Abstract
Animals foraging alone are hypothesized to optimize the encounter rates with resources through Lévy walks. However, the issue of how the interactions between multiple foragers influence their search efficiency is still not completely understood. To address this, we consider a model to study the optimal strategy for a group of foragers searching for targets distributed heterogeneously. In our model, foragers move on a square lattice containing immobile but regenerative targets. At any instant, a forager is able to detect only those targets that happen to be in the same site. However, we allow the foragers to have information about the state of other foragers. A forager who has not detected any target walks towards the nearest location, where another forager has detected a target, with a probability exp(−αd), where d is the distance between the foragers and α is a parameter characterizing the propensity of the foragers to aggregate. The model reveals that neither overcrowding (α → 0) nor independent searching (α → ∞) is beneficial for the foragers. For a patchy distribution of targets, the efficiency is maximum for intermediate values of α. In addition, in the limit α → 0, the length of the walks can become scalefree.
1. Introduction
Many organisms collectively gather resources for survival. For a lone animal looking for sources of food, the foraging efficiency is decided by its searching strategy and the distribution of the targets [1,2]. However, when others are around, the interactions between them may become important in determining the behaviour of the individuals. For example, an animal that is unsuccessful in locating food by searching on its own can instead locate other distant members who are successful and join them. Such behaviour is well documented across different species [3–5]. However, what kind of interactions would be the most advantageous for the individuals in the group? Traditionally, this problem has been studied in behavioural ecology under the paradigms of producer–scrounger and informationsharing models [5,6]. The informationsharing theory, in its canonical form, considers foragers simultaneously searching for food as well as looking for opportunities to join others [7]. In the producer–scrounger games, the simultaneous execution of the above activities is not possible [8].
With the advancement of different data logging techniques in animal experiments [9,10], it has been possible to record the trajectories of animals searching for food. The information from such experiments has allowed for detailed statistical analysis and has helped in shedding light on the possible relationship between movement patterns and efficiency of search. The emergence of heavy tails in the walklength distributions has prompted physicists to explain the foraging movement of animals in terms of Lévy walks and flights [11]. Lévy walks are characterized by move lengths (l), which are distributed according to a probability density function (for large values of l), with 1 < μ ≤ 3, and by uniformly distributed turning angles between consecutive moves. Although small in number, large move lengths occur often when an animal is executing a Lévy walk. However, the moves are not instantaneous, and an animal is assumed to traverse a distance l with a finite velocity. For μ > 3, the walks essentially become Brownian motion where large move lengths are extremely rare. In the limit, μ → 1, the motion is ballistic, where the path of the animal is rather a straight line, when observed over a finite time. The most efficient searching, by a single forager looking for targets, is supposed to happen when μ ≈ 2 [1,12,13]. However, investigations [12,14–16] have also probed the robustness of Lévy walks and have found these to be actually optimal or only marginally advantageous under conditions pertaining to the nature of the encounter with targets (destructive or nondestructive), the presence of memory, geometry of the foraging region, among others. Deviations greatly reduce the advantage of Lévy walks in favour of ballistic or Brownian motion. In addition, the presence of patchy environments may lead the foragers to use composite search strategies which also result in fattailed distributions for step lengths [17].
In general, the success with which the individuals in a group forage, as well as the emergent patterns in the movement of foragers, depends on several factors. When food is abundant, there is hardly any benefit for individuals to join others, so that encounters with food sources are increased. However, when the distribution of food is ephemeral or patchy, interaction between foragers may become important. Several field studies and theoretical models have investigated these aspects. The producer–scrounger paradigm was used to model the collective behaviour of foragers [18,19]. Cooperation in foraging was shown to develop in a stochastic environment [20]. Explanation for observed scalefree move lengths observed in spider monkeys [21] has been based on their social behaviour [22]. Such mobility patterns of foragers were modelled in [23] with the agents having unbounded velocities. The role of information transfer in the recruitment of foragers has been studied in colonies of social insects [24–26]. Very recently, the influence of communication on the foraging pattern of gazelles was investigated [27]. The study showed that communication over intermediate length scales results in faster searches.
