# European Institute for Statistics, Probability, Stochastic Operations Research and their Applications

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August 25-29, 2014

Workshop on

Population Dynamics and Statistical Physics in Synergy

 SUMMARY SPEAKERS PROGRAMME ABSTRACTS +PRESENTATIONS

SUMMARY

This workshop is part of the DFG Priority Program "Probabilistic Structures in Evolution" (DFG-SPP 1590).

Themes of the workshop are genealogical processes; multiscale genetic models; adaptive dynamics; evolutionary branching; evolution for interaction networks.

The programme will consist of 3-hour mini-courses and 1-hour talks.

## ORGANISERS

 Cristian Giardina University of Modena and Reggio Emilia Andreas Greven Universität Erlangen-Nürnberg Frank den Hollander Leiden University

## LIST OF INVITED SPEAKERS

 Mini courses: Jorge Kurchan Paris Algorithmic aspects of adaptive dynamics Christoph Richard Erlangen Combinatorial aspects of trees Talks: Siva Athreya Bangalore Ellen Baake Bielefeld Martina Baar Bonn Matthias Birkner Mainz Jochen Blath Berlin Anton Bovier Bonn Éric Brunet Paris Nicolas Champagnat Nancy Mark Dykman Michigan Lisa Hartung Bonn Alex Kamenev Minnesota Anton Klimovsky Essen Joachim Krug Cologne Marcel Ortgiese Münster Etienne Pardoux Marseille Frank Redig Delft Jan Swart Prague Anita Winter Essen

MONDAY AUGUST 25

 09.30 - 10.00 Registration 10.00 - 10.15 Opening 10.15 - 11.15 Etienne Pardoux A path-valued Markov process indexed by the ancestral mass 11.15 - 11.30 Break 11.30 - 12.30 Ellen Baake The Moran model with recombination: Type distributions on partitions, ancestry, and duality 12.30 - 14.00 Lunch 14.00 - 15.00 Matthias Birkner Ancestral lineages in locally regulated populations 15.00 - 15.15 Break 15.15 - 16.15 Jochen Blath The seed-bank coalescent

TUESDAY AUGUST 26

 09.00 - 10.30 Mini course (I) Christoph Richard Combinatorial aspects of tree-like structures (I) 10.30 - 11.00 Break 11.00 - 12.30 Mini course (I) Kurchan Algorithmic aspects of adaptive dynamics (I) 12.30 - 14.00 Lunch 14.00 - 15.00 Mark Dykman Scaling and fragility of the rates of rare events 15.00 - 15.15 Break 15.15 - 16.15 Anton Bovier Extremal Processes of Gaussian Processes Indexed by Trees 16.15 - 16.30 Break 16.30 - 17.30 Alex Kamenev 18.30 - Conference dinner

WEDNESDAY AUGUST 27

 09.00 - 10.30 Mini course (II) Christoph Richard Combinatorial aspects of tree-like structures (II) 10.30 - 11.00 Break 11.00 - 12.30 Mini course (II) Kurchan Algorithmic aspects of adaptive dynamics (II) 12.30 - 14.00 Lunch 14.00 - 15.00 Joachim Krug Fitness landscapes and adaptive evolution 15.00 - 15.15 Break 15.15 - 16.15 Éric Brunet Existence of open evolutionary paths in a rugged landscape 16.15 - 16.30 Break 16.30 - 17.30 Frank Redig Energy and wealth redistribution models

THURSDAY AUGUST 28

 09.00 - 10.00 Marcel Ortgiese The Interface of the Symbiotic Branching Model 10.00 - 10.15 Break 10.15 - 11.15 Nicolas Champagnat Adaptive dynamics and evolutionary branching in the limit of small mutations in a stochastic individual-based population model 11.15 - 11.30 Break 11.30 - 12.30 Martina Baar Stochastic individual-based models of adaptive dynamics 12.30 - 14.00 Lunch 14.00 - 15.00 Siva Athreya Dense Graph Limits under Respondent Driven Sampling 15.00 - 15.15 Break 15.15 - 16.15 Anton Klimovsky Is there more biodiversity in non-homogeneous environments than in homogeneous ones?

