Spatial and spatiotemporal volatility models are a class of models designed to capture spatial dependence in the volatility of spatial and spatiotemporal data. Spatial dependence in the volatility may arise due to spatial spillovers among locations; that is, in the case of positive spatial dependence, if two locations are in close proximity, they can exhibit similar volatilities. In this paper, we aim to provide a comprehensive review of the recent literature on spatial and spatiotemporal volatility models. We first briefly review time series volatility models and their multivariate extensions to motivate their spatial and spatiotemporal counterparts. We then review various spatial and spatiotemporal volatility specifications proposed in the literature, along with their underlying motivations and estimation strategies. Through this analysis, we effectively compare all models and provide practical recommendations for their appropriate usage. We highlight possible extensions and conclude by outlining directions for future research.

Humboldt University, Berlin

Abstract: In this talk, we introduce the continuum seed-bank model which is a natural generalization of Blath et al. (2016) and Greven, den Hollander, et al. (2022) to accommodate more general dormancy time distributions, such as a type of Pareto distribution. To this end, we first formulate an infinite-dimensional SDE and show that it has a unique strong solution, referred to as the continuum seed-bank diffusion. This diffusion process serves as a Markovian lift of a non-Markovian Volterra process. We construct a discrete-time Wright-Fisher type model with finitely many seed-banks, and demonstrate that the continuum seed-bank diffusion, under the weak* topology, is the scaling limit of the allele frequency process in a suitable sequence of such models. Furthermore, we establish a duality relation between the continuum seed-bank diffusion and a continuous-time and -state Markov jump process. The latter is the block counting process of a partition-valued Markov jump process, referred to as the continuum seed-bank coalescent. We discuss some basic properties of the coalescent process such as scaling limit interpretation, exchangeability, limiting distribution of the ancestral line, and comparisons of E[T_{MRCA}]. We prove that E[T_{MRCA}] is finite if and only if the expected dormancy time is finite. As a byproduct, we also derive the limiting distribution of the continuum seed-bank diffusion. Additionally, we show that the continuum seed-bank coalescent does not come down from infinity, and provide asymptotic bounds of E[T_{MRCA}] in certain special cases.

Likai Jiao " Continuum seed-bank modell"

Bath 

Abstract: In 2-type spatial stochastic population models exhibiting bistability, interfaces tend to form between regions consisting predominantly of one of the two types. To understand how the population evolves, we may study the dynamics of these interfaces in time. For several bistable systems, it is known from recent work that the limiting interface, under certain rescalings, evolves by a geometric evolution called mean curvature flow. This interface evolution is known to develop singularities in finite time, which imposes a short-time constraint on the convergence results.

In this talk, I will first discuss some models exhibiting this phenomenon, including a variant of the Spatial Lambda Fleming Viot model, and results concerning their interfaces. I will then discuss an ongoing work which uses tools from analysis, in particular level-set methods and the theory of viscosity solutions, to prove that interfaces in a broad class of bistable population models converge globally in time to a generalized mean curvature flow.

This is joint work with Jessica Lin (McGill)

Dr. Thomas Hughes " Interface evolution in bistable spatial population models: a global approach"

Oxford

Abstract: We introduce a broad class of spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined via the convolution of the point measure with a nonnegative kernel. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE.

A novelty of our model is that we explicitly model a juvenile phase: offspring are thrown off in a Gaussian distribution around the location of the parent, and reach (instant) maturity with a probability that can depend on the population density at the location at which they land. Although we only record mature individuals, a trace of this two-step description remains in our population models, resulting in novel limits governed by a nonlinear diffusion.

Using a lookdown representation, we retain information about genealogies and, in the case of deterministic limiting models, use this to deduce the backwards in time motion of the ancestral lineage of a sampled individual. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model.

We also investigate the behaviour of lineages for three different deterministic models of a population expanding its range as a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a porous medium equation with logistic growth.

Dr. Terence Tsui: "Looking forwards and backwards: dynamics and genealogies of locally regulated populations"

Lyon 

Abstract: Better understanding the genetic diversity within tumors is of key interest for clinicians to propose adaptive strategies. I will present a probabilistic model of tumorogenesis to provide quantitative results over time on the genetic diversity.

The trait space is modelled using a finite oriented graph. The population of cells follows a continuous time branching process. The  biological phenomenon taken into account are cell death and cell division. During each division event each daughter cell mutates, independently from each other, to another trait (using the edges of  the graph) with a certain probability.

The classical regime of « large population and rare mutation » is considered. It means that a parameter $n \in \mathbb{N}$ is used to  quantify both the decrease of the mutation probabilities, as negative powers of $n$, and also the typical size of the population, depending on $n$ as positive power of $n$, at which we are interested in  understanding the genetic composition.  The results are on the  asymptotic sizes of the subpopulations of cells. Notably, the behaviour of mutant cells will depend on whether the mutation is deleterious, neutral or selective. 

