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

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-08-02T18:02:48Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Fock space approach to Schnakenberg model.pdf: 6802590 bytes, checksum: 388dea2a463bd63474eaab61feece049 (MD5) / Made available in DSpace on 2016-08-02T18:02:48Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Fock space approach to Schnakenberg model.pdf: 6802590 bytes, checksum: 388dea2a463bd63474eaab61feece049 (MD5) 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.