Abstract

It is not exactly known why birds aggregate in coordinated flocks. The most common hypothesis proposes that the reason is protection from predators. Most of the currently developed examples of individual-based predator-prey models assume predators are attracted to the center of a highly coordinated flock. This proposed attraction of a predator to a flock would appear to be contradictory to an alternate hypothesis that flocks evolved as a protection against predation. In an attempt to resolve this apparent conflict, in this article we use a fuzzy individual-based model to study three attack tactics (attack center, attack nearest, attack isolated) and analyze the success of predation on two types of prey (social and individualistic). Our simulations revealed that social flocking (as opposed to individualistic behavior) is the optimal anti-predatory response to predators attacking mainly isolated individuals.

1 Introduction

The study of collective behavior is a fascinating field that analyzes how simple actions of an individual influence the complex global dynamics of a group. Aristotle once stated: “The whole is greater than the sum of its parts”—a statement that describes the essence of collective behavior. Typical examples of collective behavior are flocks of birds, schools of fish, and swarms of insects—phenomena that can be easily observed in nature. Collective behavior is also interesting because similar patterns emerge at smaller scales (cellular level) [6, 44].

Even though collective behavior is a common sight, it is still surrounded by mystery [27]. Several different hypotheses in the literature suggest reasons why animals sometimes coalesce into organized groups. The most common one proposes that such groups may function as an effective defense against predators [12, 22, 27, 37]. This hypothesis is supported by evidence that animals in groups may benefit from an increased probability of detecting a predator [9], individuals in groups may reduce the amount of time spent for predator vigilance [7, 42], and an individual in a large group may have a lower probability of being attacked by a predator [10]. Other hypotheses suggest that aggregating animals may benefit through higher mating efficiency, and more efficient foraging [23]. Some studies claim that fish schools or bird flocks save energy because of hydrodynamic or aerodynamic benefits [30]; however, the opinions on this matter are contradictory [3, 38, 48]. Our work focuses mainly on bird flocks; however, some of the results can also be applied to fish schools, since they have some similarities in structure and behavior, in that they both operate in a three-dimensional world [22].

Bird flocks are among the most widely observed, yet least understood, phenomena of collective behavior, mostly because of the difficulty of obtaining field data and, with the exception of a few types of urban flocks, the unpredictability of the appearance of highly organized flocks in nature [15]. Two types of highly organized bird flocks emerge in nature—cluster flocks, demonstrated by pigeons and starlings, and line flocks, such as can be seen in groups of geese flying in a vee [14]. Every evening, when birds that fly in organized groups return to their roosting areas, small flocks coalesce into giant cluster flocks, often numbering tens of thousands of birds. Birds may then perform complex aerial maneuvers before finally settling in their roosts [27]. Such behavior can often be seen every evening at the same place, so it might appear that birds flying in such flocks are actually attracting predators and making it easy for them to attack the flock, which runs counter to the idea that highly coordinated flocks evolved to reduce the impact of predation.

Complex flocking behavior can emerge if individuals follow simple rules. In 1987, Reynolds [41] published a ground-breaking article that presented the first computer flocking animation (boids). At the same time, Heppner and Grenander [16] were working on a similar project in which they modeled birds' behavior with stochastic nonlinear differential equations. In these two and most subsequent models, equations govern the behavior of the artificial animals (animats).

Our model uses fuzzy logic [51] and fuzzy-rule-based systems [5, 32], rather than equations, to develop the behavior of artificial animals.

Some current state-of-the-art models are in three dimensions and incorporate some simplified aerodynamics [17]. To make an in-depth study of shapes and patterns that emerge within the flock during a predator attack, a three-dimensional model would be required [13], but for the purpose of our study a two-dimensional model suffices, since some researchers suggest that the dimensionality of the model minimally affects the results of the simulations [18, 19, 24], and others believe that models should be as simple as possible [4].

Various authors have upgraded the basic models to add additional functionalities. Moškon et al., for example, used fuzzy logic to simulate the foraging behavior of artificial birds [33]. There are also several models that implement predators. All of them are based on Reynolds' model, and most of them use a predator that attacks the center of the flock [20, 29]. This tactic may be right for some species of real fish; a swordfish, for example, in nature typically attacks the center of a prey school. In the first attack it disperses the school, and in the following attacks it focuses on the individual fish that become separated from the rest of the group [45]. Compared to birds, however, fish may have better perception of the environment because of the lateral line, and schooling might be used to confuse the lateral line of predators [25]. Assuming that birds do not have a sense like the lateral line, avian predators in nature might not attack in such a fashion.

Others that studied predator-prey dynamics in collective behavior mostly focused on a flock's response to the predator's attack. For example, Inada et al. focused on common escape patterns that emerge [20], while Lee et al. analyzed how the size of the flocks changes during an attack [29].

In both cases the predator attacked the center of the flock. With respect to the hypothesis that flocks form as a defensive mechanism, targeting the center of the flock in hope of catching a prey might be viewed as a tactic based on pure luck. As already stated, this attack tactic might not be used by avian predators, except maybe for the first attacks, where the goal might be to disperse the prey in order to prepare for other tactics whose positive outcome was more probable. This research is not the first to propose different attack tactics; Nishimura [35, 36] was the first to study target selection mechanisms. The key differences between this research and Nishimura's studies are: (1) we use fuzzy logic; Nishimura used differential equations; (2) we model target selection through “realistic” visual perception; Nishimura expressed the probability of a prey becoming a target through a mathematical equation that does not take into account the position of the predator relative to the flock, nor its orientation; (3) in our model the prey that “sees” the predator tries to escape; in Nishimura's model it does not; and (4) we study social versus individualistic prey behavior; Nishimura studies ordered, partially disordered, and fully disordered prey motion.

