Probability and Discrete Mathematics in Mathematical Biology - IMS

Probability and Discrete Mathematics in Mathematical Biology
(14 Mar - 10 Jun 2011)

~ Abstracts ~

Modeling sexually transmitted infections: Chlamydia as a case study
Christian L. Althaus, University of Bern, Switzerland

Mathematical models of sexually transmitted infections (STIs) are traditionally described by a set of ordinary differential equations (ODEs). Such deterministic models often assume that the duration of infection and the length of sexual partnerships are exponentially distributed. In this talk, I will critically evaluate these assumptions in mathematical models of chlamydia transmission, the most common bacterial STI in developed countries. First, I will show how the duration of infection can be inferred from longitudinal data and how it affects the estimation of the transmission probability from cross-sectional data. Second, I will present data describing the characteristic distribution of sexual partnership durations. Using the individual-based modeling framework Rstisim, it is illustrated that taking into account realistic distributions of sexual partnership durations offers an alternative description of the core group concept: heterogeneity in sexual behavior simply arises as an emergent property of the stochastic simulations. The simulations allow a detailed description of the transmission dynamics of chlamydia, raising questions about the duration of naturally acquired immunity and the impact of control efforts.

How can we understand ancient evolutionary transitions such as multicellularity and cellular differentiation?
Homayoun Bagheri, University of Zurich, Switzerland

All complex life forms that grow beyond microscopic scales are multicellular. Multicellularity and differentiation are fundamental organizational themes in development, and have played a key role in the evolution of life. How does one study the deep evolutionary transitions that led to these traits? In this talk, I use bacteria as a case study to understand the origins of multicellularity in simple life forms. I summarize results from phylogenetic reconstruction and genome comparisons, theoretical models of population dynamics, strain isolation, and laboratory experiments that shed light on different aspects of the evolution of multicellularity and cellular differentiation.

Quasi-stationary distributions in biology
Andrew Barbour, University of Zurich, Switzerland

The mathematical study of quasi-stationary distributions for random processes with certain extinction was originally stimulated by work of Ewens (1963) on diffusion processes in population genetics. Since then, a substantial body of theory has been developed, with many surprising discoveries: for instance, a single absorbing process may well have many different quasi-stationary distributions. The aim of this talk is to get back closer to the biological origins of the subject. In particular, a set of conditions is presented under which the quasi-equilibrium distribution is unique, and can easily be calculated: indeed, it is then close to the `return' distribution, as discussed by Ewens but frowned upon by some subsequent authors.

(Joint work with P. K. Pollett)

Ancestry in the face of competition, v0.1: Directed random walk on the directed percolation cluster
Matthias Birkner, Johannes Gutenberg University of Mainz, Germany

The spatial embeddings of genealogies in models with fluctuating population sizes and local regulation are relatively complicated random walks in a space-time dependent random environment. They seem presently not well understood. We use the supercritical discrete-time contact process on $Z^d$ as the simplest non-trivial example of a locally regulated population model and study the dynamics of an ancestral lineage sampled at stationarity, viz. a directed random walk on a supercritical directed percolation cluster. We prove a LLN and an annealed CLT for such a walk via a regenerative approach. Furthermore, we will discuss approaches to extend these results to larger samples and to more general models that allow multiple occupancy of sites.

Based on joint work in progress with J. Cerny, A. Depperschmidt and N. Gantert.

Order-invariance and random partial orders
Graham Brightwell, London School of Economics, UK

We consider random processes that "grow"' a partially ordered set (poset) by adding a new maximal element at each step. Such processes can be thought of as growing a random poset, but they also include processes that generate the elements of some fixed poset in a random order. We are interested in processes that satisfy an "order-invariance" property: informally, conditioning on the poset that we see after some fixed number of steps, each possible order of generation is equally likely. The main result is a description of extremal order-invariant processes.

