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dx
dt
=(
A
−
By
)
x
(3.12)
dy
dt
=(
Cx
−
D
)
y
.
The solution of these equations, starting from suitable initial values of the two pop-
ulations exhibits a typical oscillation of both population sizes.
Now we show an MP analysis of the same phenomenon which provides the same
conclusion. A prey is replicated, by means of a rule
x
2
x
, in proportion to the
number of prey (their average proliferation, and the availability of resources from
the environment determine the proportionality factor). The rate of predator repli-
cation and prey death (rules
r
2
and
r
3
) depends on both the number of prey and
predators. Finally, predator death is proportional to predator population size (rule
r
4
). We do not discuss here the method for determining these parameters, which
will be explained in the following sections, however, the MP grammar of Table 3.8
provides a typical Lotka-Volterra dynamics which is depicted on the left part of
Fig. 3.4.
→
Ta b l e 3 . 8
MP grammar of Lotka-Volterra dynamics
Reactions
Regulators
r
1
:
x
→
2
x
ϕ
1
=
0
.
03
x
0001
x
2
y
r
2
:
x
→
0
ϕ
2
=
0
.
0009
+
0
.
009
xy
+
0
.
ϕ
3
=
0
.
0009
+
0
.
015
xy
+
0
.
0003
x
2
y
r
3
:
y
→
2
y
r
4
:
y
→
0
ϕ
4
=
0
.
066
y
We recall that, in accordance with our general notation, in Table 3.8, symbols
x
y
are used with different meanings. In fact, when they are used in MP-rules, they
represent metabolites, while when they are arguments of regulators, they denote
metabolite quantities in the given metabolic state.
,
3.1.2
The Brusselator (Belousov-Zhabotinsky Reaction)
Brusselator is an idealization, due to Prigogine's school [112], of a famous kind
of chemical oscillating reaction, called BZ reactions, discovered by Belousov and
then analyzed by Zhabotinsky. The best-known BZ reaction can be created with a
mixture of potassium bromate
KBrO
3
, malonic acid
CH
2
2
, and manganese
sulfate
MnSO
4
prepared in a heated solution of sulfuric acid
H
2
SO
4
. Brusselator has
the following form:
(
COOH
)
⎧
⎨
a
→
x
2
x
+
y
→
3
x
⎩
b
+
x
→
y
+
d
x
→
e
.
