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Spatial structures in simulations of animal grouping
e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 468–476
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Spatial structures in simulations of animal grouping Vincent Mirabet a,∗ , Pierre Auger a,b , Christophe Lett a,c a
Laboratoire de Biom´etrie et Biologie Evolutive (UMR 5558); CNRS; Univ. Lyon 1, 43 bd 11 nov, 69622, Villeurbanne Cedex, France UR GEODES, Institut de Recherche pour le D´eveloppement, Centre de Recherche d’Ile de France, 32, avenue Henri Varagnat, 93143 Bondy Cedex, France c UR ECO-UP, Institut de Recherche pour le D´eveloppement, Centre de Recherche Halieutique M´editerran´eenne et Tropicale, rue Jean Monnet, B.P. 171, 34203 S`ete, France b
a r t i c l e
i n f o
a b s t r a c t
Article history:
We present numerical simulations of an animal grouping model based on individual
Received 23 November 2005
behaviours of attraction, alignment and repulsion. We study the consequences on the sim-
Received in revised form
ulated groups’ internal structures, of using different functions. These different functions
18 October 2006
which are adapted from the literature define the intensity, associated with these behaviours,
Accepted 20 October 2006
as a distance function between individuals. We also investigate here the impacts of: the
Published on line 14 December 2006
number of individuals, the number of influential neighbours and the strength of the alignment behaviour on the structures. We show that homogeneous groups can be identified
Keywords:
when: the different functions used lead to a smooth transition from attraction to repulsion;
Individual based model
alignment overcomes repulsion and attraction, in particular within this transition zone; and
School
when there is a low number of influential neighbours. We also point out the fact that oth-
Group
erwise, the model results in heterogeneous internal structures, which take the form of a
Aggregation
concentration of individuals in subgroups, in lines, or at the periphery of the groups.
Animal behaviour
© 2006 Elsevier B.V. All rights reserved.
Simulation
1.
Introduction
Two main approaches have been used to model the spatial and temporal dynamics of animal grouping structures like fish schools, bird flocks, insect swarms and mammal herds. The first approach, mathematics-based, which has been reviewed by Okubo (1980, 1986), uses partial differential equations (PDE) (Holmes et al., 1994). For instance, in order to estimate the dynamics of skipjack tuna, Sibert et al. (1999) and Adam and Sibert (2002) used an advection–diffusion–reaction model, based on tagging data from the western Pacific Ocean and from the Maldives, respectively. The second approach, computer-based, deals with modelling simple behaviours at the individual level and letting the grouping structures emerge from these individual-based rules. The first studies using this
∗
Corresponding author. Tel.: +33 4 72 43 12 84. E-mail address: [email protected] (V. Mirabet). 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.10.018
approach were conducted by Reynolds (1987, 1999) to simulate the dynamics of bird flocks. He developed the concept of bird-oid, or “boid”, as being the virtual counterpart to a real bird. This approach was also used to simulate the dynamics of fish schools (e.g., Huth and Wissel, 1992, 1993; Reuter and ¨ Breckling, 1994; Stocker, 1999) and herds (Gueron et al., 1996). The two approaches have been referred to as Eulerian for the ¨ former and Lagrangian for the latter (Grunbaum and Okubo, 1994) in reference to the Eulerian and Lagrangian descriptions of fluid motion. Though Eulerian and Lagrangian models are mostly used to describe animal dynamics at large and small ¨ scales, respectively, Grunbaum (1994), Flierl et al. (1999) and Adioui et al. (2003) showed that the two approaches could be complementary. There is also a large body of literature on the formation of patterns in lattice-gas (Lagrangian) models for
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collective motion (Bussemaker, 1996; Bussemaker et al., 1997; ´ and Vicsek, 1995; Czirok ´ et al., 1999; Czirok ´ and Vicsek, Csahok ´ 2000; Deutsch, 2000; Gregoire et al., 2003) in which individuals are treated as interacting particles. Conversely, Tu and Terzopoulos (1994) used a detailed model of “artificial fishes” to simulate schooling. The individual-based rules which are typically used to simulate the dynamics of fish schools are repulsion, alignment and attraction. The individuals can move in continuous space in two (e.g., Huth and Wissel, 1992; Reuter and Breckling, 1994) or three (e.g., Reynolds, 1987; Huth and Wissel, 1993) ¨ dimensions or in discrete space also in two (e.g., Stocker, ¨ 1999; Schonfisch, 2001) or three dimensions. In each time step, every individual modifies its direction according to the positions and directions of other individuals in its neighbourhood. The individual adopts either a repulsion, alignment or attraction behaviour towards a neighbour according to the distance between them, and also updates its moving direction by averaging the influences of the neighbours. The influence of a neighbour can be measured with the function of distance (e.g., Reuter and Breckling, 1994). To introduce stochasticity in this model, one may obtain the value of the new direction from a normal distribution. Niwa (1994) showed the way the strength of randomness in individual movements influences the dynamics of a whole group, which may go from a loose hardly moving group (swarm) at high randomness, to a tight highly polarized moving group (school) at low randomness. Similar effects are produced by the strength of the alignment force (Couzin et al., 2002). Not all the neighbours may be considered in the averaging procedure, but only those corresponding to certain criteria (e.g., those in front of the individual being considered, “front priority” in Huth and Wissel, 1993). In a model of herd dynamics, Gueron et al. (1996) considered that individuals also adopt a hierarchical behaviour (e.g., priority to repulsion followed by alignment, and then attraction). By using individual-based models of animal grouping, Romey (1996) and Couzin et al. (2002) could study the way differences between individuals could explain their preferred positions within a school. It has also been shown that only a small proportion of stimulated individuals were able to entrain a whole group to a particular location (Huse et al., 2002) or direction (Couzin et al., 2005). Kunz and Hemelrijk (2003) studied the effects of individuals’ size and shape on school characteristics. Different properties of the dynamics of simulated fish schools have been successfully compared with experimental data (Huth and Wissel, 1994; Viscido et al., 2004). Slight modifications of the basic models enable to simulate fish schools avoiding obstacles, feeding, swimming along a gradient (Huth and Wissel, 1993), migrating between feeding and spawning grounds (Hubbard et al., 2004) or escaping from a predator (Vabø and Nøttestad, 1997; Inada and Kawachi, 2002). In the literature, many different ways are used to define the weight associated with attraction, alignment and repulsion behaviour, as a function of distance. Warburton and Lazarus (1991), Warburton (1997) and Parrish et al. (2002) made a substantial effort in trying to synthesise the effects of using different types of attraction–alignment–repulsion (AAR) functions on the simulated group properties by calculating a series