FRIDAY AUGUST 29

 09.00 - 10.00 Lisa Hartung Variable speed branching Brownian motion 10.00 - 10.15 Break 10.15 - 11.15 Jan Swart Balancing selection and rebellious voter models 11.15 - 11.30 Break 11.30 - 12.30 Anita Winter Tree-valued spatial Lambda-Fleming-Viot diffusions: The finite system scheme 12.30 Closing

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ABSTRACTS

Siva Athreya

Dense Graph Limits under Respondent Driven Sampling

We consider certain respondent-driven sampling procedures on dense graphs. We show that if the sequence of the vertex-sets is ergodic then the limiting graph can be expressed in terms of the original dense graph via a transformation related to the invariant measure of the ergodic sequence. For specific sampling procedures we describe the transformation explicitly.

PRESENTATION

Ellen Baake

The Moran model with recombination: Type distributions on partitions, ancestry, and duality

We consider the Moran model with recombination, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ loci (or sites), a finite number of letters (or alleles) at every site, and we do not make any scaling assumptions. In particular, we do not assume a diffusion limit. Due to the huge state space, it is notoriously difficult to find the full distribution of types in a sample. We therefore take an alternative route by concentrating on the joint probabilities of types at tuples of loci. Taking the usual genealogical approach, this is described via a process on the set of partitions of $\{1,\ldots,n\}$ (backward in time), which may be considered as a marginalised version of the ancestral recombination graph, and sheds new light on the work of A. Bobrowski, T. Wojdyla and M. Kimmel (2009). With the help of an inclusion-exclusion principle we show that the type distribution corresponding to a given partition may be represented in a systematic way, with the help of so-called recombinators. The same is true of correlation functions (known as linkage disequilibria in biology) of all orders.
Considering a graphical representation of the Moran model together with the ancestral process suggests a duality relation between the two. Indeed, there is a formal duality relation between the type distribution process forward in time and the partitioning process backward in time, where the joint probabilities of types at tuples of loci play the role of the duality function. We prove this via the representation of the duality function in terms of recombinators, and with the help of various tools from discrete mathematics, in particular, the Moebius inversion principle.
(joint work with Mareike Esser and Sebastian Probst)

Martina Baar

Stochastic individual-based models of adaptive dynamics

We consider the limiting behavior of a model for the Darwinian evolution in an asexual population characterized by a natural birth rate, a logistic death rate due to age or competition and a probability of mutation at each birth event. In this stochastic individual-based model the population is described by a measure-valued Markov process, known as BPDL process. We focus on the limit of a large population with rare mutations and small mutation effects on a long time scale. More precisely, we study this three limits simultaneously on a timescale which separates birth and death events from mutation events and we will see that on this time scale coexistence of two traits cannot occur. In other words, the population stays essentially single modal centered around a trait, which evolves continuously. In fact, we are able to prove that for this combined limit the population process converges to an ODE, known as canonical equation of adaptive dynamics. This is a joint work with A. Bovier and N. Champagnat.

PRESENTATION

Matthias Birkner

Ancestral lineages in locally regulated populations

Ancestral lineages of sampled individuals in the stepping stone model form (delayed) coalescing random walks, and can thus be viewed as a spatial coalescent process. The analogous object in spatial population models which unlike the stepping stone model do allow random fluctuations of local population sizes is a system of correlated coalescing random walks in a space-time random environment. We discuss regeneration approaches to study its long-time behaviour and implications for spatial type distributions. A simple guiding example is the discrete time contact process, with an ancestral lineage given by a directed walk on the backbone of an oriented percolation cluster.
(joint work, in part in progress, with Jiří Černý, Andrej Depperschmidt and Nina Gantert)

