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Sep 23, 2013
09/13

by
Nicolas Champagnat

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In the course of Darwinian evolution of a population, punctualism is an important phenomenon whereby long periods of genetic stasis alternate with short periods of rapid evolutionary change. This paper provides a mathematical interpretation of punctualism as a sequence of change of basin of attraction for a diffusion model of the theory of adaptive dynamics. Such results rely on large deviation estimates for the diffusion process. The main difficulty lies in the fact that this diffusion process...

Source: http://arxiv.org/abs/0903.2345v1

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Jun 28, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes with killing on $[0,\infty)$. We obtain criteria for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. Our approach is based on probabilistic and coupling methods, contrary to the classical approach based on spectral theory results. Our general criteria apply in the case where $\infty$ is entrance and 0...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1510.05794

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Jun 29, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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This article studies the quasi-stationary behaviour of population processes with unbounded absorption rate, including one-dimensional birth and death processes with catastrophes and multi-dimensional birth and death processes, modeling biological populations in interaction. To handle this situation, we develop original non-linear Lyapunov criteria. We obtain the exponential convergence in total variation of the conditional distributions to a unique quasi-stationary distribution, uniformly with...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1611.03010

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Jun 28, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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This article studies the quasi-stationary behaviour of multidimensional birth and death processes, modeling the interaction between several species, absorbed when one of the coordinates hits 0. We study models where the absorption rate is not uniformly bounded, contrary to most of the previous works. To handle this natural situation, we develop original Lyapunov function arguments that might apply in other situations with unbounded killing rates. We obtain the exponential convergence in total...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1508.03161

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Sep 19, 2013
09/13

by
Nicolas Champagnat; Amaury Lambert

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We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\ge 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta...

Source: http://arxiv.org/abs/1009.0680v1

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Jun 29, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1611.02473

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Sep 18, 2013
09/13

by
Nicolas Champagnat; Sylvie Roelly

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A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process|the conditioned multitype Feller branching diffusion are then proved. The general case is first considered, where the...

Source: http://arxiv.org/abs/0707.3504v2

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Sep 21, 2013
09/13

by
Nicolas Champagnat; Amaury Lambert

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We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate $\theta$ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time...

Source: http://arxiv.org/abs/1108.4812v1

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Jun 29, 2018
06/18

by
Julien Claisse; Nicolas Champagnat

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We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function...

Topics: Optimization and Control, Probability, Mathematics

Source: http://arxiv.org/abs/1607.08046

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Jun 28, 2018
06/18

by
Nicolas Champagnat; Benoît Henry

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We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate b. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\\theta$ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1509.06500

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Sep 22, 2013
09/13

by
Nicolas Champagnat; Sylvie Méléard

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We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This...

Source: http://arxiv.org/abs/0812.1655v1

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Sep 20, 2013
09/13

by
Nicolas Champagnat; Amaury Lambert

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The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE describing the evolution of the dominant type, called the ``canonical equation of adaptive dynamics.'' Here, in order to include the effect of stochasticity (genetic drift), we consider self-regulated randomly fluctuating populations subject to...

Source: http://arxiv.org/abs/math/0601643v4

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Jun 30, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the $Q$-process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1404.1349

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Jun 27, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one dimensional strict local martingale diffusions coming down from infinity. We prove under mild assumptions that their expectation at any positive time is uniformly bounded with...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1506.02385

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Jun 30, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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We study the quasi-stationary behavior of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka-Volterra birth and death processes and Feller diffusions with competitive Lotka-Volterra interaction. To this aim, we develop original non-linear Lyapunov criteria involving two Lyapunov functions, which apply to general Markov processes.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1704.01928

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Sep 21, 2013
09/13

by
Nicolas Champagnat; Amaury Lambert

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We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the...

Source: http://arxiv.org/abs/0706.4157v1

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Jun 29, 2018
06/18

by
Nicolas Champagnat; Denis Villemonais

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We provide an original and general sufficient criterion ensuring the exponential contraction of Feynman-Kac semi-groups of penalized processes. This criterion is applied to time-inhomogeneous one-dimensional diffusion processes conditioned not to hit 0 and to penalized birth and death processes evolving in a quenched random environment.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1603.07477

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Jul 20, 2013
07/13

by
Nicolas Champagnat; Sylvie Méléard

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The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death,...

