Biomedical Engineering Reference
In-Depth Information
In the (
u, v
) plane, we obtain
d v
v u
(
-
1)
=
α
d u
u
(1
-
v
)
The variables
u
and
v
can be separated
(1
-
v
)
(
u
-
1)
d v
=
α
d u
(7.50)
v
u
Integration of (7.50) produces the phase trajectories
α
u v u v H
(7.51)
For a given
H
, the trajectories in the phase plane are closed as illustrated in
Figure 7.27.
The diagram of Figure 7.27 shows that the two populations are linked and
form the shifted cycles of Figure 7.26. For our concerns here, we will keep in mind
that the nonlinear terms
bAB
and
dAB
represents the interactions between species
A
and
B
, especially the first term
bAB,
which represents the competition between
the species.
α
+ -
ln(
)
=
7.4 Biochemical Reactions in Microsystems
In the preceding section, we have investigated the kinetics of biochemical reactions.
However, in the reality they can seldom be considered alone without taking into
account other physical phenomena like diffusion or transport. Indeed, the reactants
are usually injected into the microchamber in which they later diffuse and react.
Thus, it is important to consider the global problem of advection-diffusion coupled
with the biochemical reaction itself. We will consider next the kinetics of these
coupled problems.
Figure 7.27
Closed phase plane trajectories from (7.45) with various
H
, corresponding to the Lotka-
Volterra system. The arrows denote the direction of change with increasing time
t
.
