The effects of motion and heterogeneity on stochastic population dynamics

Abstract

Noise has come to be accepted as a quintessential part of social and biological processes. It has transcended the misconception of being an obstacle, which hinders our understanding of `true mechanisms' hiding behind the randomness, and is now recognised as the cause of many important phenomena. Different sources of noise exist, and their combined effect is not trivial to understand. In this thesis, we contribute by studying models which combine intrinsic and extrinsic noise. We consider systems with discrete interacting components; as a consequence, they are subject to intrinsic noise. At the same time, we explore how two sources of extrinsic noise modify the time evolution of these models: motion, and agent-to-agent heterogeneity. We investigate motion of individuals in a two dimensional setting. Members of the population take positions in space and are moved by an external flow. The position of agents defines an interaction graph, so the population is structured. The interaction network is modified as the flow advects individuals in space. We choose an evolutionary dynamics setting, and study how the changing population structure alters the probability with which a mutant invades a population of wild-type individuals. We find that seemingly subtle changes in the mechanics of evolution, which implement birth and death events, can lead to significant changes in the mutant's chances of success. Therefore, we propose these differences can be used to identify the underlying mechanism in a given experimental setting. Furthermore, we debate that the commonly used term to describe the invasion process in unstructured populations, `well-mixed', is a misnomer, which must be used with care. To study agent-to-agent heterogeneity we use models of epidemics and opinion dynamics. For the latter we explore how the achievement, maintenance or alternation of consensus are affected by the presence of more than two co-evolving opinions, with potentially different conviction strengths. In the model of disease spread, we study how heterogeneity in the susceptibility and infectiousness of individuals influences the frequency and amplitude of outbreaks. In both models heterogeneity is represented by an arbitrary number of compartments that describe the `type' of the agents. To be able to simplify the mathematical description of these systems, we approach them by aggregating these compartments into bigger groups. In both cases, we find that this marginalised description provides a good approximation of the model dynamics. We are able to characterise the recurrence and severity of outbreaks using the aggregated components; similarly, we provide an analytical description of the simplified opinion dynamics, which is an approximation when conviction strengths differ, and exact when they are homogeneous.

Date of Award	1 Aug 2019
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Tobias Galla (Supervisor) & Ahsan Nazir (Supervisor)