Abstract
This thesis mainly presents new research on mathematical descriptions for stochastic interactions in multi-species systems. Oscillations have been observed in many biological competition models, e.g. the Lotka-Volterra model and the Rock-Paper Scissors (RPS) model. In this thesis, by analysing properties of these oscillations, the effects of demographic noise are demonstrated and comprehensively explained in some specific stochastic multi-species competition models. Three paradigmatic models are introduced and analysed using novel, and, generally applicable techniques.Chapter 2 considers the well-known cyclic dominance dynamics of three mutually competing populations, as in the toy model for the RPS game. In this chapter, the non-zero-sum RPS model is studied in detail, especially for the case with small mutation rate. The mean period of oscillations is computed in detail for stochastic competition models. Phenomena that are observed numerically are explained theoretically in details using techniques from Markov Processes, asymptotics, local-global maps and stochastic differential equations.
In chapter 3 the focus is on the extension to models of four species interacting. Even in the simplest case of symmetric interactions there are new possibilities for the deterministic (and stochastic) dynamics.
Chapter 4 considers the effect of a random environment of competitive interactions on the dynamics of the simplest RPS model. By coupling the zero-sum RPS model to a larger well-mixed random system, the RPS subsystem still has a sustained oscillation, but the oscillation has larger amplitude and higher frequency which is induced by the random interaction network. Via the cavity method, the changes in amplitude and frequency can be estimated quantitatively.
| Date of Award | 2 Mar 2018 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Jonathan Dawes (Supervisor) & Tim Rogers (Supervisor) |
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