Abstract:
The Brusselator model is widely used to illustrate and study basic features
of models of chemical reactions involving trimolecular steps. We provide
the necessary mathematical theory related to Reaction-Diffusion systems in
general and to the Brusselator model in particular. Specifically, the issues
of local and global existence of solutions, their uniqueness and regularity are
discussed in detail. The theoretical and numerical investigation presented
futher in the thesis provides an insight into the asymptotic behavior of the
solutions of this model, characterizing the parameter region for each of the
three qualitatively different cases: homogeneous steady state, Turing pattern
and bulk oscillations. Particular attention is given to the supercritical Hopf
bifurcation parameter domain where no substantial theory is available. This
study was largely motivated by the observations of Young, Zhabotinsky and
Epstein that Turing patterns eventually (for sufficiently small ratio of the
diffusion coefficients) dominate the Hopf bifurcation induced bulk oscillations.
In this work we confirm this observation and further establish more precisely
the shape of the boundary separating the Turing pattern domain and the bulk
oscillations domain in the parameter space. The obtained results are used in
revealing an essential mechanism generating oscillating patterns in the coupled
Brusselator model. It can be considered as a model of the reaction sequences in
two thin layers of gel that meet at an interface. Each layer contains the same
reactants with the same kinetics but with different diffusion coefficients. The
occurrence of oscillating patterns is due to the fact that for the same values of
the parameters of the model but with different diffusion coefficients the one
system can be in the Turing pattern domain while the other is in the bulk
oscillations domain. Hence, roughly speaking, one layer provides a pattern
while the other layer drives the oscillations.