According to Nishimura's study [36] the best predator tactic is to attack a peripheral target, and not an isolated target, as can be observed in nature [21, 45]. The reasons for this conclusion are two: (a) the equation Nishimura uses for a predator targeting isolated prey (Nishimura's strategy S) will not select any target if none exists that has a large enough separation from the flock; and (b) Nishimura's predator has perfect vision (i.e., it is able to perceive all prey), and it can happen that the predator will select a target that is on the opposite side of the flock and therefore requires a substantial amount of time to reach it. The first reason is unrealistic in that, of the three types of motion that Nishimura studies (ordered, partially disordered, and fully disordered), ordered motion, a rare event in nature, is a clear favorite. The second is unrealistic in that the selected target potentially might not be visible to the predator at all, due to occlusion.

The analysis of predator-prey pursuit is interesting not only from a biological perspective, but also from the perspective of control theory [34, 49]. The control theory approach could potentially represent an alternative, more mathematical approach to our study. However, we believe our fuzzy, individual-based model with its differences from Nishimura's approach permits a simulation whose behavior is closer to that of real birds.

2 Methods

The basis for our work is an existing fuzzy-logic-based bird flocking model made by Lebar Bajec et al. [28], called synflocks. An artificial animal, an animat, can be described using three qualities: (a) its perception of the environment, (b) its drives, and (c) its action selection. Perception acts as a filter for important information. Drives define desired actions that will fulfill the animal's needs. Action selection combines these actions and performs the appropriate locomotor response. Assuming the artificial universe consists only of artificial animals with no environmental factors like artificial trees, then an artificial animal's behavior is dependent mostly on the position, direction, and speed of the neighbors it perceives.

The animats in our model use the basic Reynolds drives—cohesion, separation, and alignment [41]. Cohesion simulates attraction toward flock-mates and is modeled as the animat's tendency to fly toward distant flock-mates when there are none nearby. Separation is a drive that helps the animats to avoid collisions—it forces an individual to fly away from flock-mates that are too close. With the third drive—alignment—animats coordinate their velocity (direction and speed of flight) with flock-mates. A visual representation of the drives can be seen in Figure 1. In our model the drives are described with simple linguistic if-then rules. Fuzzy logic is used for the transformation of the rules into numerical values: the desired change in direction and speed of each individual animat. More precisely, the if-then rules are used in a Mamdani fuzzy inference system [32] (see  Appendix).

Figure 1. 

(a) Perception of nearby neighbors. The black bird is the observed individual. The dark gray birds are the perceived nearby birds that influence the observed bird's behavior. The white and outlined birds are either occluded by nearer birds (shaded areas), outside the observed bird's field of vision (hatched area), or outside the number-limited range. Red arrows represent the resulting force vectors of the three basic drives—(b) alignment, (c) cohesion, and (d) separation.

Figure 1. 

(a) Perception of nearby neighbors. The black bird is the observed individual. The dark gray birds are the perceived nearby birds that influence the observed bird's behavior. The white and outlined birds are either occluded by nearer birds (shaded areas), outside the observed bird's field of vision (hatched area), or outside the number-limited range. Red arrows represent the resulting force vectors of the three basic drives—(b) alignment, (c) cohesion, and (d) separation.

The original Reynolds model and most existing models are based on metric distance. In these models every animat within a limited radius influences the behavior of the observed animat; if an animat is outside that radius, it does not have any influence. Nevertheless, recent research [1, 2] suggests that in nature only around seven nearest neighbors influence an individual. As this technique is already gaining support in current state-of-the-art models [17], we likewise use a number-limited neighborhood (topological distance) instead of a radius-limited neighborhood (metric distance). In models based on topological distance, only a fixed number of nearest animats are influential, regardless of their distance.

Our model uses topological distance and concentrates on vision as the principal means of neighborhood perception. Our animat's field of vision is 300° wide, with a blind angle of 60° directly behind it. Most current models presume that birds have “perfect” vision and do not allow for the occlusion of distant birds due to other birds flying in the flock. Yet a recent study by Kunz et al. [24] shows that obstruction of vision increases the realism of simulations. Our model thus takes into account only seven visible (non-occluded) nearby animats (see Figure 1).