In this paper, we develop a minimal model of collective foraging where the motion of the individual foragers would be random in the absence of any interaction. We investigate the effect of the interaction on the search efficiency and the spatiotemporal scaling that emerges in the movement of foragers. The foragers are walkers on a twodimensional square lattice, whereas targets are immobile but regenerative. The foragers have local information about the distribution of the targets. We are primarily interested in the emergent complexity that should result from the interactions between the foragers; therefore, we begin with a simplistic assumption about the movement of a forager independently searching for targets. We assume that such a forager, at any instant of time, has no preference while choosing the direction of movement, and hence executes a random walk (RW) on the lattice. However, the foragers possess global information about the state of other foragers, i.e. whether they have been successful in locating targets. We characterize the interaction between foragers through the parameter α. This parameter controls the propensity of a forager, who is unsuccessful in finding a target in its locality, to approach the nearest site where targets have been discovered by others. We call such a movement a targeted walk (TW). The detailed description of a RW and a TW is provided in §2.
We expect that when the targets regenerate anywhere on the lattice with uniform probability, the aggregation of foragers would provide no additional advantage when compared with independent searching. This aspect becomes clear from our results. However, when the reappearance of the targets is guided by spatial correlations which result in spatially heterogeneous distribution of the targets in patches [2,28], we find that the efficiency attains a maximum for intermediate values of α. This implies that for foragers who are momentarily unsuccessful in identifying targets, sometimes joining others who are successful can be beneficial. This is the main result of our paper. In addition, the fact that we obtain scalefree TWs under certain conditions suggests collective behaviour as one of the possible mechanisms responsible for the emergence of Lévy walks in foraging patterns found in the real world.
2. The model
In the following, we describe our model in more detail. We consider a twodimensional square lattice of size L with periodic boundary condition (PBC). The PBC is applied with regard to the movement of foragers as well as the regeneration of the targets. There are N_{F} foragers which are initially distributed randomly on this lattice. The targets (food) are regenerative such that at every time step the total (amount) is N_{T}. We define the neighbourhood of a forager as a circular region of radius R centred on it. We assume that at any instant of time, a forager is able to determine which are the foragers in its neighbourhood that have been successful in identifying and consuming food. However, a forager is only able to detect targets at the site where it has arrived. Let be the position of a forager i on the lattice at time t and let j be the nearest of the all the foragers in the neighbourhood which have consumed one target at time t. The rules governing the behaviour of i are the following:
(1) If there are targets at , then the forager stays at that site and consumes one target. Let denote the number of targets at a site (x, y) at time t. Thus, and . If at any instant of time the number of foragers at any site exceeds the number of targets available, then the targets are distributed randomly between the foragers.
(2) If the site is empty with the forager having arrived at this site by taking a random step (defined below), or the forager being there in the presence of food (at time t − 1), then the movement of the forager is decided according to the following rules.
(a) If there is no successful forager in the neighbourhood, then the forager takes a random step, i.e. becomes equal to one of the nearest neighbour coordinates or with equal probability.
(b) If there is a successful forager j in the neighbourhood, then with a probability p_{i} the forager ‘moves’ in the direction of the former and with a probability 1 − p_{i} moves out by taking a random step. Here, , where d_{ij} is the Euclidean distance between and . The movement in the direction of j implies moving to a nearest neighbour site such that the distance between and becomes less than d_{ij}. We term this movement as a targeted step in contrast to a random step and let the forager remember this distance as .
(3) If the site is empty and if the forager has arrived at this site by taking a targeted step, then the movement of the forager is decided as follows.
(a) Same as (2)(a).
(b) If there is a forager j in the neighbourhood and then the forager takes a targeted step towards j and .
(c) If there is a forager j in the neighbourhood and then same as (2)(b).

(4) This rule pertains to the regeneration of targets at every instant. Let ΔN(t) be the number of targets that have been consumed at any instant of time t. We randomly select one of the remaining N_{T} − ΔN(t) targets. At a distance d from the location of this target, a new target is placed, in a random direction. The distance d is chosen from a distribution , where γ is a parameter. This process is repeated another ΔN(t) − 1 times. Thus, the total number of targets at the beginning of the time step t + 1 again becomes equal to N_{T}.