PRESENTATION

Jochen Blath

The seed-bank coalescent

We identify a new natural coalescent structure, the seed-bank coalescent , which describes the gene genealogy of populations under the influence of a strong seedbank effect, that is, in which dormant forms' of individuals (such as seeds or spores) may jump a signicant number of generations before joining the active' population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright-Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size N . This extends earlier results of Kaj, Krone and Lascaux (2001) who show that the
genealogy of a Wright-Fisher model in the presence of a weak' seed-bank effect is still given by an ordinary (suitably time-changed) Kingman coalescent. The qualitatively new feature of the seed-bank coalescent is that ancestral lineages are independently blocked at a certain rate from taking part in coalescence events, thus strongly altering the predictions of classical coalescent models. In particular, the seed-bank coalescent does not come down from innity', and the time to the most recent common ancestor is highly elevated. This provides a genealogical explanation for the empirical observation that seed-banks drastically increase genetic variability in a population and indicates how they may serve as a buffer against other evolutionary forces such as genetic drift and selection.
(joint work (in progress) with N. Kurt, A. Gonzalez Casanova, M. Wilke Berenguer (all TU Berlin))

PRESENTATION

Anton Bovier

Extremal Processes of Gaussian Processes Indexed by Trees

Gaussian processes indexed by trees form an interesting class of correlated random fields where the structure of extremal processes can be studied. One popular example is Branching Brownian motion, which has
received a lot of attention over the last decades, non the least because of its connection to the KPP equation.
In this talk I review the construction of the extremal process of BBM (with Arguin and Kistler) and present some more recent results on “variable speed” BBM, obtained with Lisa Hartung

PRESENTATION

Éric Brunet

Existence of open evolutionary paths in a rugged landscape

An evolving population under selection explores its fitness landscape by fixing favorable mutations. If one assumes that deleterious mutations cannot be fixed, it is however not obvious that the population can find an evolutionary path leading to the fittest possible state when the fitness landscape is rugged . I will present some rigourous results (and some conjecture, and some open questions) on the existence and the number of acceptable evolutionary paths in the "House of Cards" model.

PRESENTATION

Nicolas Champagnat

Adaptive dynamics and evolutionary branching in the limit of small mutations in a stochastic individual-based population model

The mathematical study of adaptive dynamics, and more specifically of the phenomenon of evolutionary branching by which a population is driven by selective forces to sub-divide into two interacting subpopulations with different phenotypes, has been done in the last years using either an approach based on an assumption of rare mutations and large population on a stochastic individual-based model, or an approach based on a limit of small mutations on a PDE model. Both approaches suffer from irrealistic features: the first one requires a very long time scale to observe evolutionary branching; in the second one, exponentially small population densities can have a very strong impact on the future evolutionary dynamics. The goal of this talk is to present an intermediate approach that may solve these two drawbacks, consisting in applying a combination of limits of small mutations and large population on the stochastic individual-based model.

Mark Dykman

Scaling and fragility of the rates of rare events

We discuss rare events that result from large classical and quantum fluctuations in systems away from thermal equilibrium. They include extinction in population dynamics and switching between coexisting stable states in dynamical systems. We review the results on the scaling of the rates with the distance to the bifurcation points, which turns out to be different for the extinction and switching problems even for the bifurcations of the similar type. We show that the rates of rare events can discontinuously change with the change of the parameters of the system, which we call the fragility effect. This change is accompanied by an abrupt change of the most probable path followed in the rare event. We discuss the criteria for the onset of fragility. The associated breakdown of the conventional analysis and the onset of a kink on the optimal path are considered.

PRESENTATION

Lisa Hartung

Variable speed branching Brownian motion

In this talk I explain how the convergence of the extremal process of variable speed BBM is obtained when the "speed function", describing the time-inhomogeneous variance, lie strictly below their concave hull. The resulting limiting objects turn out to be universal in the sense that they only depend on the slope of the speed function at 0 and the final time t.
(joint work with A. Bovier)

PRESENTATION

Alex Kamenev

Does noisy environment facilitate extinction, or stabilize the population?