Predator and prey behaviors in our study are based on rules extracted from relevant theoretical literature and field observations. We modeled our simulations after a common scenario, where a peregrine falcon (Falco peregrinus) is attacking a flock of European starlings (Sturnus vulgaris). In horizontal flight, the most economical flight speed (as to the amount of energy spent for flight propulsion) is around 60% of the bird's maximum speed [46]. Let us call this speed the optimal cruising speed. The optimal cruising speed of a European starling is 11 m/s, and the optimal cruising speed of a peregrine falcon is 13 m/s [46]. In accordance with these values we set the maximum speed of our prey animat to 18 m/s, and the maximum speed of our predator animat to 22 m/s. Note that we presumed that the peregrine falcon was not hunting by using its characteristic hunting stoop (high-speed dive), when it can reach speeds up to 157 m/s [47]. The minimum flight speed, in nature and in our model, is around 40% of the maximum flight speed [46], which amounts to 7.2 m/s for prey animats and 8.8 m/s for the predator animat. To define the predator-prey relationship we introduced three additional drives—hide, seek, and regulate speed. The prey's behavior is governed by the three basic drives (cohesion, separation, and alignment), and in addition hide and regulate speed (see  Appendix for their explanation). The hide drive helps the prey to survive, as it forces it to fly away from the attacking predator; it was tuned so that the direction of the prey's escape matches field observations by Handegard et al. [11]. Regulate speed is only active when the hide drive is inactive—when the predator is hidden from the prey's sight. This drive encourages prey to fly with their optimal cruising speed. The predator's behavior is guided only by the seek drive (see  Appendix for its detailed explanation). With the seek drive the predator tries to catch the selected target.

We implemented three different attack tactics (see Figure 2). In our first tactic the predator attacks the center point of the seven perceived prey. This mimics the tactic in which a predator attacks the center of the flock in hope of hitting a target, but takes into account the limited amount of information available—distance, relative position, difference in speed, and difference in heading of the perceived artificial animals.

Figure 2. 

The three attack tactics: (a) the predator attacks the center of the seven perceived prey, (b) the predator attacks the nearest of the seven perceived prey, (c) the predator attacks the most isolated of the seven perceived prey. The red bird is the predator, and in the electronic version, the red arrow represents the resulting force vector of the predator's seek drive.

Figure 2. 

The three attack tactics: (a) the predator attacks the center of the seven perceived prey, (b) the predator attacks the nearest of the seven perceived prey, (c) the predator attacks the most isolated of the seven perceived prey. The red bird is the predator, and in the electronic version, the red arrow represents the resulting force vector of the predator's seek drive.

In the second tactic, the predator attacks the nearest of the seven perceived prey. The nearest prey might be the one that is also the fastest to reach, therefore making it a logical target for a predator. If a real predator chooses its prey in such a fashion, then flocking might work as a mechanism to reduce an individual's domain of danger [10]. The domain of danger is defined as the area in which the observed individual is the predator's nearest neighbor. Obviously, the average size of the domain of danger decreases if the number of birds in the flock increases, thus favoring tight, highly organized flocks. By reducing the domain of danger an individual lowers the probability of being attacked by the predator, thus possibly increasing its chances of survival.

A predator using the third tactic attacks the most isolated of the seven perceived prey. In our study the most isolated prey is the one that has the largest angular distance to its nearest neighbor. We define angular distance as the angle between a potential target and its nearest neighbor—from the predator's viewpoint. From a predator's viewpoint, isolated prey appear to have a large domain of danger because they are the most separated from the rest of the perceived prey. From the predator's perspective they would require the largest amount of time to decrease their domain of danger—time that is available to the predator to catch them. If we presume that flocking is indeed a protection mechanism, we can assume that the most isolated bird is the one that is the most vulnerable, making it a logical target for a predator.

3 Results and Discussion

To recapitulate, the predator in our model uses one of the following three attack tactics: (1) attack center (i.e., attack the center point of the seven perceived prey), (2) attack nearest (i.e., attack the nearest of the seven perceived prey), and (3) attack isolated (i.e., attack the most isolated, from the predator's point of view, of the seven perceived prey). In addition to escaping predator attacks and regulating their flight speed, the prey can exhibit two types of behavior: (1) social behavior (i.e., prey obey the cohesion, separation, and alignment drives) or (2) individualistic behavior (i.e., prey ignore the cohesion and alignment drives, but obey the separation drive to avoid collisions). In total this gives six combinations, through which we wished to answer the following questions: (1) what is the optimal predator tactic, given a certain prey behavior, (2) what is the optimal prey behavior, given a certain predator tactic.

To provide answers to these questions we ran simulations where small cluster flocks consisting of 20, 40, or 60 social or individualistic prey were attacked by a predator from eight different bearings, relative to the flock. One of the starting configurations along with the eight bearings can be seen in Figure 3. The different bearings were used to eliminate any dependence of the simulation's results on the predator's bearing (e.g., a head-on attack versus an attack from behind). This gives a total of 432 simulations (72 per configuration, i.e., selected predator attack tactic and prey behavior). Each simulation was ran for 900 steps (frames)—30 s in our visualizations. We measured the time the predator needed to catch a prey. If the predator failed to catch the prey, the time to catch was set to 900 frames. The histograms of the time to catch for all simulations can be seen in Figure 4; a more in-depth discussion is given in the following sections.

Figure 3. 

One of the starting configurations along with eight bearings of the predator's attack.

Figure 3. 

One of the starting configurations along with eight bearings of the predator's attack.

Figure 4. 

The influence of the predator's attack tactic and prey's behavior on survivability of prey. Presented are six histograms of the predator's time to catch in corresponding simulation runs (n = 72). Dark gray histograms present the time to catch in simulations with social prey, whereas light gray histograms present the time to catch in simulations with individualistic prey (cohesion and alignment drives not used). In the electronic version, red lines present the corresponding median time to catch.

Figure 4. 