Dr. Vianney Brouard: Genetic composition of supercritical branching populations under rare mutation rates

Leiden

Abstract: This talk will focus on a spatially structured interacting Moran model with seed banks in random environment. The population sizes are sampled from a translation-invariant, ergodic, uniformly elliptic field that constitutes the static random environment of the model. Under mild assumptions on the model parameters, we identify the domain of attraction of each mono-type equilibrium and establish convergence to (mono-type) equilibrium for a large class of intitally consistent type-distribution. Our result shows that, for a.s. realisation of the population sizes, the fixation probability of the entire population to a single genotype homogenizes in the long run and crucially depends on the average relative seed-bank strength in the populations

Dr S. Nandan: Spatial populations with seed banks in random evironment

JGU Mainz

Abstract: Understanding the interplay between recombination and resampling is a significant challenge in mathematical population genetics and of great practical relevance. Asymptotic results about the distribution of samples when recombination is strong compared to resampling are often based on the approximate solution of certain recursions, which is technically hard and offers little conceptual insight. We generalise an elegant probabilistic argument, based on the coupling of ancestral processes but so far only available in the case of two sites, to the multilocus setting. This offers an alternative route to, and slightly generalises, a classical result of Bhaskar and Song.

University of Warwick

Abstract: Multiple merger coalescents (MMCs) have long been suggested as appropriate models for the genetic diversity of organisms with skewed offspring distributions, such as many marine and microbial species. They predict an excess of low frequency mutations relative to the Kingman coalescent, similarly to other phenomena such as historical population growth. However, their predictions for the full site frequency spectrum differ from the Kingman coalescent even under fairly arbitrary demographic scenarios. I will describe joint work with Einar Arnason, Katrin Halldorsdottir, and Bjarki Eldon, in which we assessed the extent to which several prominent MMC families were able to explain the genetic diversity of cod populations sampled around Iceland. The Kingman coalescent provides a poor fit even under non-parametric, best-fit demographies or simple models of population structure. More surprisingly, so does the Beta-coalescent, which has often been seen as a natural candidate model for skewed offspring reproduction. Instead, an MMC model by Durrett & Schweinsberg for recurrent selective sweeps provides a remarkably good fit only a few free parameters. Our results are informative of possible mechanisms giving rise to the skewed and shallow genealogies observed among cod.

Dr. Jere Koskela: Multiple merger coalescent model selection für whole genome cod data

Jointly with Stochastik Kolloquium GU Frankfurt

Université de Grenoble

Abstract: We will introduce and study a variant of a well-known model in population genetics, namley Muller's ratchet, which is seen as one explanation of the ubiquity of sexual selection in Nature. Consider a population of N individuals, each of them carrying a type in N_0. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type k has the same selective advantage over all individuals with type k′>k, and type k mutates to type k+1 at a constant rate (in the classical Muller's ratchet, the selective advantage is proportional to k′−k). For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal (`best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a ``dual'' hierarchical multitype logistic system. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.

Charline Smadi "Quasi-equilibria and click times for a variant of Muller's ratchet"

Abstract: Organisms having a genealogy not well described by the Kingman coalescent are not rare. One example is the Atlantic cod which presents shallow genealogies and high-variance offspring number that might rather be described by a multiple merger coalescent. The aim of our work is then to find a realistic individual-based model fitting these data.

We focus on spatially structured populations undergoing localized, recurrent bottlenecks, and describe their ancestral lines. We start by presenting an individual based model introduced by González Casanova, Miró Pina, Siri-Jégousse (2022) whose genealogy is described by a Xi-coalescent known as the symmetric coalescent. We then introduce migration in this setting: we construct a multiple-island model and see how this structure affects the coalescent process. Depending on the severity and the length of the bottlenecks we derive as scaling limits different structured Xi-coalescents featuring simultaneous multiple mergers and migrations. This talk is based on ongoing work with Alison Etheridge, Jere Koskela and Maite Wilke Berenguer.

Marta Dai Pra "The effects of migration in a population model with bottlenecks"

Abstract: Dormancy mechanisms allowing individuals to enter and exit a protected state of reduced metabolic activity are ubiquituous in nature. Hence, we aim to understand the consequences of dormancy on evolutionary and ecological properties of microbial populations. In this talk, we will consider a stochastic individual-based model as proposed by Champagnat and Méléard (2011) where we incorporate competition-induced dormancy. To study the behaviour of the population over time, we derive the Polymorphic Evolution Sequence and the Canonical Equation of Adaptive Dynamics (CEAD) as scaling limits of the model. At the equilibria of the CEAD we may observe evolutionary branching, which describes the splitting of a population into distinct traits and may hence be understood as speciation. We will show a general criterion for evolutionary branching and demonstrate the effect of dormancy in a specific model. Using our mathematical tools and simulations, we also see an impact of dormancy on subsequent branchings, the speed of adaptation, species diversity and niche width. This is joint work with Jochen Blath, András Tóbiás and Maite Wilke Berenguer.

Tobias Paul "The impact of dormancy on evolutionary branching"