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of indices at individual, group and population levels. Though some of these indices are connected to the positions of individuals within groups, these works, as opposite to the present one, did not intend to study directly the outcomes of using particular functions on the simulated spatial structures. In the present work, we use five types of AAR functions adapted from the literature, and study their impact on the internal spatial structures of simulated groups. We analyse the influence of different model parameters on these structures: the number of individuals, the number of influential neighbours and the strength of the alignment behaviour. We also investigate the effect, on the structures, of adding randomness in the direction or speed of individuals. Most of the results that we provide are snapshots which show the spatial arrangement of individuals within groups at a particular moment of a simulation. Those snapshots were chosen for they represent typical structures obtained across 10–20 simulations. The present work will therefore serve as a review of different sets of AAR functions, with a focus on their impact on the simulated group formations. First of all, we emphasize the different functions and parameter values that produce “unnatural” formations, like the concentrations of individuals in subgroups, along lines, or at the periphery of the groups. Then, we provide guidelines for avoiding such situations.
2.
Methods
The simulations we have conducted consider individuals as being characterized by their positions and moving directions. Individuals move in an infinite three-dimensional space. These individuals interact with their neighbours by using behavioural rules. At each timestep, their positions and directions are updated by these rules (described below). Individuals perceive the environment around them and can sense the distance between them and other individuals. They possess a spherical field of view outside which they do not perceive the environment. Within this field, they only take into account the influences of a defined number among the nearest neighbours. Here, we do not consider any dead angle that could limit the field of view of individuals. All distances are measured in arbitrary units, and individuals move at a constant speed of 5 units per timestep. All individuals are identical and move according to the same rules so as to ensure that grouping appears and maintains itself without any pre-determined leader. In order to define changes in the moving directions of individuals, we used three interaction rules named attraction, alignment and repulsion. The attraction rule reflects the tendency of individuals to limit their isolation by searching for the proximity of siblings. This rule defines a moving direction that points toward the neighbour. Conversely, the repulsion rule indicates the tendency of individuals to maintain a private zone. It defines a moving direction that points away from the neighbour, i.e., in the opposite direction to the one defined by the attraction rule. Finally, the alignment rule ensures that any individual maintains a correlated moving behaviour with its partners within the group. It consists of adjusting the moving direction to the one of the neighbour. Romey (1996) indicated that such a rule is not necessary to produce schooling. However, in our simulations, we could observe that the absence
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Fig. 1 – Different types of attraction–alignment–repulsion (AAR) functions adapted from the literature (a) Huth and Wissel (1992); (b) Reuter and Breckling (1994)—the distance range considered here is smaller than in the original publication, hence the absence of the last domain where attraction dominates; (c) Warburton (1997) Fig. 20.1a; (d) Warburton (1997) Fig. 20.1b; (e) Duncan Crombie (quadratic functions for attraction and repulsion, unpublished). These functions define the intensity (or weight) of the behaviour an individual adopts toward a neighbour depending on the distance between them. For a mathematical description of functions used, we refer to the respective papers.
of alignment led to swarming. In most situations, an individual is surrounded by more than one neighbour in its field of view and the various influences of the different neighbours are averaged. The change in moving direction was limited to 15◦ per timestep. Other realistic angles were tested that did not significantly modify the results presented here. Most models of animal grouping, found in the literature, use attraction–alignment–repulsion rules like those defined above. The intensity of the attraction, alignment or repulsion behaviour tends to depend on the distance between the individuals concerned. The types of functions used to relate the intensity to the distance vary considerably among models. The attraction–alignment–repulsion functions, considered here, are found in (or adapted from) the literature (Fig. 1). These functions reflect the tendency of the intensity of repulsion (respectively, attraction) behaviour to decrease (respectively, increase) as distances among an individual and one of its neighbours increase. Henceforth, these functions will be referred to as AARa–e functions. At the beginning of a simulation, individuals are placed within a confined place to facilitate immediate interactions between them. Within this location their positions and moving directions are initially randomly distributed. After a transitory dynamics, the model reaches a stable state in which individuals do not modify their respective positions, and the spatial structure moves in the simulation space. This situation occurs because: our model does not include stochastic processes except for the simulations performed to investigate the effect of randomness (see the corresponding part of Section 3); it also occurs because the alignment strength is sufficiently high, and individuals move at constant speeds, so that the
necessary outcome of the model is a moving stable structure ´ (similar to the “flying crystal” referred to in Gregoire et al., 2003). Most of the results presented below are snapshots which show the internal spatial structures of simulated groups. Many simulations which differ in terms of initial positions and moving directions of individuals were performed. The snapshots presented here are intended to be representative of the typical structures obtained. These snapshots were typically obtained after 2000–20,000 simulation timesteps, which represents the time necessary for the structure to reach its steady state. Our model is programmed in C language, simulations ran on a PC computer under Linux system, and the graphical output used openGL libraries. Graphic representations which use a perspective view were chosen to give a better illustration of the group structures. On these representations, background grids have been added to facilitate a three-dimensional viewing, but have no real signification as space is continuous and infinite.