Dynamic networks in dynamic populations
Tom Britton, Stockholm University, Sweden

We study a randomly growing population (where new individuals are born and old die) in which edges between individuals appear and disappear randomly over time. A specific feature of the model is that individuals are born with a "social index" which affects how frequently they create new neighbours. For this model we study asymptotic properties valid after a long time: the degree distribution, degree correlation and a threshold condition determining whether a giant connected component exists or not. (Joint work with Mathias Lindholm and Tatyana Turova)

On optimal estimation of a nonsmooth functional
Tony Cai, University of Pennsylvania, USA

In this talk I will discuss some recent work on optimal estimation of nonsmooth functionals. These problems exhibit some interesting features that are significantly different from those that occur in estimating conventional smooth functionals. This is a setting where standard techniques fail. I will discuss a newly developed general minimax lower bound technique that is based on testing two fuzzy hypotheses and illustrate the ideas by focusing on the problem of optimal estimation of the L_1 norm of a high dimensional normal mean vector. An estimator is constructed using approximation theory and Hermite polynomials and is shown to be asymptotically sharp minimax. This is joint work with Mark Low.

Go forth and multiply?
Steve Evans, University of California at Berkeley, USA

Organisms reproduce in environments that vary in both time and space. Even if an individual currently resides in a region that is typically quite favorable, it may be optimal for it to "not put all its eggs in the one basket" and disperse some of its offspring to locations that are usually less favorable because the effect of unexpectedly poor conditions in one location may be offset by fortuitously good ones in another. I will describe joint work with Peter Ralph and Sebastian Schreiber (both at University of California, Davis) and Arnab Sen (Cambridge) that combines stochastic differential equations, random dynamical systems, and even a little elementary group representation theory to explore the effects of different dispersal
strategies.

Rates of convergence of variance-Gamma approximations via Stein's Method
Robert Gaunt, University of Oxford, UK

The Variance-Gamma distributions form a quite large family of distributions, which includes the Normal, Gamma and Laplace distrbutions as subclasses. In this talk we see how Stein's method can be used to obtain a bound on the error, in a weak convergence setting, in approximating an aympotically Variance-Gamma distributed statistic by their limiting distribution. The bound we obtain is order 1/n which is faster than the 1/n^(1/2) rate that may have been expected by the Berry-Essen theorem.

We then go on to considering the problem of approximating statistics that are asymptotically distributed as the product of k independent standard normals. We use Stein's method and a generalisation of the zero-bias coupling to acheive our approximation results.

The Lambda coalescent genealogy with selection
Bob Griffiths, University of Oxford, UK

A coalescent dual process can be derived for a class of continuous-time Cannings models with viability selection. In these models, individuals may give birth to multiple offspring whose survival depends on both the parental genotype and the brood size. In the limit of infinite population size the non-neutral Cannings models converge to a Lambda-Fleming-Viot process. The dual is a branching-coalescing process which follows the typed ancestry of genes backwards in time with real and virtual lineages. This is joint research with Alison Etheridge and Jay Taylor.

Diffusion processes and coalescent trees
Bob Griffiths, University of Oxford, UK

Wright-Fisher diffusion process models for evolution of neutral genes have the coalescent process underlying them. Models are reversible with transition functions having a diagonal expansion in orthogonal polynomial eigenfunctions of dimension greater than one, extending classical one-dimensional diffusion models with Beta stationary distribution and Jacobi polynomial expansions to models with Dirichlet or Poisson Dirichlet stationary distributions. Another form of the transition functions is as a mixture depending on the mutant and non-mutant families represented in the leaves of the infinite-leaf coalescent tree. Subordinated Wright-Fisher diffusion processes are jump diffusions. An example with an Inverse Gaussian subordinator has a fascinating connection with the Jacobi Poisson Kernel in orthogonal function theory. This is joint research with Dario Spano.

Branching diffusions in random environment
Martin Hutzenthaler, Ludwig Maximilian University of Munich, Germany

Individuals in a branching process in random environment (BPRE) branch independently of each other and the offspring distribution changes randomly over time. The variation in the offspring distribution models fluctuations in environmental conditions. Here we study the diffusion approximation of BPREs, which we denote as branching diffusion in random environment (BDRE). The one-dimensional BDRE (modelling a panmictic population) turns out to have strong analytical properties similar to Feller's branching diffusion. We obtain a phase transition for the survival probability in the subcritical regime which is well-known for BPREs. In addition we establish a phase transition in the super-critical regime which has not been reported for BPREs yet. Finally we analyse the long-time behavior of interacting branching diffusions in random environment (modelling structured populations).

