons de façon rétrospective la structure généalogique des CB-processus, puis celle des arbres de ramification, comme le fait une des approches phares de la génétique des populations moderne, dite théorie de la coalescence. Des collaborations dans divers domaines de la biologie des populations sont également exposées : génétique des populations classique, écologie des invasions, biologie de la conservation.

Source-sink systems are metapopulations of patches that can be of variable habitat quality. They can be seen as graphs, where vertices represent the patches, and the weighted oriented edges give the probability of dispersal from one patch to another. We consider either finite or source-transitive graphs, i.e., graphs that are identical when viewed from a(ny) source. We assume stochastic, individual-based, density-independent reproduction and dispersal. By studying the path of a single random disperser, we are able to display simple criteria for persistence, either necessary and sufficient, or just sufficient. In case of persistence, we characterize the growth rate of the population as well as the asymptotic occupancy frequencies of the line of ascent of a random survivor. Our method allows to decouple the roles of reproduction and dispersal. Finally, we extend our results to the case of periodic or random environments, where some habitats can have variable growth rates, autocorrelated in space and possibly in time. In the whole manuscript, special attention is given to the example of regular graphs where each pair of adjacent sources is separated by the same number of identical sinks. In the case of a periodic and random environment, we also display examples where all patches are sinks when forbidding dispersal but the metapopulation survives with positive probability in the presence of dispersal, as previously known for a two-patch mean-field model with parent-independent dispersal., Comment: 30 pages

In the first chapter, we study the conditioning of a completely asymmetric Lévy process to remain in a finite interval. The next two chapters are dedicated to continuous-state branching processes, which are time-changed Lévy processes with no negative jumps: genealogy (second chapter), from which we derive Ray-Knight type theorems, and conditioning to be never extinct (third chapter).The last chapter deals with multivariate renewal theory in two natural cases of nested random sets., Dans le premier chapitre, nous étudions le conditionnement d'un processus de Lévy complètement asymétrique à demeurer dans un intervalle fini. Les deux suivants sont consacrés aux processus de branchement à espace d'états continu, qui sont des processus de Lévy sans saut négatif changés de temps : généalogie (deuxième chapitre), dont nous dérivons des théorèmes de type Ray-Knight, et conditionnement à ne jamais s'éteindre (troisième chapitre). Enfin, le dernier chapitre traite de théorie du renouvellement multivariée dans deux cas naturels d'ensembles aléatoires emboîtés.

Assume that individuals alive at time $t$ in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of $n$ DNA sequences belonging to distinct individuals, we consider the number $S_n$ of polymorphic sites (sites at which at least two sequences differ), and the number $A_n$ of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated to coalescent point processes under this assumption. Here, $S_n$ and $A_n$ grow linearly as $n$ grows, with explicit rate. However, when the branch lengths have infinite expectation, $S_n$ grows more rapidly, e.g. as $n \ln(n)$ for critical birth--death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by $k$ individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth--death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher logarithmic series.

The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called `canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to competition in the logistic branching fashion. We start with the study of the particular case of two competing subpopulations (resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival). Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Finally, we apply a limit of weak selection (small mutation steps) to this jump process, that leads to a diffusion process of evolution, called `canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the fitness gradient.