We choose such a scheme for the regeneration of targets with a twofold view. First, it ensures that at any instant the distribution of resources is heterogeneous. The powerlaw form for P(d) generates targets distributed in wellseparated clusters with the parameter γ effectively characterizing the degree of patchiness of the distribution. A similar scheme was used to model the spacings between marine food patches in [10]. Second, the method allows, in general, the preservation of the overall statistical nature of the foraging environment during the time evolution of the model. A more realistic model would allow the simulation in an infinite lattice where the patches need not regenerate at every time step. In effect, our model provides a method to produce a foraging environment that is highly variable in space and time [20,29,30].
We call a consecutive sequence of random steps a RW and a consecutive sequence of targeted steps a TW. A forager which searches independently executes a RW until it encounters a target. However, there is a finite probability that such a forager may decide to move towards sites where foragers have already encountered food assuming that targets are clustered in space. It is natural that this probability is distance dependent as there is an energy cost associated with the movement of a forager [31]. In addition, travelling over large distances take more time during which the targeted patch becomes increasingly exploited [32]. Therefore, nearer the location of the sites, relocation to the target site is more probable. This fact is quantified through the definition of p in (2)(b). In the limit α → ∞, the trajectories of the foragers are essentially RWs. The tendency of foragers to travel towards distant locations, where targets have been discovered, increases with a decrease in the value of α. Thus, trajectories become mixtures of RWs and TWs.
Note that a TW, when initiated, is not aimed at a particular individual but towards a particular site. The variable ensures that forager has a memory of the distance to the targeted site, and therefore the instantaneous movement of the forager is in a direction so as to minimize this distance. On appearance of another site, where food gets detected, closer than the original, the TW gets directed towards it and is accordingly modified. In the case where the last targeted site is exhausted and some feeding site in the neighbourhood gets detected at a distance larger compared with the distance to the former, then the forager probabilistically adopts that target depending on the distance. This rule accounts for the fact that when a forager has adopted a strategy of moving towards targets detected by others, will not easily abandon it and return to searching independently. In addition, a forager actually approaching a cluster of sites should not get deterred if one of the sites gets emptied. A model with somewhat similar but rather deterministic rules for the movement of the foragers was considered in [33] where the effect of group size on foraging rate was studied. The randomness in our model is naturally built in by the probabilistic choice of the step directions of the foragers and the scheme used for the regrowth of the targets.
In the present study, we assume that R is larger than the system size, i.e. at any instant, a forager is informed about state of all other foragers. However, it is obvious that the actual use of this information is manifested through the distance dependent probability p. In addition, we take N_{F} = N_{T} = N which would correspond to the case of targets being rather scarce; at the same time this equality also ensures that the maximum possible value of targets encountered per unit time per forager is unity.
3. Results
In general, not only the regeneration process, but also the interaction with the foragers is supposed to influence the distribution of targets at long times. However, with the parameters used in this paper, our simulations reveal that the heterogeneity in the spatial distribution of targets at all times is qualitatively independent of α and is controlled through γ. The larger the value of γ, the more clustered the targets become and the powerlaw form for P(d) resulting in rare, but very large values of d gives rise to large separation between clusters.
In figure 1, we show the time evolution of the model for the parameters N = 128, L = 128, γ = 2.5 and α = 0. The dynamics is deterministic to a certain extent because α is zero. Initially (figure 1a), the targets appear to be distributed in two wellseparated clusters, and the foragers are mostly away from these targets. At a later time (figure 1b), the foragers are found to have aggregated over one of the clusters. This happens as soon as one of the foragers is successful in detecting a target belonging to that cluster. The trajectory of a typical forager is shown. Once the targets in the region where the foragers are present get depleted, there is a searching phase (figure 1c). The trajectory of the foragers at this stage is mostly RWs. The search ends as soon as a site in the second cluster is detected by any forager, and all the others relocate towards the site. The trajectory during this relocation mostly comprises of TWs (figure 1d,e). Eventually, the second cluster is also consumed and another searching phase follows (figure 1f). The simultaneous development of clusters of targets in other regions of the lattice is also visible in the figure.
