Applied Math Seminar: Adam Waterbury (UCSB)

Title: Approximating Quasi-Stationary Distributions with Interacting Reinforced Markov Chains

Abstract: An important question in ecology is what conditions must be met for a population of interacting species to coexist. In realistic models of such populations, after a large enough amount of time has passed, one or more of the species are sure to face extinction. However, the time that it takes for extinction to occur can be quite large, so it is natural to consider whether the population can sustain any long-term coexistence before any of the species are extinct. This metastability is captured in the notion of a quasi-stationary distribution (QSD). However, calculating the QSD of such a system can be numerically difficult, as it amounts to solving a system of nonlinear equations, which has led to a wide range of simulation-based methods that can be used to efficiently approximate QSD. In the first part of this talk I introduce two new simulation-based methods for approximating QSD that are described in terms of a large collection of interacting particles known as reinforced Markov chains. In the second part of this talk I discuss some related work studying the rare-event asymptotics of a related class of reinforced Markov chains.