Understanding the dynamics of ant colonies is a classic problem in sciences that try to understand the behavior of ``complex systems.'' It is a clear example of a system in which the complex behavior of the whole arises from the interactions of many parts, so that the whole seems to be more than the sum of its parts. What kind of behavior of the single ant creates the complex behavior of the whole colony ?
In many systems consisting of a multitude of entities, the composite behavior of the individuals creates a coordinated complex behavior of the whole system. There are two possible ways to achieve such a coordination -- the top-down approach, and the bottom-up approach. In the top-down approach, a hierarchy in the entities creates a control structure that guides the behavior of the system. This approach is for some reason is the one easier to understand, perhaps because of the similarity between the apparent structure, and its manifestation in the causal structure of events. A manifestation of this approach is found in the way an army is coordinated, through a hierarchical command structure. In the bottom-up approach, the rules followed by the individual entities bring a self-organization of the system. Most systems in nature have elements of both approaches in them. The behavior of a modern free-market economy is composed both of some hierarchy imposed by corporations and the government, and of self-organization resulting from the action of independent agents in the free market.
In this paper we present a model that is inspired by the behavior of ants in a colony. The aim is twofold. The first is to show what level of coordination might be expected from an ant colony, even under very simplistic assumptions about the behavior of the ants. Thus, it is not claimed that an ant colony is fully described by the model, rather, as the ingredients of the model are contained in an ant colony, the colony will surely be able to achieve at least a response which is as sophisticated as the one described by our model. The second aim is to show one way in which a system can perform certain tasks or computations using the bottom-up approach. This insight will be useful in studying systems other than ant colonies.
Understanding the behavior of an ant colony requires understanding of global coordination without hierarchy. Even though the reproducing ant in the colony is called the queen, empirical work shows that in many cases she does not seem to be at the top of a hierarchy which rules the colony. The behavior of the colony seems to simply arise from the behavior of the individual workers. There are many other cases in nature where we face the same phenomenon. These include the behavior of colonies of other social animals such as bees or termites, but also the understanding of the action of the immune system or the interactions of the proteins and other molecules which make up the cell.
There are several approaches to studying the behavior of a colony of social insects. One is through understanding how the behavior of the colony might be affected by pheromone trails laid by ants (see Deneubourg and Gross, 1989). Several models show how this might lead to optimal, or observed, foraging patterns of the colonies (Deneubourg et al. 1989, 1990; Gross et al. 1990; Millonas 1994). Another approach is to understand the allocation of individuals to tasks in the colony. Individuals in a colony often engage in different tasks -- foraging, nest-maintenace, brood care, etc. It has been shown that social insects may react to the environment by changing the proportion of individuals allocated to the various tasks in the colony (Calabi 1987; Gordon 1989; Robinson 1992). An example of this is recruiting ants for nest maintenance when there is a need to remove obstacles from the neighborhood of the nest. In this paper we follow the trail laid by the second approach, trying to understand mechanisms underlying the task allocation. It will, of course, be worthwhile to synthesize this work with the first approach in further research.
How does an ant colony react to its environment? Let us examine a reaction of a multi-cellular organism, such as our body, to the environment. Light signals are absorbed by photo-receptors in the retina cells. These signals are then transmitted through the central nervous system (CNS) to the brain, where they are assessed using information about the environment currently stored in the brain. A signal for a reaction is then transmitted through the CNS to regulating neurons controlling a muscle in the hand, for example. The ant colony might face a similar task. Some ants -- patrolers -- might gain information about a food source, and other ants will then need to be recruited to forage at that food source, potentially bringing into account some information about the current state of the colony, the hunger level in the brood for instance. This has to be achieved without a CNS connecting the various ants, and without a brain to store and assess the information.
Gordon et al. (1992) showed how certain behaviors of individuals in the colony may enable the colony to process information like a Hopfield net. The model assumed, however, that individual ants are able to measure global states of the colony, such as the proportion of ants allocated to certain tasks. Our model is based on the model presented by Pacala et al. (1994). In this model ants can engage in a task or be inactive. Ants doing a task can also be either ``successful'' or ``unsuccessful'' and can switch between these two according to how well the task is performed. Unsuccessful ants also have a certain chance to switch to be inactive, and successful ants had a certain chance to recruit inactive ants to their task. This is an example of how certain interactions of ants can give rise to global behavior in the colony.
In the model presented in this paper, the notion of a ``successful'' or ``unsuccessful'' ant engaging in a certain task is expanded to a general notion of a state that the ant is in. An ant in the colony can be in one of a finite number of states. Ants doing different tasks are always in different states, but ants doing the same task could also be in different states. Such a state might correspond, for example, to an ant foraging while hungry and successful. It is assumed that there are many more ants in the colony than states, so that it makes sense to talk about the fraction of ants in a certain state. The ways in which an ant can change its state are through interaction with the environment and through meeting another ant. The model is aspatial, and thus does not regard the place of ants in the colony or the pheromones left by the ants in certain places in the environment.
In Sect. 2 we present the master equation for this model, assuming an infinite number of ants in a colony. Then in Sect. 3, we show that this model can lead to a very complex behavior, including amplification, cycling, and potentially the performance of any computation that can be done by a finite Boolean network or by a finite Turing machine. We also present a different formulation of the model, in which the interaction with the environment is separated to ``measurement''. In Sect. 4 we present two examples. One is a test of how well a finite colony fits the predictions of the model. The other presents a model developed by Seeley et al. (1991) in order to show a possible solution to choosing between two alternate foraging sites in bees. It is shown how this solution can be stated within the framework of our model.