Source: http://arxiv.org/abs/math/0604041v1

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Pierre-Emmanuel Jabin

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We consider a integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between traits occur from competition for resources whose concentrations depend on the current state of the population. Following the formalism of\cite{DJMP}, we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the...

Source: http://arxiv.org/abs/1006.0803v1

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Pierre-Emmanuel Jabin

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We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space $H^{3/4}$.

Source: http://arxiv.org/abs/0904.1119v1

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Pierre-Emmanuel Jabin

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We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^p$ bounds for the solution of the corresponding Fokker-Planck PDE, which can be proved separately. This allows a great flexibility regarding the method...

Source: http://arxiv.org/abs/1303.2611v1

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Jun 28, 2018
06/18

by
Fabien Campillo; Nicolas Champagnat; Coralie Fritsch

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We present two approaches to study invasion in growth-fragmentation-death mod- els. The first one is based on a stochastic individual based model, which is a piecewise deterministic branching process with a continuum of types, and the second one is based on an integro-differential model. The invasion of the population is described by the survival probability for the former model and by an eigenproblem for the latter one. We study these two notions of invasion fitness, giving different...

Topics: Probability, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1509.08619

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Pierre-Emmanuel Jabin; Gael Raoul

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We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in two previous articles to prove the convergence to a unique stable equilibrium.

Source: http://arxiv.org/abs/1006.5415v1

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Sep 18, 2013
09/13

by
Nicolas Champagnat; Amaury Lambert; Mathieu Richard

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In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative...

Source: http://arxiv.org/abs/1209.6205v2

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Persi Diaconis; Laurent Miclo

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We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual's birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator \Pi\ can be expressed as the product of "universal" polynomials of two variables, depending on each type's size but not on the...

Source: http://arxiv.org/abs/1202.4715v1

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Jun 29, 2018
06/18

by
Fabien Campillo; Nicolas Champagnat; Coralie Fritsch

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We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we...

Topics: Probability, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1601.02516

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Sep 22, 2013
09/13

by
Nicolas Champagnat; Pierre-Emmanuel Jabin; Sylvie Méléard

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We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka-Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources...

Source: http://arxiv.org/abs/1302.0552v1

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Jun 27, 2018
06/18

by
Martina Baar; Anton Bovier; Nicolas Champagnat

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We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population $(K\to \infty)$ size, rare mutations $(u\to 0)$, and small mutational effects $(\sigma\to 0)$, proving convergence to the canonical equation of...

Topics: Populations and Evolution, Quantitative Biology, Mathematics, Probability

Source: http://arxiv.org/abs/1505.02421

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Christophe Chipot; Erwan Faou

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We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. in [6] for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic potential. The method is mathematically extended to general least-squares problems and provides an alternative approximation method in cases where the original least-squares problem is computationally not tractable, either because of its ill-posedness or its...

Source: http://arxiv.org/abs/1006.4996v1

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Jun 29, 2018
06/18

by
Nicolas Champagnat; Abdoulaye Coulibaly-Pasquier; Denis Villemonais

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We consider diffusion processes killed at the boundary of Riemannian manifolds. The aim of the paper if to provide two different sets of assumptions ensuring the exponential convergence in total variation norm of the distribution of the process conditioned not to be killed. Our first criterion makes use of two sided estimates and applies to general Markov processes. Our second criterion is based on gradient estimates for the semi-group of diffusion processes.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1603.07909

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Sep 23, 2013
09/13

by
Nicolas Champagnat; Régis Ferrière; Sylvie Méléard

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We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each...

Source: http://arxiv.org/abs/math/0510453v1

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Jun 30, 2018
06/18

by
Mireille Bossy; Nicolas Champagnat; Helene Leman; Sylvain Maire; Laurent Violeau; Mariette Yvinec

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The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g.Bossy et al 2009, Mascagni & Simonov 2004}). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized...

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1411.2304