Biology Reference
In-Depth Information
arithmetic modulo 3. That is, we can add and multiply the elements of this set accord-
ing to rules that are identical to those used to multiply real numbers. Notice that the
numbers in the set are all the remainders one can obtain under division by 3. Now
we add and multiply “modulo 3,” that is, 2 + 2 = 1 and 2
1 as well, that is the
remainder of 2 + 2 = 4 under division by 3 is 1. The number system we obtain in
this way is an example of a “finite field,” which we will use for the remainder of the
chapter. We denote the finite field with three elements by
·
2
=
F
3
, or more generally, the
finite field with
p
elements, where
p
is prime, by
F
p
.
Given rules from an ABM that describe how the agents or variables change their
states, we can find for every agent a polynomial that describes the given rule using
elements of the finite field such as {0, 1, 2} rather than {
low, medium, high
}. By
constructing a polynomial for every single agent, we obtain a set of polynomials that
describes the same behavior as the original ABM. The set is called a
polynomial
dynamical system
.
Definition 5.4 (Polynomial dynamical system).
Let
k
be a finite field and
f
1
,...,
f
n
∈
k
n
k
n
k
[
x
1
,...
x
n
]
be polynomials. Then
F
=
(
f
1
,...,
f
n
)
:
→
is an
n
-
dimensional polynomial dynamical system over
k
[
24
].
Here,
k
n
denotes theCartesian product
k
×
×···×
[
x
1
,...
x
n
]
k
k
with
n
copies of
k
.
k
denotes the ring of polynomials in variables
x
1
,...,
x
n
and coefficients in
k
.For
k
n
the function
F
is evaluated by
(
a
1
,...,
a
n
)
∈
F
(
a
1
,...,
a
n
)
=
(
f
1
(
a
1
,...,
a
n
),...,
f
n
(
a
1
,...,
a
n
)).
Example 5.5.
Let
k
= F
3
={
0
,
1
,
2
}
and
F
=
(
f
1
,
f
2
,
f
3
)
, where
f
1
=
x
1
+
x
2
f
2
=
x
1
x
2
+
x
3
+
1
x
3
.
f
3
=
x
1
+
x
2
+
3
3
3
, and iteration of
F
results in
a dynamical system.
F
has one fixed point, or steady state, namely (1, 0, 2) because
F
The PDS
F
maps points in
F
3
to other points in
F
(
1
,
0
,
2
)
=
(
f
1
(
1
,
0
,
2
),
f
2
(
1
,
0
,
2
),
f
3
(
1
,
0
,
2
))
=
(
1
,
0
,
2
)
, and note that all cal-
culation are done “modulo 3” because
F
is over
F
3
. Furthermore,
F
has a limit cycle
or oscillation of length five: applying
F
repeatedly to (0, 2, 1) yields
(
0
,
2
,
1
)
→
(
2
,
2
,
0
)
→
(
1
,
2
,
1
)
→
(
0
,
1
,
1
)
→
(
1
,
2
,
2
)
→
(
0
,
2
,
1
)
→
(
2
,
2
,
0
)
→ ···
With software packages such as ADAM [
25
], one can compute the dynamics of a
PDS and visualize it.
Most agent-based models can be translated into a polynomial dynamical system
of this type. Each variable
x
i
represents an agent, a patch, or some other entity in
the model. Each element in the finite field
k
represents a different state or condition
of the variables. The corresponding polynomials
f
i
encode the behavior of the agent
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