Structured interactions in large random ecosystems

Abstract

The central aim of statistical physics is to describe the phenomena a system exhibits at onescale, given a model for how the constituents of that system work on a more granular scale. Historically, the tools of statistical physics have been applied to physical systems where the laws governing these constituents are well understood. However, in the past 50 years, the ideas and mathematical tools of statistical physics have been applied to an increasing variety of problems far from its historical remit. This thesis comprises three closely related studies in one such discipline that has attracted the attention of modern statistical physics: theoretical ecology. Many models of complex ecosystems in theoretical ecology consider ecosystems in which species interact randomly. In these models, the statistics of the interactions are often chosen so that species are statistically equivalent. This thesis aims to understand how model ecological communities differ when this assumption is relaxed, both in general and in specific examples. In tackling this problem, this thesis extends well-known methods from the statistical physics of disordered systems to the case of structured randomness. Firstly, we investigate the effect of structured randomness on the local stability of a large complex ecosystem. We derive an explicit approximation for system stability and identify general structural features that promote or demote stability. The applicability of these findings is demonstrated through examples. Notably, the Niche model, which represents the network structure of species interactions, and the Cascade model, which captures the hierarchical nature of interspecies interactions. Next, we extend our findings to a generalized Lotka-Volterra dynamical system with structured random interaction coefficients. We extend the current theory for the properties of communities produced by these dynamics to this case, and apply it to a model which includes hierarchical interactions, where species are distinguished by their position in the hierarchy. We demonstrate that a strong hierarchical structure leads to smaller, more stable communities. Finally, we analyze a second example of a generalised Lotka-Volterra dynamical system with structured random interactions. One in which species interact on a network with an arbitrary degree distribution. Additionally, we examine the statistics of interactions between surviving species, which generally differ from the initial pools interactions. Our analytical methods provide insight into how the initial network structure imposed on the species pool evolves under dynamical constraints, revealing that dynamics produce networks with relatively fewer high-degree species than the original community, a pattern often observed in real ecological communities.

Date of Award	6 Jan 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Alessandro Principi (Supervisor) & Niels Walet (Supervisor)