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ϕ
r
(
M
is
and so the corresponding flux
s
)
of
ϕ
r
(
S
−
(
s
)=
min
{
p
r
,
y
(
s
)
|
y
∈
r
)
}·
f
r
(
s
)
.
Now, if
S
−
(
)=
r
0, then
ϕ
r
(
S
−
(
s
)=
min
{
p
r
,
y
(
s
)
|
y
∈
r
)
}·
f
r
(
s
)
=
min 0
·
f
r
(
s
)=
f
r
(
s
)=
ϕ
r
(
s
)
.
Otherwise, if
S
−
(
r
)
=
0, since the system
M
is supposed to be non-cooperative, then
S
−
(
1 and, having
S
−
(
|
r
)
|
=
r
)=
{
x
}
, we can set:
ϕ
r
(
S
−
(
s
)=
min
{
p
r
,
y
(
s
)
|
y
∈
r
)
}·
f
r
(
s
)
r
−
(
x
/|
x
)
|
=
)
·
f
r
(
s
)
ψ
(
s
)+
∑
r
∈
R
−
(
x
)
f
r
(
s
x
r
−
(
x
/|
x
)
|
=
1
)
+
∑
r
∈
R
−
(
x
)
)
·
f
r
(
s
)
−
∑
r
∈
R
−
(
x
)
f
r
(
s
f
r
(
s
r
−
(
x
x
)
|
·
|
)
x
·
ϕ
r
(
x
=
)
|
·
f
r
(
s
)=
s
)=
ϕ
r
(
s
)
.
|
r
−
(
x
|
r
−
(
x
Therefore, in any case we have
ϕ
r
(
s
)=
ϕ
r
(
s
)
.
M
are the same as the MPF system
In conclusion, the fluxes of the MPR system
M
,
whence the two systems result equivalent.
The MP graph of Fig. 3.9 with its corresponding MP grammars (MPF and MPR) of
Table 3.18 provides, in
N
, the famous Fibonacci's sequence when we start with one
unit of
A
(adult) and zero unit of
N
(newborn).
Fig. 3.9
The MP graph related to Fibonacci's MP grammars of Table 3.18
Other examples of MPR and equivalent MPF systems are given in Tables 3.19,
3.20, and 3.21.
