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Title: Balance manifolds in Lotka-Volterra systems
Author: Ching, Atheeta
ISNI:       0000 0004 8507 9810
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2019
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The Lotka-Volterra equations are a dynamical system in the form of an autonomous ODE. The aim of this thesis is to explore the carrying simplex for non-competitive Lotka-Volterra systems for the case of 2- and 3-species, where it is referred to as a balance simplex. Carrying simplices were developed by M.W. Hirsch in a series of papers. They are hypersurfaces which asymptotically attract all non-zero solutions in the phase portrait. This essentially means that all the non-trivial dynamics occur on the carrying simplex, which is one dimension less than the system itself. Many of its properties have been studied by various authors, for example: E.C. Zeeman, M.L. Zeeman, S. Baigent, J. Mierczyński. The first few chapters of this thesis explores the 2-species scaled Lotka-Volterra system, where all intrinsic growth rates and intraspecific interaction rates are set to the value 1. This simplification of the model allows for an explicit, analytic form of the balance simplex to be found. This is done by transforming the system to polar co-ordinates and explicitly integrating the new system. The balance simplex for this 2-species model is precisely composed of the heteroclinic orbits connecting non-zero steady states, along with these states themselves. The later chapters of this thesis focuses on the 3-species case. The existence of the balance simplex in particular parameter cases is proven and it is shown to be piecewise analytic (when the interaction matrix containing the parameters is strictly copositive). These chapters also work towards plotting the balance simplex so it can be visualised for the 3-species system. In another chapter, more general planar Kolmogorov models are considered. Conditions sufficient for the balance simplex to exist are given, and it is again composed of heteroclinic orbits between non-zero steady states.
Supervisor: Baigent, S. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available