Threshold phenomena in k-dominant skylines of random samples
Hsien-Kuei Hwang, Academia Sinica, Taiwan

Skylines emerged as a useful notion in database queries for selecting representative groups in multivariate data samples for further decision making, multi-objective optimization or data processing, and the k-dominant skylines were naturally introduced to resolve the abundance of skylines when the dimensionality grows or when the coordinates are negatively correlated. We first present in this talk an asymptotic vanishing property for the number of k-dominant skyline points under several random models when the dimensionality is fixed. We then present more positive results by considering the hypercubes but with growing dimensionality for which very sharp threshold phenomena will be given. (This talk is based on joint work with Wei-Mei Chen and Tsung-Hsi Tsai.)

Epidemics and rumours: the effect of network structure on transmission dynamics
Valerie Isham, University College London, UK

The basic (SIR) epidemic model for the spread of infection in a homogeneously-mixing population is a special case of a more general stochastic model used for the spread of information (a `rumour?). In both cases it is well known that there is a threshold for widespread transmission.

More generally, for both epidemics and rumours, there is particular interest in using a network to represent population structure. This ensures that some pairs of individuals are never in contact, and direct spread between them cannot occur. Natural applications are to the spread of infection or information on social networks.

In this talk, I will review simple epidemic and rumour models, and describe networks generated by a range of random mechanisms. I will then discuss the effect of different network structures on the transmission dynamics of epidemics or rumours on networks and, in particular, the effect of different network properties on thresholds for widespread transmission.

Convergence of rescaled competing species processes to a class of SPDEs
Sandra Kliem, Eindhoven University of Technology, The Netherlands

In this talk we construct a sequence of rescaled perturbations of voter processes in dimension d=1. We consider long-range interactions and show that the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows.

The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. As an example we show that the new results cover spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) for parameters approaching one.

Multi-scale analysis of the hierarchically interacting heavy-tailed Cannings models
Anton Klymovskiy, Eindhoven University of Technology, The Netherlands

We study a class of dynamical stochastic models for genetics of spatially extended populations. The models take into account the effects of migration, local reproduction, and occasional global extinction-colonisation events that affect the whole patches of the geographical space. These models are motivated by continuous time-mass limits of the Cannings model in a heavy-tailed reproduction regime (dual to the Lambda-coalescent). We employ the renormalisation group type of multi-scale analysis and show that, depending on the migration and the resampling rates, the ergodic behaviour of the process displays either (1) coexistence of several allelic types within colonies, or (2) clustering -- emergence of mono-type colonies. (Based on joint work with A. Greven, F. den Hollander, and S. Kliem.)

Survival and extinction of caring double-branching annihilating random walk
Noemi Kurt, Technische Universität Berlin, Germany

Branching annihilating random walk (BARW) is a generic term for a class of interacting particle systems in which, as time evolves, particles execute random walks, produce offspring (on neighbouring sites) and disappear when they meet other particles. Much of the interest in such models stems from the fact that they typically lack a monotonicity property called 'attractiveness', which in general makes them exceptionally hard to analyse and in particular highly sensitive in their long-time behaviour to even the slightest alterations of the branching and annihilation mechanisms. In this talk, I will review some of the models and their survival- and extinction behaviour, in particular present the so-called 'caring' double-branching annihilating random walk (cDBARW), which is one of the few variants of BARW where (under some assumptions) we can prove that, for sufficiently large branching rate, the process survives with positive probability, and for sufficiently small branching rate dies out almost surely, suggesting the existence of a phase transition. This is joint work with Jochen Blath (Berlin).

Stochastic dynamics on hypergraphs and the majority rule
Nicolas Lanchier, Arizona State University, USA

In the early seventies, Frank Spitzer and Roland Dobrushin independently introduced the theoretical framework of interacting particle systems in order to model the dynamics of physical and biological phenomena that are inherently spatial. The main objective of research in this area is to understand the macroscopic behavior and spatial patterns that emerge from the microscopic interactions that dictate the dynamics of infinite systems of components. Following the approach of Spitzer and Dobrushin, we introduce a spatially explicit version of the majority rule model proposed by socio-physicist Serge Galam. Interestingly, whereas the dynamics of traditional particle systems depend on the topology of a connected graph (the interaction network), the dynamics of spatial versions of the majority rule model depend on hypergraph structures. Numerical simulations suggest that the transition from graphs to hypergraphs significantly affects the spatial correlations of the system, and we present analytical results in support of this picture. In addition, spatial majority rules are, as far as we know, the first examples of stochastic dynamics dictated by hypergraphs and the aim of this talk is also to promote this new theoretical framework as a new modeling strategy to describe general sociological systems. This is a joint work with Jared Neufer.

