| Summary: | <p>Mathematical modelling of complex biological phenomena allows us to under-
stand the contributions of different processes to observed behaviours. Many of
these phenomena involve the reaction and diffusion of biomolecules and so we
use so-called reaction-diffusion models to describe them mathematically. By
discretising space into compartments and letting system dynamics be governed
by the reaction-diffusion master equation, it is possible to derive and simulate
a stochastic model of reaction and diffusion on an arbitrary domain. However,
there are many implementation choices involved in this process, such as the
choice of discretisation and method of derivation of the diffusive jump rates,
and it is not clear a priori how these affect model predictions.</p>
<p>To shed light on how model implementation choices affect simulation results, in
this thesis we explore how a variety of discretisations and methods for deriva-
tion of the diffusive jump rates affect the outputs of stochastic simulations of
reaction-diffusion models, in particular using Turing’s model of pattern forma-
tion as a key example. While only minor differences are observed for simple
reaction-diffusion systems, there can be vast differences in model predictions
for systems that include complicated reaction kinetics, such as Turing’s model
of pattern formation.</p>
<p>To date, very few works have considered domain growth for stochastic models.
Hence, we consider both static and uniformly growing domains and demon-
strate that domain growth can impact how the different diffusive jump rates
affect simulation results. Furthermore, we consider discretisations of the do-
main using complex meshes. Using complex meshes allows models to cap-
ture geometric characteristics of a system. We investigate the effects of using
complex meshes and the corresponding choices of diffusive jump rates on sim-
ulation results. This thesis highlights that care must be taken in using the
reaction-diffusion master equation to make predictions as to the dynamics of
stochastic reaction-diffusion systems.</p>
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