The programme will start on Wednesday, 06 Sept, at 9:00 and finish on Friday, 08 Sept, at the latest at 14:00. More details to be confirmed.

Abstracts

Martina Baar
Individual-based models of adaptive dynamics and applications to cancer immunotherapy

In this talk we study stochastic individual-based models describing Darwinian evolution of asexual, competitive populations. First, we consider a population with a large but non-constant population size characterized by a natural birth rate, a logistic death rate modeling competition, and a probability of mutation at each birth event. The population state at a fixed time is given as a measure on the space of phenotypes and the evolution of the population is described by a continuous time, measure-valued Markov process. We investigate the long-term behavior of the system in the simultaneous limits of large population size, rare mutations, and small mutational effects, showing convergence to the canonical equation of adaptive dynamics. Second, we present an extension of the model, which broadens the range of biological applications and is in particular able to describe some experiments in cancer immunotherapy, where tumors resist therapy through inflammation induced reversible dedifferentiation. The talk will focus on the mathematical aspects which arise when including phenotypic plasticity in the model.

Matthias Birkner
Ancestral lineages in locally regulated populations

Ancestral lineages in stochastic spatial population models with local regulation can be viewed as a system of correlated coalescing random walks in a dynamic random environment. We develop regeneration approaches to study its long-time behaviour and implications for the spatial distribution of types in the population. A simple guiding example is the discrete time contact process, whose ancestral lineages are directed walks on the backbone of an oriented percolation cluster. Based on joint work, in part in progress, with Jiří Černý, Andrej Depperschmidt, Nina Gantert and Sebastian Steiber.

Flo Débarre
Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

The theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. In this talk, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider spatially structured populations of fixed size N, in which two types of individuals, A and B, corresponding to two types of social behavior, are competing. Under the assumption of small phenotypic differences (weak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate this result with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures. Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher — when social interactions affect fecundities) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. Imperfect strategy transmission also alters the balance between the benefits and costs of staying next to one’s kin, leading to surprising results in subdivided populations, in that higher emigration probabilities can be favourable to the evolution of altruistic strategies.

Raphael Forien
Drift load in spatially structured populations

Suppose a particular gene is present in two different forms in a diploid population, and that individuals who carry one copy of each type leave on average more offspring than individuals who carry two copies of the same type. If the population size is large enough, a stable intermediate gene frequency is expected to be maintained, maximizing the mean fitness of the population. For smaller population sizes, because of finite sample size, the gene frequency fluctuates around its equilibrium, and the mean fitness of the population is always below its optimum. Robertson (1970) showed that for panmictic populations, the decrease in the mean fitness of the population due to small population size – called drift load – does not depend on the strength of natural selection. We show that this is no longer true for spatially structured populations and we provide estimates for drift load in populations living in one, two and three dimensional habitats. The main tool used for the proof is a central limit theorem for a measured-valued process describing the evolution of gene frequencies called the spatial Lambda-Fleming Viot process. This is joint work with Sarah Penington.

Stephan Gufler
On tree-valued Fleming-Viot processes

The Donnelly-Kurtz lookdown model contains an evolving genealogy. We can describe the genealogical tree of the population at each time by an isomorphy class of a metric measure space. In this way, we obtain a tree-valued Fleming-Viot process which is the infinite population size limit of the corresponding process read off from a Moran model. The states of tree-valued Fleming-Viot processes can be viewed as ergodic components of the genealogical distance matrices. In this talk, constructions of tree-valued Fleming-Viot processes will be discussed.

Sandra Kliem
The one-dimensional contact process and the KPP equation with noise

The one-dimensional KPP-equation driven by space-time white noise,
$\partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW$,
$t>0, x \in \mathbb{R}, \theta>0,$, $u(0,x) = u_0(x) \geq 0$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. Solutions to this SPDE arise for instance as (weak) limits of approximate densities of occupied sites in rescaled one-dimensional long range contact processes.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$, then there exist stochastic wavelike solutions which travel with non-negative linear speed. For initial conditions that are ‘’uniformly distributed in space’’, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution.