3.
Results
3.1.
Influence of the AAR functions used
In this section, we compare the spatial structures obtained when we ran the model of animal grouping with each of the five types of attraction–alignment–repulsion functions (AARa–e) presented in Fig. 1. The results presented in this section were obtained with simulations involving the same number of individuals (100) and the same number of influ-
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Fig. 2 – Simulated group structures in a 3D environment using the AAR functions of Fig. 1 with 100 individuals and 100% of influential neighbours.
ential neighbours (99). The different AAR functions lead to different typical group structures, as shown on Fig. 2. Moreover, the total volume of the different structures is similar even though the distribution of individuals within this volume is different. All the models show a tendency to form spherical groups. It is also possible to observe that individuals tend to concentrate at the periphery of these groups. This phenomenon is particularly pronounced when AARa or b functions are used (Fig. 2a and b), while AARd and e lead to most homogeneous groups (Fig. 2d and e). Another tendency identified is the formation of “subgroups” of individuals that are concentrated, for example, along lines (especially with AARc functions, Fig. 2c). With AARd and AARe functions we observe the formation of a cavity at the centre of the structure. However with AARa, AARb and AARc functions we observe a concentration of individuals along lines or planes, but still at the periphery of the group.
sidered. Therefore, the figures in each line reflect a variation in the proportion of neighbours, while the whole figure can be read as a decrease in the number of neighbours (NN). In groups of 250 individuals, we observe that a large NN leads to very structured groups (Fig. 3d). In case of a lower NN, the pattern is less obvious but tight subgroups of individuals can be identified (Fig. 3e). For even lower NN, no such pattern can be observed and the simulated group is wider (Fig. 3f). The same conclusion can be drawn for smaller groups (50 individuals, Fig. 3a–c) though the patterns may be less obvious, while patterns reinforce in the case of larger groups (Fig. 3g–i). Fig. 3a and f can be directly compared, as well as Fig. 3d and i, as they correspond to the same NN with differences in terms of group sizes. As the NN increases, subgroups tend to form, but with increasing individual number. Moreover, as the NN proportion increases, the subgroups tend to form linear structures.
3.2.
3.3.
Number of individuals and number of neighbours
In order to define the moving direction change of each individual, we test here the influence of the number of neighbours (NN) taken into account by them. Fig. 3 presents the results obtained through simulations using AARc functions, which lead to the clearest structures (see Fig. 2c). We conducted simulations with three different group sizes. Then, for each size, three different proportions of influential neighbours were con-
Strength of alignment
In some models of animal grouping proposed in the literature (e.g., Huth and Wissel, 1992, see Fig. 1a), the alignment zone is separated from the attraction and repulsion zones so that in this zone only the alignment acts on the individual behaviour. In other models, the alignment acts throughout the field of view of individuals with a constant (e.g., Duncan Crombie, unpublished, see Fig. 1e) or variable (e.g., Reuter and
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Fig. 3 – Simulated group structures in a 3D environment using AARc functions (Fig. 1c) with 50 (top), 250 (centre) or 1250 (down) individuals and 100% (left), 50% (middle) and 20% (right) of influential neighbours.
Breckling, 1994, see Fig. 1b) intensity. In both cases it is possible to diminish the influence of the alignment behaviour, either by diminishing the range of the zone or by limiting the intensity. In this section, we show the results of these two operations, in the case of the AARa and AARe functions, respectively. As the range of the alignment zone or the alignment intensity is lowered, non-homogeneous structures appear in the simulated groups (Fig. 4). However, this effect is less apparent when the number of neighbours is diminished. Whatever AAR function is used, the same type of structure is obtained when the alignment influence is weak and the NN is high. AARe functions are, however, less sensitive to this parameter than AARc and AARd, for which structures appear sooner as the alignment decreases. One structure regularly obtained in this study is characterized by its composition. In this structure, individuals positioned themselves around a central circle with a tight
subgroup of individuals above and under the circle. Fig. 4c and f both show this typical structure under two different views.
3.4.
Randomness
As no perfect move can be realised in nature, many models of animal grouping have incorporated a random factor in individual’s moving direction and/or speed (Reuter and Breckling, 1994; Huth and Wissel, 1994). We realised, therefore, two sets of simulations in order to test the effect of randomness on the group structure described just above. We added random factors of different intensities to the moving direction (Fig. 5a–c) or to the speed (Fig. 5d–f). As shown in Fig. 5, it is necessary to add a large random factor to make indistinct the structure obtained without it.