I will review population dynamics models, which incorporate an external environmental noise. The natural question is whether such a noise accelerates extinction, or works as a stabilizing factor. It appears that it can have both of these effects. I will present and explain models, where the noise facilitates extinction. Moreover, it may change the characteristic extinction time from being exponentially long in the population size into being only power-law long. On the other hand, I will present a family of models, where the strong noise is solely responsible for the population stability. (The effect bears a strong resemblance with the celebrated Coleman-Weinberg phenomenon of symmetry breaking by vacuum fluctuations in the high-energy physics.) In this latter scenario the population lifetime scales as a stretched exponential of the characteristic noise amplitude.

PRESENTATION

Anton Klimovsky

Is there more biodiversity in non-homogeneous environments than in homogeneous ones?

Consider a spatially structured population of individuals. The population is structured in a hierarchical manner into colonies, macro-colonies (= collections of colonies), macro-macro-colonies (= collections of macro-colonies), and so on. Within the colonies, the individuals reproduce under a fixed amount of resources according to the Cannings model. Next, the individuals move around from one colony to the other at given rates which depend on the geographical distance between the colonies (= migration). Furthermore, the whole macro-colonies can suffer from occasional non-local catastrophic events (e.g., droughts, forest fires). The population dynamics respects the hierarchical structure of the geographical space. Larger-scale events are less frequent that the smaller-scale ones. This results in a hierarchy of space-time scales on which the system evolves. How does the biodiversity in such a hierarchical Cannings process evolve? I will report on our findings in the case, where both the Cannings reproduction and the reshuffling-resampling mechanisms are spatially inhomogeneous. This is modelled via a random environment. It turns out that the inhomogeneities increase biodiversity (comparing to the homogeneous environment).
(joint work with Andreas Greven and Frank den Hollander)

PRESENTATION

Joachim Krug

The adaptive dynamics of an evolving population is constrained by epistatic interactions encoded in the underlying fitness landscape. In recent years empirical data sets have become available that offer glimpses into the structure of real fitness landscapes and motivate theoretical work on classic probabilistic models such as NK-landscapes and the rough Mount Fuji model. In this talk I report on recent efforts aimed at quantifying the accessibility of these landscapes under asexual adaptation. Results concerning structural properties, such as the number of local maxima and the existence of fitness-monotonic pathways, as well as on the dynamics of adaptive walks will be presented, with particular emphasis on the subtle role of genetic architecture in the NK-model. If time permits, the role of recombination in speeding up or slowing down adaptation on rugged fitness landscapes will also be briefly addressed.

PRESENTATION

Jorge Kurchan (mini course)

Population dynamics offers a practical method to compute large deviations. Even without implementing an actual program, thinking in terms of a population allows one to understand several features of large deviation functions, in particular the particular features they take in the presence of metastability. Conversely, one may think of a large population of individuals undergoing natural selection as a natural algorithm that computed the large deviations of fitness.

PRESENTATION (1)

PRESENTATION (2)

Marcel Ortgiese

The Interface of the Symbiotic Branching Model

The symbiotic branching model describes two interacting spatial populations whose evolution is given by system of correlated SPDEs. Starting from two spatially separated populations, one can consider the growth of the interface where particles of both types are present. We show that for negative correlations the system converges under a diffusive rescaling. We will discuss first properties of the limiting system and their implications for the interface.
(joint work with Jochen Blath and Matthias Hammer (both TU Berlin))

PRESENTATION

Etienne Pardoux

A path-valued Markov process indexed by the ancestral mass

We consider a Feller diffusion with a nonlinear drift which models the interaction within the population. We study the resulting path-valued Markov process, which is indexed by the ancestral population at time t=0. We write an SDE driven by a Poisson random measure for that process, and explicit its infinitesimal generator.
(joint work with Anton Wakolbinger, Frankfurt)