For the (parameter-dependent) nearest-neighbor contact process on $\mathbb{Z}$, more is known. A complete convergence theorem holds, that is, a full description of the limiting law of a solution is available, starting from any initial condition. Its proof relies in essence on the progression of so-called edge processes. In these models, edge speeds characterize critical values.

In my talk, I will introduce the two models in question (nearest-neighbor contact process and KPP-equation with noise). Then I explain in how far the concepts and techniques of the first model can be used to obtain new insights into the second model. In particular, the problems one encounters when changing from the discrete to the continuous (in space) setting are highlighted and approaches to resolve them are discussed.

Jörg Martin
A weak universality for the parabolic Anderson model

We adapt the analytical tools of paracontrolled calculus to the setup of refining lattices to give a general framework for the analysis of discrete approximations to singular SPDEs. As an application we prove that the discrete nonlinear parabolic Anderson model always scales to the linear model if the nonlinearity is weak enough.
(joint with Nicolas Perkowski)

Sarah Penington
Branching Brownian motion, mean curvature flow and the motion of hybrid zones.

Hybrid zones are interfaces between populations which can occur when two genetically distinct groups interbreed, but hybrid offspring have a lower evolutionary fitness. We can model this situation using the spatial Lambda-Fleming-Viot process. I will discuss a result on the motion of hybrid zones and also a related probabilistic proof of a known PDE result connecting the Allen-Cahn equation and mean curvature flow. The proofs rely on duality relations with a branching and coalescing random walk and a branching Brownian motion. Joint work with Alison Etheridge and Nic Freeman.

Richard Pymar
Two phase transitions in the PAM with correlated potential

The parabolic Anderson problem (PAM) is the Cauchy problem for the heat equation on the integer lattice with random potential. It is well-known that, unlike the standard heat equation, the solution of the PAM exhibits strong localisation. In particular, for a wide class of iid potentials (including Pareto potentials) it is localised at just one point. In this talk, we discuss phase transitions (between localisation and delocalisation) exhibited by the model when the potentials are chosen to be correlated. This is a joint work with Stephen Muirhead and Nadia
Sidorova.

Emmanuel Schertzer
How does geographical distance translate into genetic distance?

Geographic structure can affect patterns of genetic differentiation and speciation rates. In this talk, I will investigate the dynamics of genetic distances in a geographically structured metapopulation. We model the metapopulation as a weighted directed graph, with N vertices corresponding to N homogeneous subpopulations. The dynamics of the genetic distances is then controlled by two types of transitions -mutation and migration events between islands. Under a large population-long chromosome limit, we show that the genetic distance between two subpopulations converges to a deterministic quantity that can asymptotically be expressed in terms of the hitting time between two random walks in the metapopulation graph. Our result shows that the genetic distance between two subpopulations does not only depend on the direct migration rates between them but on the whole metapopulation structure. This is joint work with Veronica Miro Pina.

Anja Sturm
On dual processes for additive and monotone interacting particle systems and applications

In this talk we outline a general approach for obtaining pathwise dual processes for additive and monotone interacting particle systems. In the case of additive particle systems we show how this pathwise duality can be used to generate a percolation representation of the process and its dual process. In the case of monotone interacting particle systems the resulting dual process may not always be as tractable. In this talk, we apply the duality theory to a particular dynamics given by cooperative branching, coalescence (and potentially spontaneous death) and consider implications for various underlying spaces such as the integer lattice and the complete graph. Based on joint work with Jan Swart and work in progress with Tibor Mach.

Roger Tribe
One dimensional coalescing and annihilating systems as Pfaffian point processes

The Pfaffian point process property, a close relative of a determinantal point process, yields tools for analysing the one dimensional systems in the title, and some other related models. For example it allows an understanding of the gap probabilities found by Derrida et al. around 1995 by exploiting tools found in the random matrix community.