ISSN 1320-0682
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ISSN 1320-0682 |
| Source: | http://www.complexity.org.au/ci/vol06/nakamura/nakamura.html | Received: | 01/07/1998 | ||
| Vol 6: | Copyright 1998 | Accepted for publication: | 15/10/1998 |
Mari Nakamura Koichi Kurumatani
Life Electronics Research Center
Electrotechnical Laboratory (ETL)
Email: mari@etl.go.jp
Information Science Division
Electrotechnical Laboratory (ETL)
In this paper, we propose an improved ant colony model in which the foraging behavior of ants can be observed on a macro-scale as a result of micro-scale interacting behaviors among many individual ants. Simulation results of the new model have shown that the system changes its foraging behavior and selects the appropriate foraging strategy according to the food-supply rate. The mechanism for changing the foraging strategy is explained.
In a previous paper [1], we proposed a model for observing the foraging behavior of an ant colony composed of many ants whose sensitivity to stimuli was limited due to the small area surrounding them. With this model, when the ants are collecting food from sites outside of their nest, they behave based on the following rules.
An ant looking for food (i.e., in a searching task) walks randomly until it finds a food site or detects recruitment pheromone signal [2]. If it finds a food site, it changes its task to a carrying task. An ant involved in the carrying task retrieves a bit of food straight to its nest while laying recruitment pheromone on the ground. After carrying the food to the nest, it returns to the searching task. The laid recruitment pheromone gradually diffuses over a wide area. If other ants involved in the searching task detect this pheromone signal, they immediately change their task to the recruited task. Ants involved in the recruited task are attracted to the pheromone, and follow the pheromone's trail to the food site. After finding the food site, they change their task to the carrying task.
We performed a simulation of the above-described model. We observed a recruitment competition process among food sites; the process resulted in over-concentrated recruitment at a single site [1], [3]. During the competition, most of ants in the colony were assigned to the carrying or recruited tasks. This resulted in fewer ants involved in the searching task localized around the nest.
We suppose that ants were desensitized to the pheromone, in order to circumvent such an unwanted assignment.
When an ant perceives an over-concentration of recruitment around itself, it becomes desensitized to the recruitment pheromone for a certain period.
To incorporate this supposition into the model, we have established a rule stating that when an ant perceives a strong pheromone signal, it becomes desensitized. A simulation of the model was performed with this rule. We observed that when simulated with the appropriate parameter sets [1], the system organizes a stable, distributed recruitment at several food sites, and a concentrated recruitment at a single site. According to this rule, the system works to keep the strength of an organized pheromone pattern as constant as possible, and not to optimize its foraging. This is because the perception of over-concentration is determined by the presence of strong pheromone signal, not by food.
In this paper, we changed the rule of desensitization to improve the foraging efficiency according to the food supply rate. The new rule states that when an ant misses food at a food site, it becomes desensitized. Under this rule, after the ants eat up a food site, they become able to search for new sites through pheromone signals. The construction of the model is given in chapter 2. Simulations of the model were made under different food-supply conditions. The results of the simulations showed that the system organized different recruitment patterns in response to the food-supply rate, that are corresponding to the change in foraging strategies. These results are stated in Chapter 3. In chapter 4, the mechanism responsible for organizing the patterns and changing the foraging strategies is explained.
Figure 1: Mode transition rule of the desensitization model in this research.
In this revised desensitization model, the behavior of ants is determined to increase the foraging efficiency (i.e., amount of food collected by the colony). Ants change their behavior according to local situation around them, defined as below (illustrated in Fig. 1).
An ant in the trace mode follows the pheromone trail toward the corresponding food site. If it obtains or misses food at the food site, it changes its mode to the carry or the return mode, respectively. If it misses the trail, it returns to the search mode.
An ant in wander mode walks randomly without perceiving any stimulus. After a desensitization period, it returns to the search mode.
Figure 2: Signal of recruitment pheromone laid by an ant.
Recruitment pheromone laid on the ground gradually diffuses over a wider area. This diffusion is formulated as follows;
In these differential equations, P(x, y, z) denotes the density of evaporated pheromone in the air, and T(x, y) denotes the strength of the pheromone trail on the ground. The region where
is defined as an attracting zone, and the region where
is defined as an active trail, as shown in Fig. 2.
The simulated system in this paper has a nest at the center of the field and four food sites surrounding the nest. At every food site, a certain quantity of food is supplied per unit time. The other important parameters are listed below:
, and
. For
pheromone diffusion, the ground (z = 0) is a
reflecting boundary and the other boundaries are absorbing boundaries.
and
. These factors determine the time constant
of the trail evaporation and the effective range of the pheromone
diffusion.
and
, the minimum perceivable pheromone
and trail, are set so that the pheromone signal laid by an ant fades
out after a few steps.
and
.
To view simulation results as movies:
Figure 3: Concentrated recruitment to a single site. Fourteen food units are supplied per step at every site
Simulation results showed that the system organizes the following two types of recruitment patterns, in response to the food supply rate.
7 units per step)
Concentrated recruitment to a single food site is organized, as shown in Fig. 3. Most of the ants are assigned to the carrying and recruited tasks, and fewer ants are assigned to the searching task. The ratio of desensitized ants (i.e., ants in the return and wander modes) is relatively small, as shown in Fig. 5. For a larger food supply, the ratio of desensitized ants is smaller.