The influence of the predator's attack tactic and prey's behavior on survivability of prey. Presented are six histograms of the predator's time to catch in corresponding simulation runs (n = 72). Dark gray histograms present the time to catch in simulations with social prey, whereas light gray histograms present the time to catch in simulations with individualistic prey (cohesion and alignment drives not used). In the electronic version, red lines present the corresponding median time to catch.

3.1 Optimal Predator Tactic

Our simulations suggest that the best tactic for a predator attacking flocks of social prey is the attack isolated tactic (t = 3.01, df = 142, p = 0.003). It would appear that isolated prey benefit the least from the advantages of flocking. On average the predator needed 279.74 frames (SD = 72.29) to catch an isolated prey. The predator whose tactic was to attack the nearest prey needed 316.74 frames (SD = 75.76) to catch its target, and the one whose tactic was attack the center of the seven perceived prey needed 845.08 frames (SD = 165.04).

With individualistic prey the best tactic for the predator is the attack-nearest tactic (t = 5.12, df = 140, p < 0.0001). This finding makes sense, since the predator will get to the nearest prey faster than to those that are farther away. Predators that used this tactic needed, on average, 177.31 frames (SD = 33.95) to catch a prey. Predators that attacked isolated prey caught a prey in 208.06 frames (SD = 38.04). The attack-center tactic proved to be the worst, as the predator required 832.97 frames (SD = 198.89) to catch a prey.

The predator that used the attack-center tactic was, in most cases, not successful. In other words, it did not catch a prey in the 900 frames for which we ran the simulations (regardless of whether the prey was social or non-social). Thus the attack-center tactic proved to be the worst one, regardless of the prey's behavior.

As for the predator's best tactic overall, regardless of the prey's social or individualistic behavior, no difference was found between the two most successful tactics, attack nearest and attack isolated (t = 0.34, df = 264, p = 0.73).

3.2 Optimal Prey Behavior

When the predator used the attack-center tactic, no difference was found for the average time to catch between social and individualistic prey (t = 0.39, df = 137, p = 0.69). In both cases the predator was rarely successful, which is a direct result of the predator's tactic, which is based on pure luck.

With predators whose tactic actually tries to optimize the chance of a positive result, however, the average time to catch is longer when prey behavior is social than when it is individualistic—more so when the predator targets the nearest prey (t = 14.25, df = 98, p < 0.0001) than when the predator uses the attack-isolated tactic (t = 7.42, df = 108, p < 0.0001).

The best tactic for prey attacked by a predator appears to be social behavior.

3.3 Biological Relevance

Fine-scale sonar tracking of interactions among predatory fish and their schooling prey performed by Handegard et al. [11] suggests that the most successful predators attack from behind. In our simulations, when the predator attacked from the side or front, the prey always managed to escape the initial attack. The predator was successful in one of the successive attacks that occurred from behind. Attacks from behind appear to be much more successful than other strategies, as in some cases, when the predator attacked from behind, the predator's initial attack was already fruitful. An example of the corresponding chase and escape paths in the case of a single predator attacking a single prey is presented in Figure 5.

Figure 5. 

Predator attacking a single prey from three main directions: (a) behind, (b) head on, and (c) side.

Figure 5. 

Predator attacking a single prey from three main directions: (a) behind, (b) head on, and (c) side.

In 1983, Pitcher and Wyche [39] defined several escape patterns. In our simulations, when the predator attacks the center point of the seven perceived prey, prey show similar escape patterns to those in nature and existing fish school models. Indeed, we managed to spot three of the escape patterns defined by Pitcher and Wyche [39]: herd, split, and fountain. The herd pattern can be seen at the beginning of a predator approach. The herd pattern then morphs into either split or fountain. Split occurs when a large flock is divided into smaller ones. A typical characteristic of the fountain pattern is that the split flock rejoins behind the predator. These patterns can be seen in Figure 6.

Figure 6. 

Three common patterns in the artificial flock's response to a predator attack: (above) from herd (frames 30–120) tosplit (frames 120–240), (below) from herd (frames 60–330) to fountain (frames 330–510). In the snapshots below one canalso notice the phases in the flock's response: before attack (frame 60), compression (frames 60–180 and 510–600), expansion (frames 270–510), and relaxation (frames 600–690). In the electronic version, the red bird is the predator; black birds represent prey. For the sake of clarity the birds were scaled by 200% (above) and 300% (below).

Figure 6. 

Three common patterns in the artificial flock's response to a predator attack: (above) from herd (frames 30–120) tosplit (frames 120–240), (below) from herd (frames 60–330) to fountain (frames 330–510). In the snapshots below one canalso notice the phases in the flock's response: before attack (frame 60), compression (frames 60–180 and 510–600), expansion (frames 270–510), and relaxation (frames 600–690). In the electronic version, the red bird is the predator; black birds represent prey. For the sake of clarity the birds were scaled by 200% (above) and 300% (below).

A more quantifiable measure was defined by Lee et al. [29], who defined three phases that can be seen in artificial flocks during a predator attack—compression, expansion, and relaxation (Figure 6). In the first phase of an attack the flock's size decreases; this phase is called compression. In the second phase—expansion—prey try to move away from the predator. By doing this, the flock's size increases. In the last, phase prey try to regroup, so the flock's size again decreases. This last phase is called relaxation.

