Trail recruitment in ants. An ant scout (in black) which has found a food source (a), lays a pheromone odor trail on its way back to the nest (b). The recruitment trail is then perceived by nestmates (in grey) which exit the nest and follow the trail till they reach the food source (c). In turn, these recruited workers feed at the source and return to the nest reinforcing the foraging trail (d). 

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... nestmates to function as a whole and to take collective decisions. Embedded in the existence of a highly structured organization is the common—but erroneous—belief of a social leader such as the queen centralizing information and issuing commands to workers. Instead, the removal of presumed leading individuals reveals that spatio-temporal patterns still persist and thus essentially rely on interactions among nestmates or between nestmates and their environment. Without having a sense of the whole, the worker individuals collectively behave and solve daily problems such as exploration of new areas, foraging, nest moving, nest defence, waste management or brood care (for a review see [20,72]). The next sections provide a brief survey of ant patterns showing strong analogies and properties with that of SO patterns in physico-chemical systems. In physico-chemical or biochemical systems, a wide array of patterns can appear spontaneously in an initially homogeneous medium without any external driving force or influence. Likewise, ant patterns may change abruptly and orient spontaneously to one of them depending on fluctuations and initial conditions. From a practical point of view, the identification of self-organizing processes in ant societies ideally requires a two-ways approach coupling behavioural experiments and modelling. Therefore, non-linear models have been developed in which agents are designed to behave similarly to animal individuals [94,118]: they act in a probabilistic way, respond exclusively to the local information they receive from their nearby environment and follow simple decision rules coupled with feed-back loops (for further details see next sections). The level of agreement between actual and such model-predicted collective behaviour indicates the prevalence of self-organizing processes in pattern formation. However, specifically designed experiments coupled with modelling are not so common in socio-biological literature. Table 1 lists all ant patterns in which a strong self-organizing component was identified—or at least highly suspected to occur—and indicates the presumed mode of interaction responsible for the emergence of such structures. Non-linear systems consisting of simple interacting units often exhibit multiple states. Most physicists consider as a typical signature of self-organization, the occurrence of a bifurcation which is the abrupt transition of the entire system towards a new stable pattern when a threshold is crossed. For example, at a critical temperature, a ferromagnet may become demagnetized due to the disordering effect of thermal forces. Upon variations of some control parameters, a self-organized system will thus spontaneously present new types of structures whereby there is a discrete change from one state to another. As regards biological sciences, the possibility of such discrete changes is namely at the root of current views about speciation and morphological diversification in which dynamic processes of development take place in a sequential way with each step bifurcating from a previous one. Similar bifurcation phenomena are observed in ant societies which may abruptly shift from a disordered to an ordered pheromone-based foraging [9] or from a random exploration to a well-defined exploratory trail [42]. Bifurcation may take the form of symmetry-breakings as one observes the shift from an even exploitation of several food sources or an equal resting in several sites towards the collective exploitation of only one food site or the selection of a specified resting location. One should underline that in social systems such symmetry-breaking does not necessarily evoke to biologists a SO script as they can put alternative explanations forward. Among those alternatives, the coordinated movement towards a restricted area can be driven by one or a few leaders that are merely followed by nestmates or the collective choices can result from environmental heterogeneities in which each group member decides on its own to move towards the more “hospitable” place. In both cases, there is no self-organizing mechanism, no amplifying interactions at work nor assumptions of non-linearity. As soon as the existence of leaders or of environmental templates are dismissed, SO can be evoked as the main process leading to these symmetry-breakings. Experimentally, there is a simplistic way to visualize a bifurcation by confining ant foragers to a diamond-shaped bridge of which each equally long branches lead to a sucrose solution. At the beginning, starved ants evenly explore both branches of the bridge. After a while, a bifurcation occurs so that most of the foragers travel over one path only (Fig. 1(a)). Such examples of collective choice have been reported