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Fig. 4 – Simulated group structures in a 3D environment using AARa (top) or AARe (down) functions with 100 individuals and 100% of influential neighbours. The alignment zone is (a) 150 units, (b) 50 units or (c) 10 units wide. The alignment intensity is (d) 0.5, (e) 0.05 or (f) 0.005.
4.
Discussion
Most models of animal grouping, published in the literature, are based on three distinct individual behaviours, attraction, alignment and repulsion. In these models, attraction operates at long distances, repulsion at short distances, and alignment usually within a wider range of distances. However, there is some variability in the functions used in the models, namely to the way of linking the strength of these behaviours to the distance between individuals. Different types of functions can actually be used (Fig. 1) to simulate animal grouping across a wide range of parameter values. Indeed, when starting with random locations and moving directions of individuals, our simulations rapidly lead to polarized groups, except under very specific conditions. The different functions, when used under comparable conditions, lead to groups of similar shapes (roughly spherical in 3D) and volumes, but which differ in their distributions of individuals within them (Fig. 2). The number of neighbours, considered by the individuals to update their moving directions and the strength of the alignment behaviour, was shown to have major influences on these distributions. The simulated groups appeared to be particularly heterogeneous when the number of neighbours was high (Fig. 3) and the alignment weak (Fig. 4). These heterogeneities took mainly the form of an artificial concentration of individu-
als in subgroups, along lines, or at the periphery of the groups. Importantly, such structures have never been observed in nature, and have no plausible biological existence. Our simulations enabled to determine the conditions under which such artificial structures would not occur. The conclusion of these results is that to perform a simulation with correctly polarized and homogeneous groups, models of animal grouping should follow three different constraints. (i) Every other parameters being held constant, functions which present a smooth transition from attraction to repulsion (see Fig. 1e), show less tendency to produce structures. (ii) Strong alignment intensity comparatively to that of repulsion and attraction (in particular within the transition zone between attraction and repulsion), leads to less sensibility to other parameters (like NN) and (iii) a low number of neighbours as well. This last dependence is due to the impossibility, with low neighbours numbers, to find any non-biological (never observed) structure that diminishes the constraints linked to the model rules, the only acceptable structures being homogeneous. The most visible structures in the distribution of individuals, within our simulated groups, are only obtained for parameter values that may not be biologically plausible. To pretend that a fish, for example, can interact with any member of a big school is surely exaggerated. However, via their lateral lines, fish are influenced by individuals that are not their direct neighbours. Formations of subgroups of individ-
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Fig. 5 – Simulated group structures in a 3D environment using AARe functions, with 100 individuals, 100% of influential neighbours and an alignment intensity value of 1. A random factor of (a) −1◦ to 1◦ , (b) −50◦ to 50◦ and (c) −100◦ to 100◦ was added to moving direction. A random factor of (d) −1 to 1 unit, (e) −2 to 2 units and (f) −5 to 5 units was added to speed. uals within groups, as seen for instance in Fig. 3, will not necessarily be visually perceptible in simulations when few influential neighbours are used. However, such structures will influence the indices used to characterise grouping dynam-
ics (e.g., Parrish et al., 2002), especially those at the individual level, such as the nearest neighbour distance (NND). Since the structures obtained in our simulations revealed concentrations of individuals rather than regular spacing, NND is
Fig. 6 – Nearest neighbour distance (NND) calculated for different values of the strength of the alignment behaviour, i.e., (a) intensity of the alignment behaviour using AARe (as percentage of the weight presented on Fig. 1) and (b) size of the alignment zone using AARa. The median value for each dataset is indicated by the black centre lines. The first and third quartiles are the edges of the box, which is known as the inter-quartile range (IQR). Dotted lines indicate the extreme (minimal and maximal) values. Mean values were used to assess the effect of the strength of alignment on the simulated group structures (Fig. 4).