The three phases can easily be observed by tracing the artificial flock's flock size, as defined by Lee et al. [29]:
formula
where ri is the distance from the center of the flock to the ith individual, and N is the total number of animats in the flock. All of our simulations showed a similar response of the artificial flock to a predator's attack. Figure 7 visualizes how, in our simulations, the flock size changes through time, both when the artificial flock is under threat from the predator and when there is no predator nearby. The flock size is measured in body lengths; one body length equals 20 cm, which is the size of a European starling.
Figure 7. 

Plot of the flock size of an artificial flock consisting 20 animats. The black line shows the flock size over 900 simulation steps when there is no predator nearby. In the electronic version, the red line shows the flock size over 900simulation steps when the flock is under predator attack. A clear example of flock expansion is marked with a light gray background, and dark gray is used to mark an example of flock compression. As predator attacks occur one after the other, there is no clear example of flock relaxation.

Figure 7. 

Plot of the flock size of an artificial flock consisting 20 animats. The black line shows the flock size over 900 simulation steps when there is no predator nearby. In the electronic version, the red line shows the flock size over 900simulation steps when the flock is under predator attack. A clear example of flock expansion is marked with a light gray background, and dark gray is used to mark an example of flock compression. As predator attacks occur one after the other, there is no clear example of flock relaxation.

4 Conclusion

Our simulations show that the least successful predator is the one that attacks the center of the flock. They suggest that with predators whose tactic tries to optimize the chance of a positive outcome, social behavior is more advantageous than individualistic behavior, which strengthens our belief in the hypothesis that cluster flocking can be a mechanism for protection from predators.

The behavior of our artificial flocks appears to be comparable with that seen in flocks in nature. The average distance from the nearest neighbor is around four body lengths (one body length equals 20 cm) [2, 31, 40]; the response of an artificial flock to a predator attack is similar to field observations [29], and similar escape patterns to those in nature emerge [39]. Our results also show that the most successful predators attack from behind [11] and seek isolated targets [21].

Our results seem to suggest that cluster flocking around a roost is paradoxical, because although its structure might provide some protection against a predator attack, its very existence invites a predator attack, and (at least in nature) there are always isolated individuals that can be picked off. It suggests also that at least in some circumstances, which may or may not be common in nature, tight cluster flocking can be of benefit to the flock as a whole, although it does not provide absolute protection to individuals in the flock.

Acknowledgments

We sincerely thank Frank H. Heppner of the University of Rhode Island and Maja Lebar Bajec for reading early drafts of the manuscript. Our gratitude also goes to Gregor šega and Aleksandar Jurišić, who revised the statistical analysis of our results. This work was funded in part by the Slovenian Research Agency (ARRS) through the Pervasive Computing research program (P2-0395).

