Noise plays a crucial role in the formation and evolution of spatial patterns in various reaction-diffusion systems in mathematical biology and ecology. In this talk, I give two examples where noise significantly influences spatial patterning. The first example describes how patterned states can provide a refuge and prevent extinction under stressed conditions. It also illustrates the...
Collaborators:
Hannah Dopmeyer (Bielefeld University, Germany),
Fernando Cordero (BOKU University, Vienna, Austria).
Abstract:
We consider the two-type Moran model of population genetics with frequency-dependent neutral reproduction. Relying on the model's graphical representation in terms of a particle system, we establish a (factorial) moment dual. Moment duals are usually...
We give conditions for an exchangeable genealogy model, with an arbitrary sequence of population and litter sizes, to have either a unique infinite path, or a unique lineage whose descendants eventually dominate the population. Conditions relate naturally to the coalescent time scale: the second property holds iff infinite time passes on this scale, while the first property holds if a...
A modern Hopfield network can be viewed as an agent-based model as each neuron (or memory pattern) behaves like an individual agent following local update rules, and their interactions collectively produce the networkโs emergent dynamics. In this talk, I will discuss a dimension reduction technique for modern Hopfield networks.
Abstract
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type, once the population approaches the carrying capacity. Adapting techniques from [Kat91], we show that this process converges to a...
Lambda-coalescents, or multiple merger coalescents, have been extensively studied since their introduction in 1999, and were interpreted as genealogies of populations with skewed offspring distribution. More recently, multitype versions of such coalescents have garnered attention. However, it turned out that generalising the characterisation of multiple merger coalescents in terms of a finite...
Abstract
Antibody binding affinity maturation is a crucial process of the adaptive immune system. Motivated to model this process, we formulate a system of multi-type birth-death processes that can interact through their empirical distribution. We show that the empirical distribution process of the system of birth-death processes converges to a deterministic probability measure-valued...
We derive the frequency and distribution of outbreaks in an SIR model with noise. This involves techniques from matched asymptotics combined with stochastic ODEs and random maps.
Individual-based modelling has applications in biology ranging from the molecular and cellular level up to the scale of population and movement ecology, and is typically implemented with stochastic processes. This diversity of applications is underpinned by a common toolbox of analytical techniques for studying mean behaviour, fluctuations and rare events related to particle trajectories,...
