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The Scientific Way to Simulate Pattern Formation in Reaction-Diffusion EquationsCleary, Erin 09 May 2013 (has links)
For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada
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Fock space approach to schnakenberg modelLEANDRO, Marlon Oliveira Martins 24 May 2016 (has links)
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Previous issue date: 2016-05-24 / Capes / This work has as objective to study the relationship between Fock spaces, quantum
operators and chemical reaction systems used to model many stochastic processes with
biological motivation, such as population growth, heartbeats and cell metabolism. In
particular, we work with a generalization of Schnakenberg model, in its deterministic
and stochastic versions, and which is used to describe the process of cellular glycolysis.
As it is a new and at the same time a interdisciplinary area, involving knowledge in
stochastic processes, theoretical physics, chemistry and biomathematics, this text seeks to
be self-contained, including all the elements needed for the study of theme, for the purpose
of contribute to the better understanding between these diverse areas. / Este trabalho tem como objetivo estudar a relação entre espaços de Fock, operadores de
mecânica quântica e sistemas de reações químicas usados para modelar diversos processos
estocásticos com motivação biológica, como crescimento populacional, batimentos cardíacos
e metabolismo celular. Em particular, vamos trabalhar com uma generalização do modelo
de Schnakenberg, em suas versões determinística e estocástica, e que pode ser utilizado
para descrever o processo de glicólise celular. Como é uma área recente e ao mesmo tempo
interdisciplinar, envolvendo conhecimento em processos estocásticos, física teórica, química
e biomatemática, este texto procura ser autocontido, incluindo todos os fundamentos
necessários para o estudo do tema, visando contribuir para o melhor entendimento entre
estas diversas áreas.
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