Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 297

The remainder of the chapter can be summarised as follows. Sects. 2 and 3
analyse respectively the effects of delay and discreteness on blow-up behaviour
(reflecting upregulation due to a positive feedback loop) in minimal models, in
each case investigating both linear and nonlinear (cooperative) feedback. Sect. 4
explores related effects by further characterising a preexisting [ 7 , 9 ] model for
quorum sensing governed by the so-called agr operon. Finally, Sect. 5 briefly
discusses some of the implications of the results.

2 Spatio-Temporal Blow-Up Phenomena:
The Effects of Delays

The context for the current analysis is provided by two very widely studied PDE
models. Firstly, the simplest (linear) reaction-diffusion model 1 takes the form

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

o u

ot ð x ; t Þ¼ o

ð 1 Þ

wherein the linear source term embodies the simplest class of positive-feedback
mechanism, leading (for finite-mass initial data) to infinite-time blow up 2

in the

form

M
2 ð pt Þ 2 e t x 2 = 4t

x ¼ O ð t 2 Þ u as t !þ1;

2 ð pt Þ 2 X x

e t x 2 = 4t ð 2 Þ

1

u as t !þ1; x ¼ O ð t Þ;

t

for some constant M and arbitrary function X that satisfies X ð 0 Þ¼ M. Secondly,

ot ð x ; t Þ¼ o 2 u

o u

ox 2 ð x ; t Þþ u 2 ð x ; t Þ1 \x\ þ1

ð 3 Þ

is an archetypal model for nonlinear positive feedback and one that has spawned
an extensive literature concerning its finite time blow up, an effect already illus-
trated by its spatial homogeneous solution

u ð t Þ¼ 1
t c t

ð 4 Þ

for constant t c [ 0; the diffusion term in ( 3 ) turns out to have a rather limited
influence in mitigating such blow up, the competition between autoinduction

1 We for the most part adopt dimensionless forms containing the minimal numbers of parameters
in this section and the next.
2

Throughout we associate such blow up with upregulation (e.g. quorum sensing).