References

1
Ballerini
,
M.
,
Cabibbo
,
N.
,
Candelier
,
R.
,
Cavagna
,
A.
,
Cisbani
,
E.
,
Giardina
,
I.
,
Lecomte
,
V.
,
Orlandi
,
A.
,
Parisi
,
G.
,
Procaccini
,
A.
,
Viale
,
M.
, &
Zdravković
,
V.
(
2008
).
Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study.
Proceedings of the National Academy of Sciences
,
105
(
4
),
1232
1237
.
2
Ballerini
,
M.
,
Cabibbo
,
N.
,
Candelier
,
R.
,
Cavagna
,
A.
,
Cisbani
,
E.
,
Giardina
,
I.
,
Orlandi
,
A.
,
Parisi
,
G.
,
Procaccini
,
A.
, &
Viale
,
M.
(
2008
).
Empirical investigation of starling flocks: A benchmark study in collective animal behaviour.
Animal Behaviour
,
76
(
1
),
201
215
.
3
Bill
,
R. G.
, &
Herrnkind
,
W. F.
(
1976
).
Drag reduction by formation movement in spiny lobsters.
Science
,
193
(
4258
),
1146
1148
.
4
Czirók
,
A.
, &
Vicsek
,
T.
(
2000
).
Collective behavior of interacting self-propelled particles.
Physica A: Statistical Mechanics and Its Applications
,
281
(
1
),
17
29
.
5
da Silva Peixoto
,
M.
,
de Barros
,
L. C.
, &
Bassanezi
,
R. C.
(
2008
).
Predator–prey fuzzy model.
Ecological Modelling
,
214
(
1
),
39
44
.
6
Deisboeck
,
T. S.
, &
Couzin
,
I. D.
(
2009
).
Collective behavior in cancer cell populations.
Bioessays
,
31
(
2
),
190
197
.
7
Elgar
,
M. A.
(
1989
).
Predator vigilance and group size in mammals and birds: A critical review of the empirical evidence.
Biological Reviews of the Cambridge Philosophical Society
,
64
(
1
),
13
33
.
8
Engel
,
R. W.
,
Kane
,
M. J.
, &
Tuholski
,
S. W.
(
1999
).
Individual differences in working memory capacity and what they tell us about controlled attention, general fluid intelligence, and functions of the prefrontal cortex.
In A. Miyake & P. Shah (Eds.)
,
Mechanisms of active maintenance and executive control.
Cambridge, UK
:
Cambridge University Press
.
9
Galton
,
F.
(
1871
).
Gregariousness in cattle and men.
MacMillan's Magazine
,
23
(
136
),
353
357
.
10
Hamilton
,
W. D.
(
1971
).
Geometry for the selfish herd.
Journal of Theoretical Biology
,
31
(
2
),
295
311
.
11
Handegard
,
N. O.
,
Boswell
,
K. M.
,
Ioannou
,
C. C.
,
Leblanc
,
S. P.
,
Tjøstheim
,
D. B.
, &
Couzin
,
I. D.
(
2012
).
The dynamics of coordinated group hunting and collective information transfer among schooling prey.
Current Biology
,
22
(
13
),
1213
1217
.
12
Hart
,
P. J.
, &
Freed
,
L. A.
(
2005
).
Predator avoidance as a function of flocking in the sexually dichromatic Hawaii akepa.
Journal of Ethology
,
23
(
1
),
29
33
.
13
Hemelrijk
,
C. K.
, &
Hildenbrandt
,
H.
(
2011
).
Some causes of the variable shape of flocks of birds.
PLoS One
,
6
(
8
,
e22479
.
14
Heppner
,
F. H.
(
1974
).
Avian flight formations.
Bird Banding
,
45
(
2
),
160
169
.
15
Heppner
,
F. H.
(
1997
).
Three-dimensional structure and dynamics of bird flocks.
In J. K. Parrish & W. M. Hamner (Eds.)
,
Animal Groups in Three Dimensions.
Cambridge, UK
:
Cambridge University Press
.
16
Heppner
,
F. H.
, &
Grenander
,
U.
(
1990
).
A stochastic nonlinear model for coordinated bird flocks.
In S. Krasner (Ed.)
,
The ubiquity of chaos.
Washington, DC
:
AAAS Publications
.
17
Hildenbrandt
,
H.
,
Carere
,
C.
, &
Hemelrijk
,
C. K.
(
2010
).
Self-organized aerial displays of thousands of starlings: A model.
Behavioral Ecology
,
21
(
6
),
1349
1359
.
18
Huth
,
A.
, &
Wissel
,
C.
(
1992
).
The simulation of the movement of fish schools.
Journal of Theoretical Biology
,
156
(
3
),
365
385
.
19
Huth
,
A.
, &
Wissel
,
C.
(
1994
).
The simulation of fish schools in comparison with experimental data.
Ecological Modelling
,
75–76
,
135
146
.
20
Inada
,
Y.
, &
Kawachi
,
K.
(
2002
).
Order and flexibility in the motion of fish schools.
Journal of Theoretical Biology
,
214
(
3
),
371
387
.
21
Ioannou
,
C. C.
,
Guttal
,
V.
, &
Couzin
,
I. D.
(
2012
).
Predatory fish select for coordinated collective motion in virtual prey.
Science
,
337
(
6099
),
1212
1215
.
22
Krause
,
J.
, &
Ruxton
,
G. D.
(
2002
).
Living in groups.
Oxford, UK
:
Oxford University Press
.
23
Krebs
,
J. R.
, &
Davies
,
N. B.
(
1993
).
An introduction to behavioural ecology
(3rd ed.).
Oxford, UK
:
Blackwell Scientific Publications
.
24
Kunz
,
H.
, &
Hemelrijk
,
C. K.
(
2012
).
Simulations of the social organization of large schools of fish whose perception is obstructed.
Applied Animal Behaviour Science
,
138
,
142
151
.
25
Larsson
,
M.
(
2009
).
Possible functions of the octavolateralis system in fish schooling.
Fish and Fisheries
,
10
(
3
),
344
353
.
26
Lebar Bajec
,
I.
(
2005
).
Fuzzy model for a computer simulation of bird flocking
.
Ph.D. thesis, University of Ljubljana, Faculty of Computer and Information Science, Ljubljana, Slovenia
.
27
Lebar Bajec
,
I.
, &
Heppner
,
F. H.
(
2009
).
Organized flight in birds.
Animal Behaviour
,
78
(
4
),
777
789
.
28
Lebar Bajec
,
I.
,
Zimic
,
N.
, &
Mraz
,
M.
(
2005
).
Simulating flocks on the wing: The fuzzy approach.
Journal of Theoretical Biology
,
233
(
2
),
199
220
.
29
Lee
,
S.-H.
,
Pak
,
H. K.
, &
Chon
,
T.-S.
(
2006
).
Dynamics of prey-flock escaping behavior in response to predator's attack.
Journal of Theoretical Biology
,
240
(
2
),
250
259
.
30
Lissaman
,
P. B.
, &
Shollenberger
,
C. A.
(
1970
).
Formation flight of birds.
Science
,
168
(
934
,
1003
.
31
Major
,
P. F.
, &
Dill
,
L. M.
(
1978
).
The three-dimensional structure of airborne bird flocks.
Behavioral Ecology and Sociobiology
,
4
(
2
),
111
122
.
32
Mamdani
,
E. H.
(
1974
).
Application of fuzzy algorithms for the control of a simple dynamic plant.
In
Proceedings of IEEE 1974
(pp.
121
159
).
33
Moškon
,
M.
,
Heppner
,
F. H.
,
Mraz
,
M.
,
Zimic
,
N.
, &
Lebar Bajec
,
I.
(
2007
).
Fuzzy model of bird flock foraging behavior.
In
Proceedings of the 6th EUROSIM Congress on Modelling and Simulation
(
Vol. 2
, pp.
1
6
).
Vienna
:
ARGESIM
.
34
Nahin
,
P. J.
(
2007
).
Chases and escapes: The mathematics of pursuit and evasion.
Princeton, NJ
:
Princeton University Press
.
35
Nishimura
,
S. I.
(
2000
).
Studying attention dynamics of a predator in a prey-predator system.
In
Proceedings of Artificial Life 7
(pp.
377
342
).
Cambridge, MA
:
MIT Press
.
36
Nishimura
,
S. I.
(
2002
).
A predator's selection of an individual prey from a group.
Biosystems
,
65
(
1
),
25
35
.
37
Nishimura
,
S. I.
, &
Ikegami
,
T.
(
1997
).
Emergence of collective strategies in a prey-predator game model.
Artificial Life
,
3
(
4
),
243
260
.
38
Partridge
,
B. L.
, &
Pitcher
,
T. J.
(
1979
).
Evidence against a hydrodynamic function for fish schools.
Nature
,
279
(
5712
,
418
.
39
Pitcher
,
T. J.
, &
Wyche
,
C. J.
(
1983
).
Predator-avoidance behaviours of sand-eel schools: Why schools seldom split.
In D. Noakes, D. G. Lindquist, & J. A. Ward (Eds.)
,
Predators and prey in fishes.
Berlin
:
Springer
.
40
Pomeroy
,
H.
, &
Heppner
,
F. H.
(
1992
).
Structure of turning in airborne rock dove (Columba livia) flocks.
The Auk
,
109
(
2
),
256
267
.
41
Reynolds
,
C. W.
(
1987
).
Flocks, herds and schools: A distributed behavioral model.
ACM SIGGRAPH Computer Graphics
,
21
(
4
),
25
34
.
42
Sadedin
,
S. R.
, &
Elgar
,
M. A.
(
1998
).
The influence of flock size and geometry on the scanning behaviour of spotted turtle doves, Streptopelia chinensis.
Austral Ecology
,
23
(
2
),
177
180
.
43
Sherry
,
D. F.
, &
Vaccarino
,
A. L.
(
1989
).
Hippocampus and memory for food caches in black-capped chickadees.
Behavioral Neuroscience
,
103
(
2
),
308
318
.
44
Spector
,
L.
,
Klein
,
J.
,
Perry
,
C.
, &
Feinstein
,
M.
(
2003
).
Emergence of collective behavior in evolving populations of flying agents.
In E. Cantu-Paz et al. (Eds.)
,
Proceedings of GECCO 2003.
Berlin
:
Springer-Verlag
.
45
Pavlov
,
D. S.
, &
Kasumyan
,
A. O.
(
2000
).
Patterns and mechanisms of schooling behavior in fish: A review.
Journal of Ichthyology
,
40
(
2
),
163
231
.
46
Tennekes
,
H.
(
1997
).
The simple science of flight: From insects to jumbo jets.
Cambridge, MA
:
MIT Press
.
47
Tucker
,
V. A.
(
1998
).
Gliding flight: Speed and acceleration of ideal falcons during diving and pull out.
The Journal of Experimental Biology
,
201
(
3
),
403
414
.
48
Usherwood
,
J. R.
,
Stavrou
,
M.
,
Lowe
,
J. C.
,
Roskilly
,
K.
, &
Wilson
,
A. M.
(
2011
).
Flying in a flock comes at a cost in pigeons.
Nature
,
474
(
7352
),
494
497
.
49
Wei
,
E.
,
Justh
,
E. W.
, &
Krishnaprasad
,
P. S.
(
2009
).
Pursuit and an evolutionary game.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
,
465
(
2105
),
1539
1559
.
50
Wiederman
,
S. D.
, &
O'Carroll
,
D. C.
(
2013
).
Selective attention in an insect visual neuron.
Current Biology
,
23
(
2
),
156
161
.
51
Zadeh
,
L. A.
(
1965
).
Fuzzy sets.
Information and Control
,
8
(
3
),
338
353
.