We assume that the cost involved in foraging is proportional to the distance travelled by the foragers. The efficiency of searching [1], defined as the ratio of the total number of targets consumed to the total distance travelled by all the foragers, is given by 3.1where is the number of walkers at time t. The statistics is collected in the stationary state for time steps in each configuration and denotes the average over 200 configurations. In figure 2a, we plot η as a function of α. For N = 512 and L = 512, we are able to compare the efficiency of the searches for target distributions at different degrees of patchiness characterized by γ. When the targets regenerate randomly across the lattice, we find η to increase with α and then saturate to a value in the limit of α → ∞. Thus, searching independently is overall beneficial. However, for a patchy distribution of targets, the scenario is found to be very different. For example, when γ = 2.5, the maximum in the efficiency is found to occur around much smaller values of α. This signifies that the optimal strategy in this case is a mixture of independent searching and joining others. The reason being the following. When targets are clustered in space, we expect that a forager travelling to a region where targets have already been discovered increases its own chance of encountering a target. However, indiscriminately taking such decisions may not be beneficial. When a large fraction of the foragers perform TWs to a single cluster, this delays the discovery of other clusters in space. In addition, by the time the foragers reach a distant region, it becomes depleted of targets. Therefore, the most efficient strategy for encountering targets is joining others when the clusters are discovered nearby and opting to search independently otherwise. Unlike the scenario described in figure 1 (α = 0), an intermediate strategy may amount to the simultaneous discovery and exploitation of more than one cluster or faster discovery of newly generated clusters. The results in the case of γ = 2.0 and 3.5 are found to be qualitatively similar. Additionally, for γ = 3.5, when the targets are extremely clustered in space, the efficiency in the limit α → 0 is found to be larger than that of the α → ∞ limit unlike for lesser values of γ. This is because the regenerated clusters appear very close to each other and as such aggregation allows the foragers to move from one cluster to another with least amount of exploration. A comparison between different values of N = 256, 512 and 1024 shows that the maximum value of efficiency is larger for larger values of N. We quantify this effect by a scaling collapse of η versus α plots for five different values of N in figure 2b with L = 512 and γ = 2.5. The collapse reveals that the width of the maximum in η scales as , where β_{1} = 0.15 and the maximum value of the efficiency, η_{m}, scales as with β_{2} = 0.70. We define α_{m} as the value of α for which η = η_{m}. The values α_{m} for which the collapse becomes possible are plotted against N in the inset of figure 2b. The plot shows that with ζ = 0.65.
Interestingly, near the maximum of efficiency, we find that there is a maximum for the fraction of walkers executing TWs. In figure 3a, we plot the averaged quantity, f_{t}, defined as the ratio of the number of foragers executing TWs to the total number of walkers at any instant. As expected f_{t} goes to zero as α → ∞, because the foragers do not follow each other. In addition, f_{t} is large as α → 0. The activity of the foragers in this limit is similar to the scenario illustrated in figure 1. The dominant activity is searching (through RWs) between the discovery of clusters and, also, the simultaneous discovery of clusters is rare. The TWs only occur in short bouts. For intermediate values of α, when there is a possibility that a cluster can be detected while another one is being exploited, the TWs take place more frequently. This results in f_{t} having a maximum. When targets regenerate randomly across the lattice, the TWs are not beneficial in terms of the efficiency as already seen from figure 2a. Decrease in the value of α favours the increase in aggregation of the foragers through TWs, whereas the aggregation reduces the number of forager–target encounters. This competition results in the maximum of f_{t} for random regeneration. The effect of random reappearance of targets coupled to extreme aggregation for α → 0 and the fact that we use a PBC gives rise to a stable moving band in the steady state as shown in figure 4a. The band travels parallel to either of the axes.
When a forager comes across a site (x, y) having targets, the time spent at that site, before it moves away, is in general equal to the number of targets present, h(x, y) (unless other foragers are also present at the site). Large values of γ result in sites with . As a result, the presence of a forager at such a site is of longer duration. Now, if α is also small then such a forager, at any instant, draws towards the site a large number of foragers who execute TWs. During the time the former spends consuming targets, the TWs of the latter align. Such an alignment of the paths of a set of moving foragers generates an overall pattern of a moving column. Such a case is illustrated in figure 4b where two such perpendicular columns are seen giving rise to a wedge. Such a formation persists and as a whole travels until the adjacent targets are not depleted.
