Stable configurations for population and social dynamics

This dissertation investigates global and local minima in two models: the Lotka--Volterra model for population dynamics and a tractable polarized opinion social dynamic model. This dissertation contains stability results of the Lotka--Volterra model when induced by a cycle graph food web network. Results such as orbits, chaos and the probability of stability are given. A result showing convexity of the weighted connections of the food web is sufficient for global stability is given as well. Stability results of food webs which are perturbed from the cycle graph are explored as well for comparison. This dissertation goes on to investigate how algebraic relationships within the community matrix predict stability for the generalized Lotka--Volterra model. In particular, it is shown that there is a strong relationship between the transversal eigenvalues with respect to a subset of the species in a system and the Schur compliment of the Jacobian at the interior fixed point with the submatrix determined by the same subset of species. This relationship gives an alternate proof to many well known results. This dissertation also analyzes the global and local stability of an opinion dynamic model which consists of a W-well potential and a graph Laplacian for coupling. The global minimizers and their lack of confinement to an orthant are investigated. The number of local minimizers are also investigated for various W-well potentials. This dissertation investigates the different types of bifurcations that can be seen depending on the differential properties of the W-potentials.

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

2019

Author dc:creator

• Livesay, Michael Richard

Contributors dc:contributor

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

• population dynamics, opinion dynamics, local stability

Statement dc:rights

• Copyright 2019 Michael Livesay

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/104735

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/104735