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expected to decrease as the structures form. This hypothesis is confirmed by the result shown on Fig. 6 for two AAR functions. NND was calculated for different values regarding the strength of the alignment behaviour, i.e., size of the alignment zone (AARa) or intensity of the alignment behaviour (AARe). While the alignment behaviour gets weaker, structures start to form in the simulated groups (Fig. 4) and NND diminishes (Fig. 6). The fact of reducing the intensity of the alignment behaviour, by using AARe functions, appear to lead to a regular logarithmic decrease of the NND value (Fig. 6a). Thus, the reduction of the size of the alignment zone, by using AARa functions (Fig. 6b), induces a slower diminution in the values of NND. We purposely considered a simple grouping model so as to focus on the implications of using specific attraction–alignment–repulsion functions on the simulated group structures. Additional processes, taken into account in former models, may diminish these structures. For example, it is accepted that introducing stochasticity in the individual movements is somehow more biologically relevant. In nature, no individual can perform perfectly similar behaviour when similarly stimulated, and as a consequence errors occur. The stochasticity introduced in the model would translate this imperfection. As far as we could check (see Fig. 5), the use of a stochastic procedure to update individual moving directions similar to that used in former works (e.g., Huth and Wissel, 1993; Inada and Kawachi, 2002; Viscido et al., 2004) did not suppress spatial structures, though they were less visible. Other processes that may suppress, diminish or modify the simulated structures, still have to be explored. In our simulations, we regularly observed lines of individuals (e.g., Fig. 3h). We believe that this is due to the definition of the repulsion and attraction forces: when individuals are distributed along lines, the repulsion force associated to neighbours on the right side (oriented to the left) is largely compensated by the one associated to neighbours on the left side (oriented to the right). The same can be claimed concerning the attraction behaviour as the orientation of the force is opposed. The symmetry existing in the way attraction and repulsion forces are defined, one pointing in the opposite direction of the other, is probably largely responsible for the formation of structures observed. For example, we could observe that the structure identified when the alignment influence is weak (Fig. 4c and f) has its origin in that symmetry. Indeed, when the attraction and repulsion behaviours dominate, each individual will tend to be positioned in a location where the two tendencies are cancelling each other. Looking at members within the tight subgroups of the structure allowed us to separate the other members of the group in three categories. The ones belonging to the same subgroups are strongly repulsive for the individual considered, the ones belonging to the other subgroup are attractive, and, finally, group members of the circle are tangentially positioned at the attraction–repulsion limit, and have therefore no influence. Another way of breaking the symmetry within the group is to add a dead angle, which is biologically relevant, especially for fish. The fact that all individuals move at the same speed also probably participates to the rapid outcome of crystal-like structures. Such structures may occur less in models that are not built according to the assumption of constant speed, like those based on a Newto-
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nian approach (e.g., Viscido et al., 2004, 2005). However, some preliminary simulations that we have conducted with such a model induced the same structures under specific conditions. Finally, though individuals tend to join groups with members sharing similar characteristics with themselves (McRobert and Bradner, 1998; Hoare et al., 2000), individuals are not identical within groups. Individuals who display different behaviours, depending on internal parameters or external stimuli, surely influence the whole group dynamics (Romey, 1996; Huse et al., 2002; Couzin et al., 2002, 2005) and structure. One can argue that it is not necessarily relevant to simulate homogeneous and highly polarized groups. For example, Viscido et al. (2004) showed that the cohesion and polarity of small fish schools in experimental tanks was low, and that alignment had to be weak for simulations to reproduce the properties of the real schools. Within large pelagic fish schools, areas of low (e.g., vacuoles) or high (e.g., nucleus) ´ densities of individuals can be found (Freon et al., 1992; Gerlotto and Paramo, 2003; Paramo et al., 2003). Our results showed that such heterogeneities may be the consequences of local interactions among neighbours. However, this is more likely to occur in simulated fish schools when the number of neighbours is high, while in real schools it is believed that vacuoles, for example, result from local interactions involving few ´ neighbours (see Fig. 14A in Freon et al., 1992). Different species tend to have distinct grouping strategies as the number of individuals increases. For example, Barange et al. (1999) suggested that sardine schools become denser, whereas anchovy schools expand, when the number of fish increases. Nøttestad et al. (1996) also reported that herring schools sizes and densities ´ change much with their activity. Freon et al. (1992) studied the way fish school shape and internal structure changes when overpassed by a vessel. We therefore intend to use models of animal grouping to study the determinism of such behaviours.