The observation of different patterns encouraged us to investigate the general nature of collective motion. We measured the order parameter popularly used to characterize motion of flocks of selfpropelled particles [34]. As the motion of the foragers is restricted to the lattice, the order parameter is 3.2where denotes the fraction of foragers moving towards the positive xdirection at any instant and likewise. A large value of Φ would imply that, on the average, the foragers are moving in the same direction, at any instant of time. Temporal and configuration averaging is denoted by and is similar to the calculation of η. In figure 3b, the plot Φ versus α is shown. We find that the maximum of η is accompanied by a transition from a regime where the value of Φ is dominated by fluctuations, owing to random movement of the foragers, to higher values of Φ. The higher values of Φ occur when foragers migrate from one cluster to another. However, the actual values of the order parameter are low because of the movement on the lattice and the fact that just after the depletion of a cluster, the foragers diffuse in all directions.
The influence of the collective searching on the movement of individual foragers is evidenced in the distribution of lengths of TWs. The tendency to search independently is less when α is small and long TWs are probable. In figure 5, we plot the probability density function, P(l), of length of TWs of length l. We find that TWs become scalefree in the limit of α → 0 for patchy distributions. For N = 512, L = 512, γ = 2.5 and α = 0, we find with μ = 2.80 (using regression analysis [35]). As α increases, the TWs become shorter and the distribution falls off faster. These are shown in figure 5a. For random distribution of targets and extremely clustered distribution (γ = 3.5) which correspond to moving bands and columns, respectively, there is a deviation from power law (figure 5b). In the limit α → 0, we generally find that scalefree TWs arise for values of N which are either equal to L or larger but close to L (figure 5c,d). For the distribution is fattailed but not a power law. In the case of N < L, the clusters are exploited so fast that long TWs become rare.
The fact that fattailed distributions of TWs result in the limit α → 0 also raises the question as to whether the primary cause behind the long TWs is the choice of the interaction radius R being larger than the system size. To investigate this, we study the effect of decreasing R on the nature of P(l) when α = 0. If there are N noninteracting random walkers in a region having area L^{2}, then the average separation between them is of the order of r_{0}, where, . We consider the values of R in units of r_{0} such that where m is a positive integer. Having ensures that at any instant, on the average, a forager remains informed about the state of n other foragers where . In figure 6a, we plot the values of n for five different values of m when L = 512 and N = 512. The powerlaw fit reveals with ν ≈ 2.23. In figure 6b, we plot the distributions P(l) corresponding to the different values of m. We also show P(l) corresponding to the case when n = N, that is, R is larger than the system size. Interestingly, we find that the development of the power law starts when n is still much smaller than N. Below n ≈ 4 (m = 2) the fall in P(l) is fast, and from n ≈ 20 (m = 4) and above, the fall becomes rather slow and the powerlaw region becomes prominent. Thus, even when a forager is only interacting with a small number (when compared with the total number of foragers) of other foragers in its locality, long TWs may arise. The reason behind this is the fact that a TW once initiated may continue even when the originally targeted site gets depleted of targets. On the detection of targets at some site within a radius R from the current position of the forager, the TW gets directed towards that site (rules (3)(b) and (c) of the model).
The interdependence between the foragers is also evident in the temporal behaviour of the model. When targets are clustered there is usually a waiting time corresponding to the searching phase between the discovery of clusters. We define τ as the time between successive encounters with targets by the group as a whole. In figure 7, we plot the probability density function P(τ). With the decrease in the value of α, the states of the foragers become increasingly correlated. This is reflected in the powerlaw nature of P(τ). For N = 512 = L and γ = 2.5, we find with δ = 1.67 for values of α between 5.0 × 10^{−3} and 0 (figure 7a). We find the powerlaw form for P(τ) to be retained for larger values of γ as well. The value of γ becoming less finally leading to a random distribution of targets makes P(τ) fall faster (figure 7b). Faster decay of P(τ) also occurs when N > L (figure 7c,d). This is because when both targets as well as foragers are abundant the long waiting times become rare.