The Axelrod model for the dissemination of culture revisited
Nicolas Lanchier, Arizona State University, USA

The Axelrod model is a stochastic process based on the voter model which, in addition to social influence, also accounts for homophily, the tendency to interact more frequently with individuals which are more similar. Each individual is characterized by a set of cultural features, and pairs of neighbors interact at a rate proportional to the number of features they share, which results in the interacting pair having one more cultural feature in common. The Axelrod model has been extensively studied during the past ten years based on numerical simulations and simple mean-field treatments while there is a lack of analytical results for the spatial model. This talk gives rigorous clustering and coexistence results about the one-dimensional system that confirm some of the conjectures formulated by statistical physicists and social scientists.

An algorithmic approach to quantify reticulation in evolution
Simone Linz, University of Tuebingen, Germany

The 200th anniversary of Charles Darwin?s birth was celebrated widely around the world in 2009. Darwin?s pioneering work laid the foundation for our current understanding of evolution and, ever since, biologists have been interested in the reconstruction of evolutionary trees. However, trees do not always best represent evolution. For example, a comparison of gene trees representing a set of taxa and reconstructed for different genetic loci sometimes reveals conflicting tree topologies. These discrepancies may be due to reticulation events such as horizontal gene transfer and hybridization. Although reticulation is widely accepted as a driving force in the innovation and evolution of genomes, its impact on the evolutionary process and the phylogeny of species remains controversial. Consequently, a basic problem, which is known to be NP-hard, is to quantify the extent to which reticulation has influenced the evolutionary history. In this talk, we will present an exact approach to calculate the minimum number of reticulation events for when the input are rooted binary or rooted non-binary phylogenetic trees.

Vertices of high degree in the preferential attachment tree
Malwina Luczak, London School of Economics, UK

The preferential attachment tree is the most basic model of evolving web graphs and social networks. At each stage of the process, a new vertex is added and joined to one of the existing vertices, with each vertex chosen with probability proportional to its current degree. In probability theory, this is also known as a Yule process.

Much is known about this model, including the fact that the numbers of vertices of each small degree follow a ``power law''. Here we study in detail the degree sequence of the preferential attachment tree, looking at the vertices of large degrees as well as the numbers of vertices of each fixed degree.

Our method is based on bounding martingale deviations, using exponential supermartingales.

This is joint work with Graham Brightwell.

Uncovering latent structure in valued graphs: a variational approach
Mahendra Mariadassou, INRA (French National Institute for Agricultural Research), France

As more and more network-structured data sets are available, the statistical analysis of valued graphs has become common place. Looking for a latent structure is one of the many strategies used to better understand the behavior of a network. Several methods already exist for the binary case. We present a model-based strategy to uncover groups of nodes in valued graphs. This framework can be used for a wide span of parametric random graphs models and allows to include covariates. Variational tools allow us to achieve approximate maximum likelihood estimation of the parameters of these models. We provide a simulation study showing that our estimation method performs well over a broad range of situations. We apply this method to analyze host--parasite interaction networks in forest ecosystems.

Random modeling of adaptive dynamics for sexual populations
Sylvie Méléard, École Polytechnique, France

We study models describing the evolution of a sexual (diploid) population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. We take into account the genetics of the reproduction. Each individual is characterized by two allelic traits and the associated phenotypic trait. The population is described as a point measure valued process with support on the genotype space. Its dynamics is a birth and death dynamics with selection and Mendelian rule in the reproduction and competition between individuals. Allelic mutations may occur during the reproduction events. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of the scales, and if the mutation steps are small, the population process is approximated in a long time scale by a jump process jumping from a monomorphic homozygote equilibrium to another one. This study involves a three-types diploid nonlinear dynamical system, which is studied using small perturbations of a neutral case. This work shows in particular that at this scale and before an evolutionary singularity, heterozygote superdominance cannot occur. This work is a joint work with Pierre Collet (Ecole Polytechnique) and Hans Metz (Leiden University).

