Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 298

driving a local increase in u and diffusion acting to smear out such a peak being an
intrinsic feature of such models.
Here we incorporate delays in feedback, typically associated in the autoregu-
lation of gene expression with the time needed for transcription and translation of
further copies of the relevant transcription factor but also attributable to intra-
cellular transport, for example. Thus in generalising ( 1 ) we investigate

ot ¼ o 2 u

o u

ox 2 ð x ; t Þþ u ð x ; t T Þ1 \x\ þ1;

ð 5 Þ

where T [ 0 corresponds to the delay time and we lump the entire signalling and
genetic machinery into the single variable u ð x ; t Þ , which could be thought of as the
concentration of a signalling molecule within a population of cells.
In the spatially homogeneous case, solutions to ( 5 ) can be written as a super-
position over modes of the form

u ð t Þ¼ A ð k Þ e kt ; k ¼ e kT

ð 6 Þ

for constants A, the large time behaviour being dominated by the real root for k,
whereby

k 1 T as T ! 0 ;

ð 7 Þ

k 1

T ln T

as T !þ1;

i.e. the longer the delay, T , the lower the growth rate; it is this real root that we
denote by k in what follows. The large-time behaviour in the spatially structured
case ( 5 ) then follows readily: setting

u ð x ; t Þ¼ e kt v ð x ; t Þ

and treating v to be slowly varying in t (an assumption whose validity is readily
confirmed a posteriori) implies

ot o 2 v

ð 1 þ kT Þ o v

ox 2

and hence

u ð 1 þ kT Þ 2 M
2 ð pt Þ 2

e kt ð 1 þ kT Þ x 2 = 4t x ¼ O ð t 2 Þ; as t !þ1; ð 8 Þ

so that the delay has the effect of reducing the effective diffusivity as well as the
growth rate. The behaviour as t !þ1 with x ¼ O ð t Þ is also instructive and can
be characterised by applying the Liouville-Green (JWKB) method in the form

u a ð x ; t Þ e f ð x ; t Þ

ð 9 Þ