4. Discussion
Our model demonstrates that the presence of interactions within a group of foragers can help maximize the efficiency of searching. The decision of an individual as well as the cost of joining are assumed to depend on the distances between the foragers. The optimal strategy, from which the individuals benefit, is a mixture of searching independently and joining other foragers. The dependence of the efficiency on the parameter α and the fact that α is the inverse of a length scale supports the notion that the exchange of information above a length scale may not be really useful for maximizing the individual success rates. We also find that such interactions can lead to scalefree walks by the foragers in certain limits. Interdependence between the foragers also correlates the states of the foragers manifested by powerlawdistributed waiting times. The model also shows that under certain conditions steady or transient patterns of collective motion can emerge. We conclude that our model can help understand overall foraging patterns of animal groups in terms of simple strategies at the level of the individual animals.
Foraging experiments, using clumped distributions for food sources, with different species of birds such as pigeons [36], sparrows [8], spice finches [37] and great tits [38] have revealed that individuals closely watch their conspecifics, take cues from their activity, or are attracted by their vocalizations, and home in towards the latter when the latter discovers food. The artificial arrangements of food are substitute for patchy distributions found in the actual environments. Such distributions are also ephemeral in the sense that a source starts depleting once it is detected. A realworld example involves birds preying on fish schools. The school can only be detected close to the water surface and may become undetectable when it submerges to larger depths [39]. Similar foraging behaviour has also been observed in fish [40], primates [41], human populations [42] and other mammals [43]. However, the case when the majority of the foragers consistently keep relocating to the region where food has been discovered is, in general, found to be disadvantageous. Observation on greenfinches [44] with depletable patch conditions have revealed that such behaviour leads to a decrease in the average food intake with the increase in the group size. Similar results were also obtained in [37]. The disadvantage is largely attributed to the suppression of simultaneous discovery of multiple patches which is very true for our model. In addition, in our model, the decrease in efficiency with increase in number of foragers can be seen from figure 2a in the limit of α → 0 and γ = 2.5. In a study on the Mongolian gazelle [27], it was found that the gazelles exchange information about locations of resources at such distances that result in the optimal search. The authors also discussed possible disadvantages (other than search efficiency) that may result from communication at very short and long distances. Remarkably, the movement of gazelles was found to be Brownian. A study on terns [45] noted that even when the observed maximum distance of approach between terns trying to locate fish schools was 1500 m, the most frequent joinings occurred at distances less than 500 m. The fact that large excursions may occur because of social interactions was found in the case of foraging groups of red colobus monkeys [46]. Formation of transient but regular flocking patterns has been observed in large groups of cormorants [39]. As soon as a fish school is detected by a bird, searching independently, the other birds aggregate at the location. After feeding, the pattern disintegrates and the cormorants revert back to moving independently. In general, the interactions between individuals that come into play during foraging would be shaped through evolution [47]. However, it is not necessary that in all species the interactions would always result in the most efficient search by the individuals [48]. In addition, it may be possible that during the controlled experiments the animals may learn by experience during the trials. Our present model characterizes the interaction though the parameter α, such that we investigate the efficiency by tuning this parameter, when the resource distribution is both heterogeneous and ephemeral. In future, we would like to extend the present model such that α becomes an evolvable trait.
The precise rules used in our modelling scheme may not be unique. We believe that probabilistic decision making rules such as those based on quorum responses [49–51] or Bayesian estimation [52,53] would have yielded qualitatively similar results. A natural extension of the model would be to consider Lévy walks for motion of foragers searching independently. It would be also interesting to incorporate flock cohesion [34,54] in the model. Incorporating a tendency of nearby foragers to align their motion and form flocks may allow us to find an optimal group size for efficient foraging. Such exercises may also provide guiding principles in design of artificial foraging swarms [55] where information sharing becomes a mode of cooperation [56,57].
Funding statement
We acknowledge support from the FP7 ERC COLLMOT grant. K.B. acknowledges the support from the BITS Research Initiation Grant Fund.
Acknowledgements
K.B. is grateful to A. K. Nandi for helpful discussions during the course of this research. K.B. also thanks V. Guttal for helpful discussions.
 Received June 25, 2014.
 Accepted August 1, 2014.
 © 2014 The Author(s) Published by the Royal Society. All rights reserved.