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one of the
I
i
C
, i.e., one of the neighbors' concentrations level is the highest
possible. In this case,
f
=
(
l
)
=
l
+
i
. Indeed, the right hand of the first line evaluates to
p
−
1
l
i
for the
i
such that
I
i
=
C
(1
−
(
C
−
I
i
)
)
l
i
is
l
i
if and only if (
I
i
=
C
). If there are
several neighbors with concentration level
C
should evaluate to the neighbor
with the smallest index. This is assured by multiplying with
i
−
1
j
,
f
(
l
)
1
, which
evaluates to 0, if a neighbor with a smaller index has concentration
C
. The second
and third line in Eq. (
5.9
) are equivalent to the first row, as they describe movement to
p
−
0
(
C
−
I
j
)
=
patches with concentration levels lower than
C
.Theterm
k
=
m
+
1
i
=
0
(
p
−
1
assures that the second summand evaluates to 0 if a patch has a higher concentration
than
m
. The proof for the downhill movement (Eq.
5.8
) is similar and left as an
exercise.
Exercise 5.31.
Based on the uphill and downhill polynomial, construct a polynomial
f
k
−
I
i
)
Exercise 5.32.
Consider a 13 by 13 grid with torus topology. A rabbit moves two
steps up and two to the right at every iteration. Construct a polynomial that describes a
rabbit's movement. Use the polynomial to simulate the movement of a rabbit starting
in the center of the grid.
(
l
)
that evaluates to the maximum concentration of its eight neighbors and itself.
Exercise 5.33.
For the rabbit in Exercise
5.32
, after how many iterations do you
expect it to reach its starting position again? Confirm your answer by evaluating the
polynomial.
Exercise 5.34.
Construct the polynomial as in Exercise
5.32
for a grid of arbitrary
size
n
by
n
.
Exercise 5.35.
Consider a grid where each patch is covered with a
low, medium
,
or
high
amount of grass. At each iteration, the rabbit moves to the neighboring patch
with the most grass, i.e., to one of the eight neighboring patches or it stays on the
same patch. Construct a polynomial describing the rabbit's movement based on the
amount of grass on the neighboring patches.
Exercise 5.36.
Consider the grid from Exercise
5.35
. Rabbits eat grass, and the
amount of grass on a patch decreases by one level for every iteration that a rabbit
occupies it, i.e., a patch with
high
grass changes to
medium
grass, if occupied by a
rabbit,
medium
to
low
, and
low
remains
low
. Construct a polynomial that describes
the amount of grass on a patch.
We provide these general polynomials and simplification techniques to aid in the
transformation of an ABM into a PDS. Whereas large agent-based models may be too
complex for efficient analysis, we hope that the algebraic structure of a polynomial
dynamical system can be used to expedite computation of optimal control.
5.10
SUMMARY
Agent-based models provide a very intuitive and convenient way to model a variety of
phenomena in biology. The price we pay for these features is that the models are not
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