Appendix: Supplementary Material

In our model the animat is defined as a three-stage process [26]: perception, drives, and action selection. For each animat the perception stage extrapolates data about the universe (other animats) that is available to the observed animat. In the drives stage, based on these data, a set of actions is computed that would fulfill individual needs of the modeled artificial animal. The action selection stage merges the actions and executes the locomotor response, thus changing the state of the universe.

A.1 Perception

In our model the universe consists of animats only; the universe is in essence a list of animats. To extrapolate data about the universe, as a first step, the observed animat is removed from this list, and the animats remaining in the list are sorted based on their distance from the observed animat. Animats hidden by those that are closer to the observed animat and ones that are outside its field of view are then filtered out. The nearest seven animats remaining in the list provide the data about the universe. Only information about the relative position (i.e., angular offset with respect to the observed animat's heading), distance, speed difference, and heading difference is passed on to the next stage.

Our approach differs from other existing models of number-limited neighborhoods (topological distance) [1] in that it always takes into account a fixed number of nearest neighbors. Other models typically implement the number-limited neighborhood as a variable radius-limited neighborhood [13]. In a variable radius-limited neighborhood, the observed animat will increase its radius of perception if in the previous step it perceived less than the specified number of animats, and decrease it if in the previous step it perceived more than the specified number of animats. This approach in truth varies the number of perceived animats in every step of the simulation, but limits it to the specified number. The principal reason this approach is taken is probably the speed of computation. Its drawback is that the observed animat could potentially perceive all of the other animats, provided these were distributed so that they were all at the same distance from the observed animat.

