Dynamics of multiple populations interactions

The main focus of this thesis is to study the dynamical systems of interacting species and populations. In the first part, we consider the generalized Lotka-Volterra equations that describe multi-species interactions, including amensalism, commensalism, mutualism, competition, and predator-prey. In particular, we revisit the results presented in [30] where we investigated the asymptotic stability of the generalized Lotka-Volterra equations with respect to the underlying network topologies, such as trees and complete graphs. It was of interest to study the coexistence (no local extinction of species) equilibrium being locally stable. We find that the techniques in proofs and stability results for the tree networks are applicable to a community described by a modified Lotka Volterra model of predator-prey interaction. We then consider specific cases where the models are modified Lotka-Volterra competition equations, exhibiting bistable dynamics which are ubiquitous in ecology. In particular, we consider a competitive model from [48] which accounts for crowding effects – high population density increases the mortality rate. It was concluded from [48] that crowding promotes the coexistence of competing species. We confirm this analytically by studying the (local and global) stability of the coexistence equilibrium and the partial extinction equilibrium. In the second part, we concentrate on infectious disease dynamics. We introduce an epidemiological model with a minimal number of parameters and study the initial stages of the COVID-19 epidemic in Belgium. Among the reported COVID-19 data, we use only hospitalization data to estimate the model parameters and evaluate the basic reproductive number R0. As the states of the model are partially observable, we perform sensitivity and uncertainty analyses in various ways in order to assess the confidence in parameter estimates. Prior to the estimation procedure, we first analyze the structural identifiability of the parameters in the model to ensure that unique solutions for the parameters. We also conduct global sensitivity analysis to investigate the influencing parameters to the model observables. After the estimation, we examine the practical identifiability of the model parameters numerically. While not all model parameters are practically identifiable, it is found that R0 is structurally and practically identifiable.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Park, Shinhae

Contributors dc:contributor

• Rapti, Zoi
• DeVille, Lee
• Bronski, Jared
• Zharnitsky, Vadim

• asymptotic stability
• mathematical models
• bistable dynamics
• epidemic models
• Covid-19
• identifiability analysis
• input-output relations
• profile likelihood

Statement dc:rights

• Copyright 2022 Shinhae Park

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115721

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115721