Cemetery Organization and Brood Care

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Many ant species, for example Lasius niger and Pheidole pallidula, exhibit the behavior of clustering corpses to form cemeteries [127] (cited in [77]) Each ant seems to behave individually, moving randomly in space while picking up or depositing (dropping) corpses The decision to pick up or drop a corpse is based on local information of the ant s current position This very simple behavior of individual ants results in the

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172 Cemetery Organization and Brood Care

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emergence of a more complex behavior of cluster formation It was further observed that cemeteries are sited around spatial heterogeneities [77, 563] A similar behavior is observed in many ant species, such as Leptothorax unifasciatus, in the way that the colony cares for the brood [77, 912, 913] Larvae are sorted in such a way that di erent brood stages are arranged in concentric rings Smaller larvae are located in the center, with larger larvae on the periphery The concentric clusters are organized in such a way that small larvae receive little individual space, while large larvae receive more space While these behaviors are still not fully understood, a number of studies have resulted in mathematical models to simulate the clustering and sorting behaviors Based on these simulations, algorithms have been implemented to cluster data, to draw graphs, and to develop robot swarms with the ability to sort objects This section discusses these mathematical models and algorithms Section 1721 discusses the basic ant colony clustering (ACC) model, and a generalization of this model is given in Section 1722 A minimal ACC approach is summarized in Section 1723

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Basic Ant Colony Clustering Model

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The rst algorithmic implementations that simulate cemetery formation were inspired by the studies of Chr tien of the ants Lasius niger [127] Based on physical experie ments, Chr tien derived the probability of an ant dropping a corpse next to an n-cluster e as being proportional to 1 (1 p)n for n 30, where p is a tting parameter [78] Ants cannot, however, precisely determine the size of clusters Instead, the size of clusters is determined by the e ort to transport the corpse (the corpse may catch on other items, making the walk more di cult) It is likely that the corpse is deposited when the e ort becomes too great On the basis of of Chr tien s observations, Deneubourg et al [200] developed a model e to describe the simple behavior of ants The main idea is that items in less dense areas should be picked up and dropped at a di erent location where more of the same type exist The resulting model is referred to as the basic model Assuming only one type of item, all items are randomly distributed on a twodimensional grid, or lattice Each grid-point contains only one item Ants are placed randomly on the lattice, and move in random directions one cell at a time After each move, an unladen ant decides to pick up an item (if the corresponding cell has an item) based on the probability Pp = 1 1 +

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(1743)

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where is the fraction of items the ant perceives in its neighborhood, and 1 > 0 When there are only a few items in the ant s neighborhood, that is << 1 , then Pp approaches 1; hence, objects have a high probability of being picked up On the other hand, if the ant observes many objects, that is >> 1 , Pp approaches 0, and the probability that the ant will pick up an object is small

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17 Ant Algorithms

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Each loaded ant has a probability of dropping the carried object, given by Pd = 2 +

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(1744)

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provided that the corresponding cell is empty; 2 > 0 If a large number of items is observed in the neighborhood, ie >> 2 , then Pd approaches 1, and the probability of dropping the item is high If << 2 , then Pd approaches 0 The fraction of items, , is calculated by making use of a short-term memory for each ant Each ant keeps track of the last T time steps, and is simply the number of items observed during these T time steps, divided by the largest number of items that can be observed during the last T time steps If only one item can be observed during each time step, = n /T , where n is the number of encountered items The basic model is easily expanded to more than one type of item Let A and B denote two types of items Then equations (1743) and (1744) are used as is, but with replaced by A or B depending on the type of item encountered

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