Ali, Adnan (2012) Stochastic pattern formation in growth models with spatial competition. PhD thesis, University of Warwick.

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Abstract

The field of stochastic growth encompasses various different processes which
are ubiquitously seen across the physical world. In many systems, stochasticity
appears quite naturally, where inherent randomness provides the right setting for the
tone of motion and interaction, whose symphony leads to the surprising emergence
of interesting patterns and structure. Although on the microscopic scale one can be
overwhelmed by the randomness arising from the fluctuating interactions between
components, on the macroscopic scale, however, one is mesmerized by the emergence
of mathematical beauty and symmetry, leading to complex structures with fractal
architecture.
Competition between components adds an extra degree of complexity and
leads to the possibility of critical behaviour and phase transitions. It is an important
aspect of many systems, and in order to provide a full explanation of many natural
phenomena, we have to understand the role it plays on modifying behaviour. The
combination of stochastic growth and competition leads to the emergence of interesting
complex patterns. They occur in various systems and in many forms, and
thus we treat competition in growth models driven by different laws for the stochastic
noise. As a consequence our results are widely applicable and we encourage the
reader to find good use for them in their respective field.
In this thesis we study stochastic systems containing interacting particles
whose motion and interplay lead to directed growth structures on a particular geometry.
We show how the effect of the overall geometry in many growth processes
can be captured elegantly in terms of a time dependent metric. A natural example
we treat is isoradial growth in two dimensions, with domain boundaries of competing
microbial species as an example of a system with a homogeneously changing
metric. In general, we view domain boundaries as space-time trajectories of particles
moving on a dynamic surface and map those into more easily tractable systems
with constant metric. This leads to establishing a generic relation between locally
interacting, scale invariant stochastic space-time trajectories under constant and
time dependent metric. Indeed “the book of nature is written in the language of
mathematics” (Galileo Galilei) and we provide a mathematical framework for various
systems with various interactions and our results are backed with numerical
confirmation.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Stochastic processes, Stochastic systems
Official Date:	September 2012
Dates:	Date Event September 2012 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Grosskinsky, Stefan
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	ix, 144 leaves : illustrations.
Language:	eng

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