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Workshop YEP VIII 2011 "Stochastic Models for Population Dynamics" March 14-18, 2011
MONDAY, MARCH 14
TUESDAY, MARCH 15
WEDNESDAY, MARCH 16
THURSDAY, MARCH 17
FRIDAY, MARCH 18
Agnes Backhausz A random model of publication activity A discrete model, inspired by
publication activity, is introduced. It includes an increasing number of
objects (authors) equipped with positive weights, which also increase with
time. We start with one author with random initial weight. At each step a
randomly chosen group of authors produces a new publication. The weights of
the members of this group are increased by random quantities. Then a new
author comes with random initial weight. The probability distributions are
chosen such that this becomes a preferential attachment model. It is proven
that the empirical weight distribution converges weakly with probability 1,
and the limit law has a regularly varying tail. Methods of martingale theory
and renewal theory are applied in the proofs [1, 2]. References Roman Berezin Survival and Extinction of the Contact Random Walk We study a nearest neighbour contact process fused with an independent random walk in d ≥3. It is known that the critical value of the birth rate of the contact process starting with a single particle, necessary for inde nite survival approaches 1 as a suitable scaling parameter approaches in nity. The main point of interest is to nd sharp asymptotic for how close this critical value is to 1. This result give further a clue to improving the results of Konno (1995) for the Contact Processes with Rapid Stirring. Ted Cox Limit theorems for voter model perturbations We present joint work with Rick Durrett and Ed Perkins on limit theorems for a class of interacting particle systems (spatial competition models) we call voter model perturbation. This work includes a measure-valued limit theorem and a hydrodynamic type limit theorem. In each case it is possible to "invert" the limit and prove survival and/or coexistence results for several interesting competition models. Andrej Depperschmidt Ancestry in the face of competition: Directed random walk on the directed percolation cluster The spatial embeddings of genealogies in models with
fluctuating population sizes and local regulation are complicated random
walks in a space-time dependent random environment (RWRE). Such RWRE are
presently not well understood. We consider the supercritical discrete time
contact process on Zd which is the simplest non-trivial
example of a locally regulated population model. We study the RWRE performed
by the ancestral lineage of an individual sampled from the invariant
distribution. We prove a LLN and an annealed CLT via a regenerative
approach. Leif Döring Limit Properties of Mutually Catalytic Branching Models A 2-parameter class of mutually catalytic branching models interpolating between various probabilistic models such as the voter process, the stepping stone model, and a parabolic Anderson model shall be discussed. The parameters govern correlations and strength of the underlying branching mechanism. Based on an observation taken from a recent work with Jochen Blath and Alison Etheridge, the focus lies on negative correlations. Surprisingly, this allows a "in a nutshell" presentation of results and proofs for limit results for finite and infinite branching rate processes Patsy Haccou (Problem Session) Stochastic models in Ecology and Evolution
We will consider models of Sandra Kliem Convergence of Rescaled Competing Species Processes to a Class of SPDEs We construct a sequence of rescaled perturbations of voter processes in dimension d=1 whose approximate densities are tight. We combine both long-range models and fixed kernel models in the perturbations. In the case of long-range interactions only, 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. As an example we show that the results cover the stochastic spatial Lotka-Volterra model for parameters approaching one. Amaury Lambert Population dynamics and evolution in the trait space We will start with a brief study of some well-known deterministic models for populations structured by types. Then we will set up a general framework for probabilistic models of structured populations. In these models, all types, but one, eventually become extinct, so we will (prefer to) assume the presence of mutations. Our goal is to understand the evolution of such populations, seen as the sequential substitution of ancient types by recent (often more fit) types. We will first study different sorts of large population limits and explain how they are used by biologists, in addition to simplifying our problem. We will also display different concepts of a nearly neutral population. The last part will consist in showing how the recent field of adaptive dynamics uses these tools to model evolution. Mickaël Launay Interacting Urn Models The aim of this paper is to study the asymptotic
behavior of strongly reinforced interacting urns with partial memory
sharing. The reinforcement mechanism considered is as follows: draw at each
step and for each urn a white or black ball from either all the urns
combined (with probability p) or the urn alone (with probability 1−p)
and add a new ball of the same color to this urn. The probability of drawing
a ball of a certain color is proportional to wk where k
is the number of balls of this color. The higher the p, the more
memory is shared between the urns. The main results can be informally stated
as follows: in the exponential case wk =
