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Title: Noise-induced bistability and stochastic patterns
Author: Biancalani, Tommaso
Awarding Body: University of Manchester
Current Institution: University of Manchester
Date of Award: 2013
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This thesis presents a mathematical analysis of two classes of behaviours which occur in systems of populations: noise-induced bistability and stochastic patterning. Both behaviours have their origins in the intrinsic stochasticity possessed by a population system due to the discreteness of the individuals: the intrinsic noise. In the study of noise-induced bistability, we analyse a system which exhibits switching between two states. These states do not correspond to fixed points of the corresponding system of deterministic equations, but instead are the states at which the system stochasticity is minimal or vanishing. This feature suggests that the mechanism is intrinsically different to the traditional paradigm of bistability, in which a system with two stable fixed points is subject to noise. Through our mathematical analysis we highlight some characteristic properties of the dynamics, suggesting a way to distinguish, in a real system, the presence of noise-induced bistable states from other types of bistability. Stochastic patterning arises when noise acts on a reaction-diffusion system which exhibits pattern formation via an instability of the homogeneous state. If the system is close to the onset of the instability, whilst still in the stable regime, then patterning occurs due to a combination of stochastic agitation and the exponential decay of the underlying stable homogeneous state. We investigate the case of the stochastic travelling waves on both regular lattices and complex networks. In both cases, a complete analytical treatment is provided via the power spectra of fluctuations. The spirit of the thesis is to propose a simple model which is representative of an observed behaviour, and then solve the model analytically. Numerical simulations are used throughout to verify the accurateness of the analytical approximations. Thus the analytical treatments constitute the core of the work and have two purposes. They are explanatory, in the sense that they help to develop intuition about how the noise leads to a certain behaviour. Moreover, they give quantitative understanding, as we provide the explicit expressions for various quantities (stationary distributions, mean times, etc.). In some cases, the formulas that we have obtained do not rely on the details of the model, so that we would expect them to fit experimental data. In other cases this is not so, yet the analytical treatment may give insight on how to attack more realistic models.
Supervisor: Mckane, Alan Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available