Coalescent processes derived from compound Poisson population models
Martin Möhle, University of Tübingen, Germany

The compound Poisson subclass of Cannings population models is analyzed. The models in the domain of attraction of the Kingman coalescent are characterized and it is shown that compound Poisson models are never in the domain of attraction of any other continuous-time coalescent process. Results are obtained characterizing which of these models are in the domain of attraction of a discrete-time coalescent with simultaneous multiple mergers of ancestral lineages.

Reconstruction of Markov processes and networks in phylogenetics and genetics
Elchanan Mossel, University of California at Berkeley, USA

We will discuss some basic stochastic genetic models and related inference problems. A rough outline of the lectures follows:

1. Stochastic Phylogenetic and genetic models.
2. Genetic distances and their estimates.
3. Ancestral reconstruction.
4. Phylogenetic reconstruction.
5. Mixtures, gene trees and other advanced topics.

Talagrand's concentration inequality
Daniel Paulin, National University of Singapore

Talagrand's concentration inequality states that if X_1, ... X_n are independent complex variables with |X_i|<=1, and F is a 1-Lipschitz convex function, then for any t, P(|F(X)-MF(X)|>t) < C exp(-c*t^2), here MF(X) is the median of F(X).

In this talk, I will prove Talagrand's concentration inequality using the original idea of Talagrand, i.e. by induction on the dimension n.

The tree-valued Fleming-Viot process with mutation and selection
Peter Pfaffelhuber, University of Freiburg, Germany

We construct a tree-valued Markov process describing the evolution of genealogical relationships in populations of constant size given by a Moran model or the limiting Fleming-Viot process including mutation and selection. We shed some light on the state space of the process, and state the well-posedness of the corresponding martingale problem, prove ergodicity and a Girsanov-type result. As an application, the construction allows to compute genealogical distances of a pair of individuals under weak selection. This is joint work with Andrej Depperschmidt and Andreas Greven.

Limit theorems for chain-binomial population models
Phil Pollett, University of Queensland, Australia

I will describe a class of chain-binomial models of use in studying metapopulations (population networks). Limit theorems will be presented for time-inhomogeneous Markov chains that share the salient features of these models: a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure. Recent work on metapopulations with patch-dependent extinction probabilities will be discussed.

[This is joint work with Fionnuala Buckley and Ross McVinish]

Growth rate in simple and complex epidemic models with applications to pandemic influence
Andrea Pugliese, University of Trento, Italy

Deterministic homogeneous epidemic models have been successfully fit to several data of incidence of infectious diseases; often, the exponential growth rate r is estimated, and sometimes the overall epidemic size, and the results are reported in terms of the inferred reproduction ratio R0. I examine a possible reason for the success of this approach, and the consequent interpretation of R0, in terms of more complex epidemic models. First, I show equations for R0 and the final size in deterministic multigroup models. Then, I consider their stochastic counterparts: through simulations, I compare the values of r and R0 found for these models with those obtained for a homogeneous model fitted to simulated data; and I discuss possible heuristics as the number of subgroups is large. Finally, I look at the output of complex simulation models used for modelling pandemic influenza, attempting to interpret the estimates of R0 in terms of multigroup models.
This is a report of ongoing work with Antonella Lunelli (University of Trento) and Gianpaolo Scalia Tomba (University of Roma Tor Vergata).

Statistics for alignment-free sequence comparison
Gesine Reinert, University of Oxford, UK

Large-scale comparison of the similarities between two biological sequences is a major issue in computational biology. We discuss some commonly used statistics for their task and introduce two new statistics; using Stein's method we also discuss their asymptotic behaviour. Furthermore we study their power under suitable alternatives which are inspired by next generation sequencing.

This is joint work with David Chew, Shih-Yen Ku, Yihui Luan, Fengzhu Sun, Lin Wan, Mike Waterman, and Zhiyuan Zhai.

Stein's method in high dimensions
Adrian Roellin, National University of Singapore

We develop Stein's method for high-dimensional Gaussian approximation of smooth functionals of families of not necessarily independent random variables. This allows to obtain results that go beyond the standard univariate CLT. It also allows for the study of universality under dependence in many contexts. We give applications to urn models, the Currie-Weiss model and last-passage percolation on thin rectangles.

On convergence of entropy of convolution of max stable laws and their max domains
Ali Saeb, University of Mysore, India

Max stable laws are limit laws of linearly normalized partial maxima of independent, identically distributed (iid) random variables (rvs). These are analogous to stable laws which are limit laws of normalized partial sums of iid rvs. In this paper, we study entropy limit theorems for convolution of max stable laws and their max domains under linear normalization.