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Niwa, H.S., 1994. Self-organizing dynamic model of fish schooling. J. Theor. Biol. 171, 123–136. Nøttestad, L., Aksland, M., Beltestad, A., Fernø, A., Johannessen, A., Misund, O.A., 1996. Schooling dynamics of Norwegian spring spawning herring (Clupea harengus L.) in a coastal spawning area. Sarsia 80, 277–284. Okubo, A., 1980. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag. Okubo, A., 1986. Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv. Biophys. 22, 1–94. Paramo, J., Gerlotto, F., Oyarzun, C., 2003. Schooling in a pelagic fish: an approach in three dimensions using vertical scanning multibeam sonar. J. Acoust. Soc. Am. 114, 2300. ¨ Parrish, J.K., Viscido, S.V., Grunbaum, D., 2002. Self-organized fish schools: an examination of emergent properties. Biol. Bull. 202, 296–305. Reuter, H., Breckling, B., 1994. Self-organization of fish schools: an object-oriented model. Ecol. Model. 75/76, 147–159. Reynolds, C.W., 1987. Flocks, herds and schools: a distributed behavioral model. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques. ACM Press, New York, pp. 25–34. Reynolds, C.W., 1999. Steering behaviors for autonomous characters. In: Proceedings of the 1999 Game Developers, Conference, pp. 763–782. Romey, W.L., 1996. Individual differences make a difference in the trajectories of simulated schools of fish. Ecol. Model. 92, 65–77. ¨ Schonfisch, B., 2001. Simple individual based models of movement, alignment and schooling behaviour. Futur. Gener. Comp. Syst. 17, 873–882. Sibert, J.R., Hampton, J., Fournier, D.A., Bills, P.J., 1999. An advection-diffusion-reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (Katsuwonus pelamis). Can. J. Fish. Aquat. Sci. 56, 925–938. ¨ Stocker, S., 1999. Models for tuna school formation. Math. Biosci. 156, 167–190. Tu, X., Terzopoulos, D., 1994. Artificial fishes: physics, locomotion, perception, behavior. In: Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques. ACM Press, New York, pp. 43–50. Vabø, R., Nøttestad, L., 1997. An individual based model of fish school reactions: predicting antipredator behaviour as observed in nature. Fish. Oceanogr. 6, 155–171. ¨ Viscido, S.V., Parrish, J.K., Grunbaum, D., 2004. Individual behavior and emergent properties of fish schools: a comparison of observation and theory. Mar. Ecol. Prog. Ser. 273, 239–249. ¨ Viscido, S.V., Parrish, J.K., Grunbaum, D., 2005. The effect of population size and number of influential neighbors on the emergent properties of fish schools. Ecol. Model. 183, 347–363. Warburton, K., 1997. Social forces in animal congregations: interactive, motivational, and sensory aspects. In: Parrish, J.K., Hamner, W.M. (Eds.), Animal Groups in Three Dimensions. Cambridge University Press, pp. 313–336. Warburton, K., Lazarus, J., 1991. Tendency-distance models of social cohesion in animal groups. J. Theor. Biol. 150, 473–488.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Spatial structures in simulations of animal grouping Vincent Mirabet a,∗ , Pierre Auger a,b , Christophe Lett a,c a
Laboratoire de Biom´etrie et Biologie Evolutive (UMR 5558); CNRS; Univ. Lyon 1, 43 bd 11 nov, 69622, Villeurbanne Cedex, France UR GEODES, Institut de Recherche pour le D´eveloppement, Centre de Recherche d’Ile de France, 32, avenue Henri Varagnat, 93143 Bondy Cedex, France c UR ECO-UP, Institut de Recherche pour le D´eveloppement, Centre de Recherche Halieutique M´editerran´eenne et Tropicale, rue Jean Monnet, B.P. 171, 34203 S`ete, France b
a r t i c l e
i n f o
a b s t r a c t
Article history:
We present numerical simulations of an animal grouping model based on individual
Received 23 November 2005
behaviours of attraction, alignment and repulsion. We study the consequences on the sim-
Received in revised form
ulated groups’ internal structures, of using different functions. These different functions
18 October 2006
which are adapted from the literature define the intensity, associated with these behaviours,
Accepted 20 October 2006
as a distance function between individuals. We also investigate here the impacts of: the
Published on line 14 December 2006
number of individuals, the number of influential neighbours and the strength of the alignment behaviour on the structures. We show that homogeneous groups can be identified
Keywords:
when: the different functions used lead to a smooth transition from attraction to repulsion;
Individual based model
alignment overcomes repulsion and attraction, in particular within this transition zone; and
School
when there is a low number of influential neighbours. We also point out the fact that oth-
Group
erwise, the model results in heterogeneous internal structures, which take the form of a
Aggregation
concentration of individuals in subgroups, in lines, or at the periphery of the groups.
Animal behaviour
© 2006 Elsevier B.V. All rights reserved.
Simulation
1.
Introduction
Two main approaches have been used to model the spatial and temporal dynamics of animal grouping structures like fish schools, bird flocks, insect swarms and mammal herds. The first approach, mathematics-based, which has been reviewed by Okubo (1980, 1986), uses partial differential equations (PDE) (Holmes et al., 1994). For instance, in order to estimate the dynamics of skipjack tuna, Sibert et al. (1999) and Adam and Sibert (2002) used an advection–diffusion–reaction model, based on tagging data from the western Pacific Ocean and from the Maldives, respectively. The second approach, computer-based, deals with modelling simple behaviours at the individual level and letting the grouping structures emerge from these individual-based rules. The first studies using this
∗
Corresponding author. Tel.: +33 4 72 43 12 84. E-mail address: [email protected] (V. Mirabet). 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.10.018
approach were conducted by Reynolds (1987, 1999) to simulate the dynamics of bird flocks. He developed the concept of bird-oid, or “boid”, as being the virtual counterpart to a real bird. This approach was also used to simulate the dynamics of fish schools (e.g., Huth and Wissel, 1992, 1993; Reuter and ¨ Breckling, 1994; Stocker, 1999) and herds (Gueron et al., 1996). The two approaches have been referred to as Eulerian for the ¨ former and Lagrangian for the latter (Grunbaum and Okubo, 1994) in reference to the Eulerian and Lagrangian descriptions of fluid motion. Though Eulerian and Lagrangian models are mostly used to describe animal dynamics at large and small ¨ scales, respectively, Grunbaum (1994), Flierl et al. (1999) and Adioui et al. (2003) showed that the two approaches could be complementary. There is also a large body of literature on the formation of patterns in lattice-gas (Lagrangian) models for