PRESENTATION

Frank Redig

Energy and wealth redistribution models

We consider stochastic models where a configuration x_i, i\in V is updated via binary exchanges along edges ij, of the form x’_i = u(x_i+x_j), x’_j= (1-u)(x_i+x_j) (energy redistribution model) or x’_i =l x_i + (1-l) u(x_i+x_j), x’_j= l x_j + (1-u)(x_i+x_j) (wealth distribution model).
Here u is a random variable on [0,1], and l\in [0.1] (the propensity) is a parameter. Examples of such models include the KMP (Kipnis, Marchioro Presutti) model (when u is uniform), and kinetic exchange models of markets (such as the Angle model). We ask (and answer) three questions
1. When are the invariant measures product measures?
2. When is there a discrete dual process, and when are the duality functions factorizing over the vertices?
Finally, we discuss the relation between such models and diffusion processes of Wright-Fisher type.
(joint work with Pasquale Cirillo and Wioletta Ruszel (Delft))

PRESENTATION

Christoph Richard (mini course)

Combinatorial aspects of tree-like structures

Exactly solvable models from statistical physics serve as a crude but tractable approximation to real-world phenomena. In particular, solvable subclasses of self-avoiding walks on a lattice have been used to understand the behaviour of polymers. From a technical perspective, this involves enumerative combinatorics and
singularity analysis of generating functions. Focussing on planar models, we review some techniques from that area, which can also be applied to similar tree-like combinatorial structures.

PRESENTATION (1)

PRESENTATION (2)

Jan Swart

Balancing selection and rebellious voter models

In biological populations, individuals that are genetically different from their neighbours often have a selective advantage, since being different reduces competition with neighbours and vulnerability to pests and
diseases. This effect, called balancing selection, is presumably an important evolutionary force creating biodiversity. Mathematically, it can be modeled with a `rebellious' voter model where individuals like to hold
a different opinion than their neighbors. In this talk, I will review the progress that has been made in the mathematical study of such models in recent years and also mention some major remaining open problems. (joint work with Anja Sturm (Göttingen) and Karel Vrbenský (Prague)).

PRESENTATION

Anita Winter

Tree-valued spatial Lambda-Fleming-Viot diffusions: The finite system scheme

We study the evolution of genealogies for interacting spatially structured Λ-Fleming-Viot diffusions.
These are the limit processes of individual-based population models where individuals carry a type, and are replaced by descendants of possibly very sizable offspring. The spatial interaction is due to migration through geographic space. We show that the dual to these tree-valued spatial Λ-FV diffusions are tree-valued spatial Λ-coalescents. We then study the populations on large tori in $Z^d$ with $d\ge 2$. Depending on the rescaling we find global features which are universal for all Λ-dynamics and local features which heavily depend on the measure Λ.
(joint work with Andreas Greven and Anton Klimovsky)

PRACTICAL INFORMATION

### ●      Venue

Eurandom, Mathematics and Computer Science Dept, TU Eindhoven,

Den Dolech 2, 5612 AZ  EINDHOVEN,  The Netherlands

Eurandom is located on the campus of Eindhoven University of Technology, in the (4th floor) (about the building). The university is located at 10 minutes walking distance from Eindhoven main railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).
Accessibility TU/e campus and map.

### ●      Registration

Registraton is closed.

### ●      Accommodation

For invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements.

We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer.

For other hotels around the university, please see: Hotels (please note: prices listed are "best available").

More hotel options can be found on the webpages of the , Postbus 7, 5600 AA Eindhoven.

### ●      Travel

For those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location.

Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl

The University  can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.

### ●      Conference facilities : Conference room, Metaforum Building  MF11&12

The meeting-room is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.

### ●      Conference Secretariat

Upon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the day-to-day running of the conference: registration, issueing certificates and receipts, etc.

### ●      Cancellation

There is no registration fee, but should you need to cancel your participation after January 2, 2014, we will be obliged to charge a no-show fee of 30 euro.

### ●      Contact

Mrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl

The organisers acknowledge the financial support/sponsorship of:

#### Last updated 24-09-14, by PK

P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100
e-mail: info@eurandom.tue.nl