On the Greedy Algorithm for Mean Field Traveling Salesman Problem
Farkhondeh Sajadi, Indian Statistical Institute, India

Joint work with Antar Bandyopadhyay.

On the one dimensional "learning from neighbours" model
Anish Sarkar, Indian Statistical Institute, India

We consider a model of a discrete time "interacting particle system" on the integer line where in?nitely many changes are allowed at each instance of time. We describe the model using chameleons of two di?erent colours, viz ., red (R) and blue (B). At each instance of time each chameleon performs an independent but identical coin toss experiment to decide whether to change its colour or not.

We show that starting from any initial translation invariant distribution of colours the process converges to a limit of a single colour, i.e., even at the symmetric case there is no "coexistence" of the two colours at the limit.

Moreover we show that starting with an i.i.d. colour distribution with density p of one colour (say red), the limiting distribution is all red with probability which is continuous in p and for p "small" it is bigger than p. The last result can be interpreted as the model favours the "underdog".

Linking network structure and stochastic dynamics to neural activity patterns involved in sleep-wake regulation
Deena Schmidt, Ohio State University, USA

Sleep and wake states are each maintained by activity in a corresponding neuronal network, with mutually inhibitory connections between the networks. In infant mammals, the durations of both states are exponentially distributed, whereas in adults, the wake states yield a heavy-tailed distribution. What drives this transformation of the wake distribution? Is it the altered network structure or a change in neuronal dynamics? What properties of the network are necessary for maintenance of neural activity on the network and what mechanisms are involved in transitioning between sleep and wake states? We explore these issues using random graph theory, specifically looking at stochastic processes occurring on random graphs, and also by investigating the accuracy of predictions made by deterministic approximations of stochastic processes on networks.

Modeling the genealogy of populations using coalescents with multiple mergers: A survey
Jason Schweinsberg, University of California at San Diego, USA

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in time. Under standard assumptions, this process can be modeled by Kingman's coalescent, in which each pair of lineages merges at rate one. However, if some individuals have large numbers of offspring or if the population is affected by selection, then many ancestral lineages may merge at one time. In this talk, we will introduce the family of coalescent processes with multiple mergers and discuss some circumstances under which populations can be modeled by these coalescent processes. We will also describe how genetic data, such as the number of segregating sites and the site frequency spectrum, would be affected by multiple mergers of ancestral lines, and we will discuss the implications for statistical inference.

Urn models on one-dimensional integer lattice
Debleena Thacker, Indian Statistical Institute, India

This is a joint work with Antar Bandyopadhyay, Indian Statistical Institute, New Delhi.

Large graph limit and Volz' equations for an SIR epidemic spreading on a configuration model graph
Viet Chi Tran, Université Lille 1, France

We consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. This allows us to provide a rigorous proof of the equations obtained by Volz (2008) where the spread of the epidemics is summed up into 5 ordinary differential equations only.

Historical particle systems for population models with past dependence and logistic competition
Viet Chi Tran, Université Lille 1, France

A population structured by traits is considered, where individuals give birth and die according to rates that may depend on their past (ages or ancestral lineages allowing to model social interactions as cooperative breeding...) A logistic-like competition is also taken into account, where dead ancestors may also exert competition pressure via resource depletion. We construct a historical particle system and are interested in large population limits, when individuals have small masses and allometric demographies (short lives and reproduction times). In the limit, we obtain historical superprocesses as the ones considered by Perkins (1995) and which describes the evolving genealogies. This is a joint work with Sylvie Meleard.

The rate of Muller's ratchet: A new formula for an old problem
Anton Wakolbinger, Goethe University Frankfurt, Germany

In a Wright-Fisher population of size N in which weakly deleterious mutations of a multiplicative fitness disadvantage of (1-alpha) are accumulated at rate lambda per individual per generation, what is the speed at which the mean number of mutations per individual in the population increases? For a Fleming-Viot version of this system, we give some heuristics revealing that N*alpha*exp(-lambda/alpha) is a quantity that is critical for this speed, and we corroborate this by an asymptotic formula based on martingale arguments. This is joint work with Peter Pfaffelhuber.

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