Symmetry-breaking bifurcations in compartmental-reaction diffusion systems with comparable diffusivities
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Pelz, Merlin
issued: 2024-07-25T16:26:18Z
issued: 2024
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns typically occur from spatially uniform states when a slowly diffusing "activator" species reacts with a much faster diffusing "inhibitor" species. Thus, one key long-standing question is how to robustly obtain pattern formation in the biologically often realistic case where the time scales for diffusion of the interacting species are comparable. For introduced finite and infinite 1-D and 2-D coupled PDE-ODE bulk-cell models, we investigate symmetry-breaking bifurcations that can emerge when spatially segregated dynamically active compartments, the "cells", are coupled by a PDE bulk diffusion field with comparable diffusivities and degradation rates that is both produced and absorbed by the entire cell population. In the singular limit of a small common cell radius, we construct steady-state solutions and corresponding eigenperturbations and formulate a nonlinear matrix eigenvalue problem that determines the linear stability properties of the steady-states. As regulated by the ratio of the membrane reaction rates on the cell boundaries, we show for various specific prototypical intracellular reactions that our 1-D and 2-D coupled PDE-ODE models admit symmetry-breaking bifurcations from symmetric steady-states defined by a common intracellular steady-state for all cells, leading to linearly stable asymmetric patterns (cell specialization), even when the bulk diffusing species have comparable or possibly equal diffusivities. For a version of our proposed model in ℝ² and ℝ³ with one diffusing species, a hybrid asymptotic-numerical theory is developed by deri…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/88694
note: Symmetry-Breaking Bifurcations inCompartmental-Reaction Diffusion Systems withComparable DiffusivitiesbyMerlin PelzB.Sc. in Technomathematics, University of Bayreuth, 2016M.Sc. in Mathematics, Technical University of Munich, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)July 2024© Merlin Pelz, 2024The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Symmetry-Breaking Bifurcations in Compartmental-ReactionDiffusion Systems with Comparable Diffusivitiessubmitted by Merlin Pelz in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Michael Jeffrey Ward, Professor, Mathematics, UBCSupervisorMiranda Holmes-Cerfon, Associate Professor, Mathematics, UBCSupervisory Committee MemberEric Cytrynbaum, Associate Professor, Mathematics, UBCUniversity ExaminerCarl Ollivier-Gooch, Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Daniel Coombs, Professor, Mathematics, UBCCo-SupervisorAnotida Madzvamuse, Professor, Mathematics, UBCSupervisory Committee MemberBrian Wetton, Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractOriginating from the pioneering study of Alan Turing, the bifurcation analysispredicting spatial pattern formation from a spatially uniform state for diffusingmorphogens or chemical species that interact through nonlinear reactions is acentral problem in many chemical and biological systems. From a mathematicalviewpoint, one key challenge with this theory for two component systems isthat stable spatial patterns typically occur from spatially uniform states when aslowly diffusing “activator” species reacts with a much faster diffusing “inhibitor”species. Thus, one key long-standing qu…
has type: Thesis/Dissertation
date issued: 2024-11
is shown at: 10.14288/1.0444814
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Ward, Michael J. (Michael Jeffrey)
contributor: Coombs, Daniel
type: Text
identifier URI: http://hdl.handle.net/2429/88694