for a wide variety of ant species which recruit nestmates towards new food resources by the laying of a chemical trail [32,118]. The basic scheme of interactions is similar for all these species and can be described as follows (Fig. 2). As soon as one ant has succeeded in discovering a food source, it goes back to the nest and lay a chemical trail. The trail pheromone then triggers the exit of additional foragers and guides them as an Ariane thread to the food source. After feeding, each recruited ant can in turn reinforce the foraging trail and stimulate other nestmates to forage. This trail reinforcement results in a non-linear increase of foragers’ population which can lead to a self-organized bifurcation and the exploitation of one single food source (for further details on the formation of bifurcation see also Section 3.2 of this review). One should bear in mind that a discretized environment (e.g. diamond-shaped bridge) is a quite convenient way to visualize bifurcation but is far from natural conditions. In a more continuous environment, symmetry breakings will rather occur in a cascade leading to the emergence of a branched network of foraging trails. Bifurcation and symmetry-breakings are features common to a variety of ants’ collective behaviour. For example, in panic conditions, ants confined to a cell with two symmetrically located exits, prefer to escape through one of the exits if alarm is created by adding a repellent fluid [1]. Likewise when emigrating, weaver ants will shift from an even distribution to a spatial concentration and a self-assembling of tenths of workers into chains at only one location [35]. In the context of nest building, the topology of underground galleries—which varies from a tree-like structure to a highly connected tunnelling network—results from a dynamics of excavation which is clearly non-linear and from several bifurcation events [17,18]. The probability for an ant to dig out a sand grain increases in places where other ants had previously dug, leading to the formation of new bifurcations along extending galleries and to the selection of one digging site among the several initiation sites that were first randomly excavated [103]. Although living systems are complex, they can be highly ordered in time. There is a plethora of biological oscillations such as the periodic firing in neurons, the biochemical cycle of glycolytic pathway, the life cycle of the cellular slime mold or the coupled oscillations of populations of preys and predators [59,115,136]. All the above examples are different from biological clocks associated with circadian or daily rhythms. They are not periodic by virtue of some external periodic forcing function and they can be reasonably described as autonomous oscillators. Temporal patterns also exist in ant societies: oscillatory processes appear in widely varying contexts and can have periods from a few seconds to hours and even days and weeks. Recent studies found out that activity rhythms occur over short time scales, activity being defined here as movement of any type within ant colony [12,23,25]. Indeed, the average daily activity of nestmates is neither constant nor random but displays periodic bursts: one ant initially becomes active, and then activity spreads to neighboring workers before gradually dying out. Fourier analyses of activity patterns in these colonies reveal periodic components with activity peaks occurring at regular time interval [51] (Fig. 1B). Since nature provides many types of pacemakers such as diurnal light-dark cycles, one may first think about external stimuli to explain such a rythmic activity. However, it was shown that synchronized activity within ant colonies can emerge simply from mutual activation through direct physical contacts. Active ants are effective in stimulating inactive ones without an exogenous trigger. The periodicity of colony activity relies either on ants remaining active for a minimum time [113] or on active workers becoming quiescent for a refractory period during which they cannot be reactivated [63]. The fundamental outcome of this autocatalytic process is that individuals become rhythmic as a group, even though they show no intrinsic rhythm. Such synchronization could be adaptive when several active workers are required for a task (e.g. foraging, brood care, defense or nest building) to be performed efficiently, although this explanation has still to be experimentally validated. Temporal patterns may also occur over long time scale such as the periodic alternation of static and nomadic phase in colonies of army ants. During the static phase, the queen lays thousands of eggs in a brief span of time, within a few days, and the colony remains in one bivouac site. When pupae derived from the previous batch of eggs develop into adults, the appearance of tens of thousands of new workers induces an increase in the general activity level as well as in the intensity of swarm raids. Then, the colony enters a nomadic phase during which it starts emigrating at the end of each day’s foraging. The migratory phase continues as long as brood remains in the larval stage. As soon as they pupate, the emigration stops and a new ...