When food supply
thirteen units per step, no ant becomes desensitized because food supply at a single site exceeds maximum amount of foods collected by the colony per step.
6 units per step)
Stable distributed recruitment to several food sites is organized, as shown in Fig. 4. Ants in the colony are assigned to the searching task, as well as to the carrying and recruited tasks. The ratio of desensitized ants is relatively large. For a smaller food supply, both the ratio of desensitized ants and the number of food sites at which ants are recruited are larger, as shown in Fig. 5. In this case, the amount of food collected by the colony per step is larger than the food supply at a single site.
When food supply rate is between three to six units per step, average collected foods, average rate of each mode ants are almost constant.
With food supply ;SPMlt; two units per step, ants eat up all supplied foods, and rate of desensitized ants is large.
From the viewpoint of mathematical biology, the system autonomously changes its foraging strategy in response to the food supply, to increase the food-collecting efficiency. When the food supply is sufficient (case I), most ants go and return between a single food site and their nest, without wasting their time on random-walking. When the food supply is insufficient (case II), however a wide dispersion of many ants in the wander mode maintains recruitment at several food sites, from which the colony carries food.
Figure 4: Distributed recruitment to several sites. Four food units are supplied per step at every site.
The mechanism responsible for controlling the changes in recruitment patterns and foraging strategies can be explained as follows. In a model without desensitization supposition, severe recruitment competition has been observed between food sites for search mode ants around the nest (details will be explained in 4.2), even with small food-supply rate. With the new desensitization rule, desensitization is induced to distribute search mode ants over the field against strong pheromone signals. As a result, wander mode ants spread over the field. After the desensitization period, these wander mode ants return to the search mode and are recruited to several food sites (details will be explained in 4.3). This interferes with the recruitment competition process, and causes stable distributed recruitment to several food sites.
The changes in organized patterns and foraging strategies are determined by the ratio of wander mode ants affecting the recruitment competition process. When the food supply is insufficient, the ratio is large enough to suppress the competition process, and when the food supply is sufficient, the ratio is too small to interfere with the process.
Figure 5: Recruit pattern and foraging behavior versus food supply
At first, we consider the case of no desensitized ants, to explain the mechanism for the recruitment competition process (as illustrated in Figs. 6-1 and 6-2). In this section, the model is assumed to have only two food sites, to simplify the explanation.
In Fig. 6-1, S denotes the distribution of search mode ants who have just changed their mode from carry mode. S concentrate around the nest as illustrated in Fig. 6-1. The nest is located at (0, 0), and the ends of two trails are located at
. The intersections between S and the two attracting zones at time i are defined as
and
where
. x(i) is defined as the drift of the boundary between
and
.
and
, the strength of the trail to food site 1 and that to site 2 after a certain period of time, grow proportional to
and
, i.e., numbers of ants recruited to both sites.
and
are proportional to
and
. Then, we obtain the following equation.
Let u(i) denote
.
and
are determined as 1 - S(x(i)) and S(x(i)). Then, u(i) is determined by the following equation.
Let P(x) denote the pheromone density at the boundary. P(r) is proportional to f(r) (r: distance from the pheromone trail, f(r): a positive, monotonically decreasing function of r). Then, P(x) is determined as follows;
and u = f(h+x)/f(h-x) are drawn
graphically in Fig. 6-2. As S(-h)=0 and S(h)+1, the curves cross
as shown. This figure indicates that x is finally driven to
and u is finally driven to 0 or
. This means that the
result is concentrated recruitment to either site.
Figure 6: Mechanism for change of a recruitment pattern and foraging behavior.
Next, we consider the other case that all search mode ants return from
desensitization (illustrated in Fig. 6-3), to explain how
desensitization supposition enables stable distributed recruitment.
In Fig. 6-3, S denotes the distribution of search mode ants who have
just changed their mode from the wander mode. S is non-zero over a
wide area of the field.
and
denote the intersections between S and the two attracting zones at time i. In this case, there are two important differences from the previous case.
and
cannot cover the total S (
). That is, many search mode ants are located outside of
the attracting zones; gradually, they become trapped in the attracting
zones after random-walking. and
is too small when compared with
or
. Accordingly, the interference between the attracting zones is negligible when considering the growing process of the attracting zones.
In this case, each attracting zone grows independently until reaching the upper limit determined by the number of search mode ants returned from desensitization. This results in distributed recruitment to all sites equally.
In this paper, the model is simulated under static food-supply conditions. This model will show better foraging behavior when simulated in dynamic food-supply condition, where search of new food sites is essential to foraging [4].
In this model, desensitization induces negative feedback from macro-scale foraging behavior of the system to micro-scale ants behavior. Desensitization induces noise into the system on demand, in order to control system's behavior properly. Similar mechanism for changing foraging strategies according to noise in ants' behaviors was proposed [5]. This system can escape from local minimum (overconcentrated state) by use of noise (desensitized ants). Some stochastic process may allow us to control of such a nonlinear distributed system.
In conclusion, an improved ant colony model that can increase the efficiency of foraging was proposed. In this model, the system shows change in recruitment patterns corresponding to change of foraging strategies, according to the food supply rate, and maintains a high food collecting efficiency.
One of authors, M. Nakamura, owes special thanks to Dr. Y. Kakui and Dr. T. Akiba, her colleagues in ETL, and to Mr. Y. Adachi in Kanazawa Institute of Technology.
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