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collective motion (Bussemaker, 1996; Bussemaker et al., 1997; ´ and Vicsek, 1995; Czirok ´ et al., 1999; Czirok ´ and Vicsek, Csahok ´ 2000; Deutsch, 2000; Gregoire et al., 2003) in which individuals are treated as interacting particles. Conversely, Tu and Terzopoulos (1994) used a detailed model of “artificial fishes” to simulate schooling. The individual-based rules which are typically used to simulate the dynamics of fish schools are repulsion, alignment and attraction. The individuals can move in continuous space in two (e.g., Huth and Wissel, 1992; Reuter and Breckling, 1994) or three (e.g., Reynolds, 1987; Huth and Wissel, 1993) ¨ dimensions or in discrete space also in two (e.g., Stocker, ¨ 1999; Schonfisch, 2001) or three dimensions. In each time step, every individual modifies its direction according to the positions and directions of other individuals in its neighbourhood. The individual adopts either a repulsion, alignment or attraction behaviour towards a neighbour according to the distance between them, and also updates its moving direction by averaging the influences of the neighbours. The influence of a neighbour can be measured with the function of distance (e.g., Reuter and Breckling, 1994). To introduce stochasticity in this model, one may obtain the value of the new direction from a normal distribution. Niwa (1994) showed the way the strength of randomness in individual movements influences the dynamics of a whole group, which may go from a loose hardly moving group (swarm) at high randomness, to a tight highly polarized moving group (school) at low randomness. Similar effects are produced by the strength of the alignment force (Couzin et al., 2002). Not all the neighbours may be considered in the averaging procedure, but only those corresponding to certain criteria (e.g., those in front of the individual being considered, “front priority” in Huth and Wissel, 1993). In a model of herd dynamics, Gueron et al. (1996) considered that individuals also adopt a hierarchical behaviour (e.g., priority to repulsion followed by alignment, and then attraction). By using individual-based models of animal grouping, Romey (1996) and Couzin et al. (2002) could study the way differences between individuals could explain their preferred positions within a school. It has also been shown that only a small proportion of stimulated individuals were able to entrain a whole group to a particular location (Huse et al., 2002) or direction (Couzin et al., 2005). Kunz and Hemelrijk (2003) studied the effects of individuals’ size and shape on school characteristics. Different properties of the dynamics of simulated fish schools have been successfully compared with experimental data (Huth and Wissel, 1994; Viscido et al., 2004). Slight modifications of the basic models enable to simulate fish schools avoiding obstacles, feeding, swimming along a gradient (Huth and Wissel, 1993), migrating between feeding and spawning grounds (Hubbard et al., 2004) or escaping from a predator (Vabø and Nøttestad, 1997; Inada and Kawachi, 2002). In the literature, many different ways are used to define the weight associated with attraction, alignment and repulsion behaviour, as a function of distance. Warburton and Lazarus (1991), Warburton (1997) and Parrish et al. (2002) made a substantial effort in trying to synthesise the effects of using different types of attraction–alignment–repulsion (AAR) functions on the simulated group properties by calculating a series
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of indices at individual, group and population levels. Though some of these indices are connected to the positions of individuals within groups, these works, as opposite to the present one, did not intend to study directly the outcomes of using particular functions on the simulated spatial structures. In the present work, we use five types of AAR functions adapted from the literature, and study their impact on the internal spatial structures of simulated groups. We analyse the influence of different model parameters on these structures: the number of individuals, the number of influential neighbours and the strength of the alignment behaviour. We also investigate the effect, on the structures, of adding randomness in the direction or speed of individuals. Most of the results that we provide are snapshots which show the spatial arrangement of individuals within groups at a particular moment of a simulation. Those snapshots were chosen for they represent typical structures obtained across 10–20 simulations. The present work will therefore serve as a review of different sets of AAR functions, with a focus on their impact on the simulated group formations. First of all, we emphasize the different functions and parameter values that produce “unnatural” formations, like the concentrations of individuals in subgroups, along lines, or at the periphery of the groups. Then, we provide guidelines for avoiding such situations.
2.
Methods
The simulations we have conducted consider individuals as being characterized by their positions and moving directions. Individuals move in an infinite three-dimensional space. These individuals interact with their neighbours by using behavioural rules. At each timestep, their positions and directions are updated by these rules (described below). Individuals perceive the environment around them and can sense the distance between them and other individuals. They possess a spherical field of view outside which they do not perceive the environment. Within this field, they only take into account the influences of a defined number among the nearest neighbours. Here, we do not consider any dead angle that could limit the field of view of individuals. All distances are measured in arbitrary units, and individuals move at a constant speed of 5 units per timestep. All individuals are identical and move according to the same rules so as to ensure that grouping appears and maintains itself without any pre-determined leader. In order to define changes in the moving directions of individuals, we used three interaction rules named attraction, alignment and repulsion. The attraction rule reflects the tendency of individuals to limit their isolation by searching for the proximity of siblings. This rule defines a moving direction that points toward the neighbour. Conversely, the repulsion rule indicates the tendency of individuals to maintain a private zone. It defines a moving direction that points away from the neighbour, i.e., in the opposite direction to the one defined by the attraction rule. Finally, the alignment rule ensures that any individual maintains a correlated moving behaviour with its partners within the group. It consists of adjusting the moving direction to the one of the neighbour. Romey (1996) indicated that such a rule is not necessary to produce schooling. However, in our simulations, we could observe that the absence
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Fig. 1 – Different types of attraction–alignment–repulsion (AAR) functions adapted from the literature (a) Huth and Wissel (1992); (b) Reuter and Breckling (1994)—the distance range considered here is smaller than in the original publication, hence the absence of the last domain where attraction dominates; (c) Warburton (1997) Fig. 20.1a; (d) Warburton (1997) Fig. 20.1b; (e) Duncan Crombie (quadratic functions for attraction and repulsion, unpublished). These functions define the intensity (or weight) of the behaviour an individual adopts toward a neighbour depending on the distance between them. For a mathematical description of functions used, we refer to the respective papers.
