Cellular Automata Simulation 219
sugar
sugar
green mountain
green mountain
FIGURE 7.30: Two sugar (gre e n) mountains in the Sugarscape model.
incorporation of concepts such as breeding, pollution, culture, combat, trade,
and dise ase.
The notatio ns and definitions regarding Sugarscape presented in this sec-
tion are based on [33].
7.10.1 A simp le Sugarscape model
Agents live in a spa c e named Sugarscape, which is a two-dimensional
square lattice toroida lly connected at the top, bo ttom, and left and right
edges. Each lattice point contains sugar, and the maximum quantity of sugar
is set on a per- point basis. The amount of sugar increases at a pre defined rate
(cf. regeneration rules G
α
and G
∞
, which are described below). The agents
collect sugar from lattice points, and the sugar at each point is res tored to its
maximum if not collected by age nts for a short period of time. Sugar is thus
maximized at all points at the initial iteration step.
Figure 7.30 shows an example Sugarscape model. Green mountains are
located at the northeast (top right) and southwest (bottom left) corners of
the lattice, where areas of the mountains with greater elevation repre sent
larger amounts of sugar, with the maximum quantity at the peaks set to 4.
In contrast, the areas toward the northwest and southeast corners eventually
become completely devoid of sugar. Note tha t since the lattice is toroidally
connected, when an agent crosses the right (left) edge, it reappears on the
left (right) edge. We will now explain this simple environment for the initial
model.
Initially, the quantity of sugar everywhere increases according to the simple
rule described next [33, p. 23]. Various other rules for sugar gr owth have been
devised, and the rule given here is extended in later sections.
Sugarsca pe growth rule G
α
:
The sugar grows at a rate of α units per step, where the maximum growth is
such that the upper limit of the quantity for that p osition is attained.
220 Agent-Based Modeling and Simulation with Swarm
In particular, the following rule is known as the instantaneous growth
rule [33, p. 27].
Sugarsca pe growth rule G
∞
:
Sugar instantaneously grows to the maximum quantity for that position.
Two important agent features are their vision and metabolic rate. These
features are different for each agent a nd can be genetically transferred from
parent to child. Therefore, although the agents consume (burn) sugar at each
step, the consumption o f s ugar depends on the metabolic rate. In the following
exp e riment the metabolic rate for the population forms a uniform distribution
with a minimum value of 1 and a maximum of 4. The vision of the agents is
limited to the four directions of the lattice (up, down, left, and right), and
the agents a re unable to see across a diago nal. Furthermore, the distribution
of vision is uniform, with a minimum of 1 and a maximum of 6 , a nd an agent
with a v ision of 3 can se e up to three units away in each direction.
The agents thus accumulate sugar while moving around in the S ugarscape ,
and each agent is capable of s tockpiling an unlimited amount of sugar.
Agents move in accordance with the rule given below [33, p. 25], in which
they process local information (such as the current amount of sugar at each
position) within their field of view, and subsequently compute their order of
preference for relocation.
Agent movement rule M:
• Survey the lattice in all four directions (up, down, left, and right) and
search for positions within the field of vision containing the larges t quan-
tity of sugar and no other agents.
• If more than o ne such position ex ists, choose the nearest.
• Move to the new position.
• Collect all the sugar at the new position.
Agents can move only once at each step of the iteration, and the order in
which agents move is random. When they a rrive at the new position, their
sugar reserve is increased by the amount of sugar at the new position minus
the amount of metabo lized sugar. Metabolized sugar cannot be accumulated,
and if an agent’s reserve becomes zero or negative, the agent dies of starvation
and disappears from Sugarscape.
In the fo llowing discussion, if Sugarscape follows growth rule E and the
agents fo llow movement rule A, the set of experimental conditions is deno ted
(E,A). For example, if sugar is replenished by the instantaneous regeneration
rule and the ag e nts move in accordance with rule M, then the experimental
conditions are given by (G
∞
,M).
