  Gene expression time delays, modelling the complex biological processes of
gene transcription and translation, have been shown to play an important role
in cellular dynamics. Time delays, motivated by the gene expression process,
can also greatly affect the behaviour of reaction-diffusion systems. In this
dissertation, we explore their effects on Turing pattern mechanisms. By
incorporating time delays, modelled as both a fixed parameter and as a
continuous distribution, into classical reaction-diffusion systems that exhibit
Turing instabilities, we investigate the changing behaviour of these systems.
We find that an introduction of increasing time delay increases the time taken
for spatially inhomogeneous patterns to stabilise, and the two are related
linearly. We also present results to show, through a linear stability analysis,
that an increasing time delay can act both to expand or shrink the Turing space
of a certain reaction-diffusion mechanism, depending on the placement of
time-delayed terms. Significantly, we find that modelling time delays as a
continuous distribution has a negligible impact on qualitative or quantitative
aspects of the results seen compared with a fixed time delay of the mean of the
distribution. These findings serve to highlight the importance of considering
gene expression time delays when modelling biological patterning events, as
well as requiring a complete understanding of the cellular dynamics before
attempting to apply Turing mechanisms to explain biological phenomena. The
results also suggest, at least for the distributions considered in this
dissertation, that fixed delay and distributed delay models have almost
identical dynamics. This allows one to use simpler fixed delay models rather
than the more complicated distributed delay variants.