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... of a travelling wave. In Eciton burchelli , a new swarm raid develops each day out of the nest: aroused workers start exiting the bivouac, expanding in all directions and soon, one sector becomes favoured by all raiders that enter new ground. As regards their direction, the successive raids rotate about the bivouac site systematically so that foraging overlap between successive days was minimized ([52,65] but see also [110]). Likewise, a spectacular radial displacement of foraging trails was observed in some harvester ant species (Fig. 1(c)). Indeed, the main foraging trail makes a more or less complete revolution by rotating with a period of several days to three weeks around the nest entrance like the hand of a clock [105]. A mathematical model shows that this self-organized clock pattern can be generated from the concurrent effects of trail-laying foragers which recruit nestmates and of the environment which becomes locally depleted of food resources around the foraging column [64]. Wave-like events in the swarms of army ants or in the foraging columns of harvester ants confirm that self-organized movements are more structured—both spatially and temporally—than could be expected in the absence of leaders or centralized control. Disturbance patterns may occur within a group of ants and show striking similarities with disturbances reported in self-organized critical systems—also called “avalanches” [4]. For instance, as a sand pile grows by the slow addition of individual sand grains at random position, the slope become steeper up to a critical value. If more sand is added, it is likely to slide off and to trigger an avalanche whose size is given by the number of sliding grains. The power law f ∼ x − a that describes the frequency f of avalanche size x is considered as the signature of a self-organized criticality. Similar endogenous disturbances were described within a population of Argentine ant workers feeding at a food source. Without any external disturbance, a transient departure from the food source can be triggered by an initial contact between a wandering ant and feeding nestmates [66]. Likewise, under certain experimental conditions, an aggregate of Argentine ant workers may form at the end of a rod and a droplet of around 40 ants may fall down without any external disturbance (Fig. 1(d)). When the flux of incoming ants is sufficiently high, this process of droplets’ fallings can continue for hours [120]. The magnitude of these displacements (i.e. in the number of leaving or dropping ants) broadly follows a power law distribution characteristic of self-organized critical systems. These abrupt changes in the size of groups of ants fit well to avalanche models and are quite similar to the observed criticality of sand piles [76]. Although these disturbances may reduce food exploitation because foraging ants are interrupted, they could give to ants’ group the advantage of rapid alarm communication and escape behaviour. These analogies between physico-chemical and social patterns strongly suggest that self-organizing processes shape many life-sustaining activities of social insects. Nevertheless, few studies overcome the descriptive level and provide a quantitative and heuristic approach to pattern formation in ant societies. The best-achieved research in this field concerns the formation of foraging patterns. Indeed, the sight of hundredths of ants moving together in raiding swarms or following accurately a trail network till a restricted food area while neglecting other nearby food sources has easily prompted the “how” question among researchers. Therefore, key-properties of self-organizing systems will be addressed in the next section of this review through case examples exclusively drawn from ant foraging context, even though our conclusions apply to other social structures and group-living organisms. Many physico-chemical patterns are well-known to appear spontaneously when competing driving forces banish featureless uniformity. Likewise, two opposite types of interactions among group members—positive and negative feed-backs—both contribute to the emergence of social structures. It is widely recognized that the identification as well as the understanding of these feed-back mechanisms provides a key to understanding how insect societies display complex collective patterns and organize their workforce efficiently. As regards positive feed-backs, various forms are encountered among group members that can often be reduced to the rule of thumb “Do as your neighbor”. In particular, information-laden signals can release a specific behavioural response in the receiver which may, in turn, produce a signal identical to the one he received. Such positive feed-back loop will ultimately lead to the propagation and amplification of information and/or behaviour. In the case of ant foraging, positive feedbacks often take the form of pheromone trails deposited by ants that have found a profitable food source (Fig. 2). Social amplification through the laying of a recruitment trail can be easily evidenced in a Y shaped setup in which ant foragers have the choice between two branches of equal length each leading to a food source [Fig. 3, [7,30,117]]. The probability of choosing one branch 1—here called p 1 —at a certain time depends on the number of ants having already laid a trail on this branch 1 and can be described by the following ...