The number of objects that can be stored in working memory in humans and other mammals is small (four to seven) [1, 8]. The hippocampus, the structure in the brain primarily responsible for working memory, is generally similar in birds and mammals [43]. Our use of seven nearest animats in the perceptual world of the prey and predator is thus based on research on working memory. In addition, once the predator selects its target, in our model, it filters out all other potential prey, mimicking selective attention [50].

A.2 Drives

The drives that result in actions that would fulfill the animat's individual needs are modeled using Mamdani fuzzy rules [32]. For a detailed description of the cohesion, separation, and alignment drives, as well as a brief explanation of how a specific action is computed, the reader is invited to refer to [28]. An in-depth description is available in [26].

The data extrapolated in the previous stage are fuzzified as singletons and used as inputs for the fuzzy rules. The fuzzy variables, membership functions, and control rules for the hide drive can be seen in Figure 8. Figure 9 presents the regulate-speed drive, and Figure 10 presents the seek drive. The fuzzy rules are evaluated per individual perceived animat and the fuzzy outputs aggregated.

Figure 8. 

Detailed description of the hide drive. On the membership function charts that describe speed, the value 1 represents the appropriate maximum speed (predator's or prey's), while −1 represents the negative value of the appropriate maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

Figure 8. 

Detailed description of the hide drive. On the membership function charts that describe speed, the value 1 represents the appropriate maximum speed (predator's or prey's), while −1 represents the negative value of the appropriate maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

Figure 9. 

Detailed description of the regulate speed drive. On the membership function charts that describe speed, the value 1 represents the prey's maximum speed, and −1 represents the negative of the prey's maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

Figure 9. 

Detailed description of the regulate speed drive. On the membership function charts that describe speed, the value 1 represents the prey's maximum speed, and −1 represents the negative of the prey's maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

Figure 10. 

Detailed description of the seek drive. On the membership function charts that describe speed, the value 1 represents the predator's maximum speed, and −1 represents the negative of the predator's maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

Figure 10. 

Detailed description of the seek drive. On the membership function charts that describe speed, the value 1 represents the predator's maximum speed, and −1 represents the negative of the predator's maximum speed. While these are the desired changes in speed, the action selection stage ensures that the current speed of the observed individual never falls below the appropriate minimum speed (predator's or prey's).

The fuzzy outputs are then defuzzified and a force vector computed. This force vector represents the action that will fulfill the individual need. It gives the direction and magnitude of the individual drive (the desired change in speed and direction).

A.3 Action Selection

In the action selection stage the force vectors resulting from individual drives are merged together and a resulting force vector computed. The merging is achieved through a simple weighted sum of the individual force vectors. The resulting force vector is interpreted as a Newtonian force, based on which the observed animat's speed, heading, and position are updated.

A.4 Parameter Values

All of our model's parameters were either extrapolated from relevant theoretical literature (speed, field of view, etc.) or tuned (fuzzy membership functions, action selection mixing weights, etc.) so that the resemblance of the displayed behavior to that observed in nature was visually as close as possible. For example, the action selection mixing weights have been configured so that the simulations visually resemble (as closely as possible) field observations of flocking behavior. In the prey's case these were 5, 5, 3, 4, and 1 for cohesion, separation, alignment, hide, and regulate speed, respectively. Table 1 presents all of the remaining parameters of the model.

Table 1. 

Default parameter values.

Parameter
Description
Default value
Δt Time step 1 frame ( s) 
T Maximum length of one simulation 900 frames (30 s) 
vaM Maximum speed of the predator animat 22 m/s 
vam Minimum speed of the predator animat 8.8 m/s 
vpM Maximum speed of prey animats 18 m/s 
vpm Minimum speed of prey animats 7.2 m/s 
φ Animat field of view 300° 
n Number of influential nearest neighbors 
wc Weight for cohesion drive 
ws Weight for separation drive 
wa Weight for alignment drive 
we Weight for hide drive 
wrs Weight for regulate-speed drive 
wh Weight for seek drive 
l Body length 0.2 m 
 
 Fuzzification Singleton 
 Fuzzy conjunction Product 
 Fuzzy disjunction Probabilistic sum 
 Fuzzy implication Product 
 Fuzzy aggregation Probabilistic sum 
 Defuzzification Center of gravity 
Parameter
Description
Default value
Δt Time step 1 frame ( s) 
T Maximum length of one simulation 900 frames (30 s) 
vaM Maximum speed of the predator animat 22 m/s 
vam Minimum speed of the predator animat 8.8 m/s 
vpM Maximum speed of prey animats 18 m/s 
vpm Minimum speed of prey animats 7.2 m/s 
φ Animat field of view 300° 
n Number of influential nearest neighbors 
wc Weight for cohesion drive 
ws Weight for separation drive 
wa Weight for alignment drive 
we Weight for hide drive 
wrs Weight for regulate-speed drive 
wh Weight for seek drive 
l Body length 0.2 m 
 
 Fuzzification Singleton 
 Fuzzy conjunction Product 
 Fuzzy disjunction Probabilistic sum 
 Fuzzy implication Product 
 Fuzzy aggregation Probabilistic sum 
 Defuzzification Center of gravity 

Author notes

Contact author.

∗∗

Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia. E-mail: jure.demsar@fri.uni-lj.si (J.D.); ilb@fri.uni-lj.si (I.L.B.)