Biology Reference
In-Depth Information
2
2
y
2
(
1
−
(
x
−
2
)
)(
1
−
)
+
0
+
1
2
2
(
1
−
(
x
−
2
)
)(
1
−
(
y
−
2
)
)
=
x
+
y
.
Converting an agent-based model into a polynomial dynamical system provides
us with a conceptual advantage, since rather than being limited to working with a
computer simulation as our only means of analysis, methods and theory from abstract
algebra and algebraic geometry can be used. For example, one might be interested in
the steady states of a model, i.e., all the configurations of the system that do not change
over time. Written as a polynomial dynamical system
F
k
n
k
n
, these states are
:
→
exactly the solutions to
F
(
x
)
=
x
,the
n
-dimensional system of equations
f
1
(
x
)
=
x
1
,...,
f
n
(
x
)
=
x
n
in
k
[
x
1
,...,
x
n
]
[
25
]. Equivalently, the solutions are the points in
the variety
of the ideal generated by the polynomials
f
1
(
x
)
−
x
1
,...,
f
n
(
x
)
−
x
n
.
V
For an introduction to varieties, we recommend [
27
].
When the polynomials describing the biological system have a special structure,
other analysis methods are available. For example, for conjunctive networks, i.e., a
PDS over
F
2
where each polynomials is a monomial, the dynamics can be completely
inferred by looking at the dependency graph or wiring diagram of the PDS [
28
]. We
can determine the fixed point and limit cycle structure of this PDS without using any
simulation.
We have mentioned
Conway's Game of Life
, a cellular automaton as a special
case of agent-based models earlier in this chapter. Cells are agents that die or come
to life based on a rule including their eight neighbors [
5
]. The Game of Life can
be translated into a polynomial dynamical system. Each variable
x
i
represents a cell
on the grid. Each polynomial
9
2
→ F
2
depends on
x
i
's eight neighbors and
itself and describes whether
x
i
will be dead (0) or alive (1) at the next iteration given
the values of its neighboring cells. The fixed points of this system correspond to
still
lives
such as blocks and beehives, two cycles to
oscillators
, e.g., blinkers and beacons.
Working with the mathematical representation of the game, one can for example study
still lives by using concepts from invariant theory using the symmetry of the rules as
group actions.
Exercise 5.29.
Using Lagrange's interpolation formula (
5.1
), construct the polyno-
mial
f
1
: F
f
i
: F
9
2
→ F
2
describing the behavior of cell
x
1
with neighbors
x
2
,...,
x
9
, i.e.,
the polynomial
f
1
(
should evaluate to 0 or 1 (dead or alive) accord-
ing to the rules of the Game of Life given the state of the cell
x
1
and its eight
neighbors.
x
1
,
x
2
,...,
x
9
)
In the next section, we provide some polynomials for common rules in agent-
based models. By providing such rules, we hope to simplify the process of translating
an ABM to a PDS. The polynomials can be used as given or as a starting point to
construct functions that represent more complex behavior.
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