p k, if Tobias Müller Random geometric graphs If we pick points X1,....., Xn at random from d-dimensional space (i.i.d. according to some probability measure) and fix a r > 0, then we obtain a random geometric graph by joining points by an edge whenever their distance is < r. I will give a brief overview of some of the most important results on random geometric graphs and then describe some of my own work on Hamilton cycles, the chromatic number, and the power of two choices in random geometric graphs. Francesca Nardi An upper bound for front propagation velocities inside moving populations Abstract Leonid Petrov Infinite-dimensional diffusions related to the two-parameter Poisson-Dirichlet distributions The one-parameter infinitely many neutral alleles diffusion model was introduced by Ethier and Kurtz in 1981. This is a family of infinite-dimensional diffusion processes preserving the one-parameter Poisson-Dirichlet distributions PD(θ). It is known that these diffusions are approximated by certain random walks on partitions (a Moran-type discrete population model). A natural algebraic combinatorial interpretation of these random walks is given which allows to extend the infinite-dimensional diffusions to the case of the two-parameter Poisson-Dirichlet distributions PD(α, θ). The Markov generator for the two-parameter family of infinite-dimensional diffusions is explicitly computed. |
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Lorenz Pfeifroth Frogs in a random environment In the first part in this talk we consider the frog model on the integers where the underlying random walk, which every active frog performs, is an arbitrary nearest neighbor Markovian random walk on Z with drift to the right. The question, we are interested in, is if the origin is visited infinitely often by active frogs with probability 1 or not. We give a necessary and sufficient condition that this will happen. Also we present a 0-1 law for this model. In the second part we consider the frog model in a random environment which is the mixture of the normal frog models. We give recurrence criteria for such a model and derive 0-1 law too and show that the recurrence of such a model only depends on the distribution of the starting configuration of frogs and not on the distribution of the jumping probability of the underlying random walk. Thomas Rippl Pathwise uniqueness for the stochastic heat equation with Holder continuous coefficient: the colored-noise-case We consider the stochastic heat equation in \R_+ \times \R^d with multiplicative noise. The noise is white in time and colored in space with correlation kernel $k(x,y) \leq const ( |x-y|^{- \alpha } +1)$ for a fixed $\alpha \in (0, 2 \wedge d)$ and $x,y \in \R^d$. We prove that if the noise coefficient is Holder-continuous of order \gamma and satisfies $\alpha < 2(2\gamma -1)$ then the equation has a pathwise unique solution. This was conjectured by Mytnik and Perkins in 2009. Pieter Trapman SIR epidemics on random intersection graphs We consider a model for the spread of a stochastic
SIR (Susceptible --> Infectious --> Removed) epidemic on a network of
individuals described by a random intersection graph. The number of cliques
a typical individual belongs to follows a mixed-Poisson distribution, as
does the size of a typical clique. Infection can be transmitted between two
individuals if and only if they belong to the same clique. An infinite-type
branching process approximation (with type being given by the length of an
individual's infectious period) for the early stages of an epidemic is
developed and made fully rigorous by proving an associated limit theorem as
the population size tends to infinity. This leads to a threshold parameter
R*
, so that in a large population an epidemic with few initial
infectives can give rise to a large outbreak if and only if
R*
> 1. A law of large numbers for the size of such a large outbreak is proved
by exploiting a single-type branching process that approximates the
susceptibility set of a typical individual. Amandine Véber Temporal and spatial scales in geographically structured population models The aim of this course is to review some classical models of geographically structured populations, with a particular view towards questions of population genetics. We shall first consider the island and stepping stone models, which assume that the global population is split into discrete communities connected through a constant exchange of migrants. Then, we shall present a rather new model for evolution in a continuum. For all these frameworks, our main interest will be to understand the space- and time-scales on which the action is taking place, in order to describe the pattern of genetic correlations they produce and to find appropriate criteria to discuss their relevance to some "real" situations. Yinna Ye Asymptotic behavior of the survival probability for a critical branching process in markovian environments Feng Yu Problems in population genetics involving selection, recombination and migration In this problem session, I will start with the classic model of a population of constant size under the influence of selection and mutation. Then I will go onto Petry's island/continent model involving migration, selection and recombination. I will show that in the large population limit, effective migration rate to the island is reduced due to effects of recombination and selection, and give Petry's approximate formula for effective migration rate. This approximation formula is not accurate for finite populations, and finding one is a challenging open problem. This leads naturally to the final topic of discussion, which is the genic view of parapatric speciation, where selection, recombination and migration all play important roles. I will present only intuition on the final topic, which still awaits proper mathematical analysis.
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