Let us conduct a simulation with the (G
1
, M) rule set. Applying these rules
to a group of agents distributed at random, agents located at p ositions with
Cellular Automata Simulation 221
One step later 10 steps later 50 steps later
FIGURE 7.31: Agent aggregation in (G
1
, M).
(a) population (b) average and std. values (vi-
sion, metabolism)
FIGURE 7.32: Population and feature changes in (G
1
, M).
little sugar exhaust their food and die, and only agents in positions relatively
rich in sugar survive. Moreover, since agents rely on their field of view to
discover and move to positions rich in sugar, the agents aggregate at the two
mountains of sugar after 50 steps (Fig. 7.31).
After 50 steps, the population of agents has been reduced to ab out 100
from a value of 40 0 at the start of the simulation, and altho ugh the average
metabolic rate for all agents is initially 2.5, this value converges to about 1.
However, no improvement in vision is found even after 100 steps. Thus, vision
can be considered as not being of prime importance in this environment, or
perhaps vision is already sufficiently advanced in this case (Fig. 7.32).
Next, let us experiment with the (G
∞
,M) rule set. In this environment,
the survival rate is low fo r agents with high metabolic rates and poor vision.
Therefore, such agents die while the remaining agents settle at the most suit-
able locations and form a stable state. As a result, it can be obser ved that
rather than aggregating at the peaks of the two mountains, the agents arrange
themselves in a formation resembling contour lines along the mountain ridg e s
(terraces) of Sugarscape [33]. Epstein and Axtell explained this phenomenon
as originating from the vision of the a gents:
222 Agent-Based Modeling and Simulation with Swarm
Specifically, suppose you are an agent with vision 2 and you are
born on the terrace of suga r height 2, just one unit south o f the
sugar terra c e of level 3. With vision 2, you can see that the nearest
maximum sugar po sition is on the ridge of the sugar terrace of
height 3, so, obeying rule M, you go there and collect the sugar.
Since there is instant growback, no point on the level 3 sugar
terrace is an improvement; and with vision of only 2, you cannot
see the higher terrace of sugar level 4. So you stick on the ridge
[33, p. 2 8].
This description suggests that while Sugarsc ape constitutes a simple envi-
ronment, by conducting demonstrative simulations of complex sys tems, Sug-
arscape is an exceedingly effective tool for studying human society. Let us
consider one such experiment in detail in the fo llowing paragraphs.
7.10.2 Life and birth
To increase the model dynamics, agents are born with a certain lifespan
between 60 and 100 iterations that is selected at random fr om a uniform
distribution. Since all agents perish after 100 steps under these conditions, a
rule for replacing agents is also added to the rule set [33, p. 33].
Agent replacement rule R[a, b]
An agent that dies is replaced by a new agent with an age of 0 and a random
set o f genes, lifespan (with an interval set a s [a, b]), initial res e rve, and loc ation.
Explicitly, since a single dead agent is replaced by a single agent generated
at random, the population is maintained at 400. The implementation of this
rule re sults in small fluctuations in the average values of the agents’ attributes
owing to the addition of the random attribute values of the new agent. Under
these conditions, the metabolic rate decrea ses rapidly, whereas vision re mains
almost unchanged (Fig. 7.33). Additionally, although the agents accumulate
the surplus s ugar and thus possess considerable reserves, the average stored
quantity appears to settle a round 30 due to the limited lifespan o f the agents.
The maximum size of a reserve is about 100 (Fig. 7.34).
7.10.3 Breeding
The last basic setting is a breeding r ule. Agents are divided into males
and females, and adja c e nt agents of opposite sex breed and produce offspring.
However, for successful breeding, the following conditions must be satisfied.
• Age limit: bo th males a nd females ca n start breeding when they are
between 12 and 15 steps of age, and breeding stops between 40 and 50
for females and between 50 and 60 for males. Each agent is assigned
Cellular Automata Simulation 223
FIGURE 7.33: Fe atures (vision, metabolism) in (G
1
, {M, R
[60,100]
}) (aver-
age and std. values).
(a) Average and std. values (b) Maximum values
FIGURE 7.34: Wealth in (G
1
, {M, R
[60,100]
}).
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