of alignment led to swarming. In most situations, an individual is surrounded by more than one neighbour in its field of view and the various influences of the different neighbours are averaged. The change in moving direction was limited to 15◦ per timestep. Other realistic angles were tested that did not significantly modify the results presented here. Most models of animal grouping, found in the literature, use attraction–alignment–repulsion rules like those defined above. The intensity of the attraction, alignment or repulsion behaviour tends to depend on the distance between the individuals concerned. The types of functions used to relate the intensity to the distance vary considerably among models. The attraction–alignment–repulsion functions, considered here, are found in (or adapted from) the literature (Fig. 1). These functions reflect the tendency of the intensity of repulsion (respectively, attraction) behaviour to decrease (respectively, increase) as distances among an individual and one of its neighbours increase. Henceforth, these functions will be referred to as AARa–e functions. At the beginning of a simulation, individuals are placed within a confined place to facilitate immediate interactions between them. Within this location their positions and moving directions are initially randomly distributed. After a transitory dynamics, the model reaches a stable state in which individuals do not modify their respective positions, and the spatial structure moves in the simulation space. This situation occurs because: our model does not include stochastic processes except for the simulations performed to investigate the effect of randomness (see the corresponding part of Section 3); it also occurs because the alignment strength is sufficiently high, and individuals move at constant speeds, so that the
necessary outcome of the model is a moving stable structure ´ (similar to the “flying crystal” referred to in Gregoire et al., 2003). Most of the results presented below are snapshots which show the internal spatial structures of simulated groups. Many simulations which differ in terms of initial positions and moving directions of individuals were performed. The snapshots presented here are intended to be representative of the typical structures obtained. These snapshots were typically obtained after 2000–20,000 simulation timesteps, which represents the time necessary for the structure to reach its steady state. Our model is programmed in C language, simulations ran on a PC computer under Linux system, and the graphical output used openGL libraries. Graphic representations which use a perspective view were chosen to give a better illustration of the group structures. On these representations, background grids have been added to facilitate a three-dimensional viewing, but have no real signification as space is continuous and infinite.
3.
Results
3.1.
Influence of the AAR functions used
In this section, we compare the spatial structures obtained when we ran the model of animal grouping with each of the five types of attraction–alignment–repulsion functions (AARa–e) presented in Fig. 1. The results presented in this section were obtained with simulations involving the same number of individuals (100) and the same number of influ-
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Fig. 2 – Simulated group structures in a 3D environment using the AAR functions of Fig. 1 with 100 individuals and 100% of influential neighbours.
ential neighbours (99). The different AAR functions lead to different typical group structures, as shown on Fig. 2. Moreover, the total volume of the different structures is similar even though the distribution of individuals within this volume is different. All the models show a tendency to form spherical groups. It is also possible to observe that individuals tend to concentrate at the periphery of these groups. This phenomenon is particularly pronounced when AARa or b functions are used (Fig. 2a and b), while AARd and e lead to most homogeneous groups (Fig. 2d and e). Another tendency identified is the formation of “subgroups” of individuals that are concentrated, for example, along lines (especially with AARc functions, Fig. 2c). With AARd and AARe functions we observe the formation of a cavity at the centre of the structure. However with AARa, AARb and AARc functions we observe a concentration of individuals along lines or planes, but still at the periphery of the group.
sidered. Therefore, the figures in each line reflect a variation in the proportion of neighbours, while the whole figure can be read as a decrease in the number of neighbours (NN). In groups of 250 individuals, we observe that a large NN leads to very structured groups (Fig. 3d). In case of a lower NN, the pattern is less obvious but tight subgroups of individuals can be identified (Fig. 3e). For even lower NN, no such pattern can be observed and the simulated group is wider (Fig. 3f). The same conclusion can be drawn for smaller groups (50 individuals, Fig. 3a–c) though the patterns may be less obvious, while patterns reinforce in the case of larger groups (Fig. 3g–i). Fig. 3a and f can be directly compared, as well as Fig. 3d and i, as they correspond to the same NN with differences in terms of group sizes. As the NN increases, subgroups tend to form, but with increasing individual number. Moreover, as the NN proportion increases, the subgroups tend to form linear structures.
3.2.
3.3.
Number of individuals and number of neighbours
In order to define the moving direction change of each individual, we test here the influence of the number of neighbours (NN) taken into account by them. Fig. 3 presents the results obtained through simulations using AARc functions, which lead to the clearest structures (see Fig. 2c). We conducted simulations with three different group sizes. Then, for each size, three different proportions of influential neighbours were con-
Strength of alignment
In some models of animal grouping proposed in the literature (e.g., Huth and Wissel, 1992, see Fig. 1a), the alignment zone is separated from the attraction and repulsion zones so that in this zone only the alignment acts on the individual behaviour. In other models, the alignment acts throughout the field of view of individuals with a constant (e.g., Duncan Crombie, unpublished, see Fig. 1e) or variable (e.g., Reuter and
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Fig. 3 – Simulated group structures in a 3D environment using AARc functions (Fig. 1c) with 50 (top), 250 (centre) or 1250 (down) individuals and 100% (left), 50% (middle) and 20% (right) of influential neighbours.
Breckling, 1994, see Fig. 1b) intensity. In both cases it is possible to diminish the influence of the alignment behaviour, either by diminishing the range of the zone or by limiting the intensity. In this section, we show the results of these two operations, in the case of the AARa and AARe functions, respectively. As the range of the alignment zone or the alignment intensity is lowered, non-homogeneous structures appear in the simulated groups (Fig. 4). However, this effect is less apparent when the number of neighbours is diminished. Whatever AAR function is used, the same type of structure is obtained when the alignment influence is weak and the NN is high. AARe functions are, however, less sensitive to this parameter than AARc and AARd, for which structures appear sooner as the alignment decreases. One structure regularly obtained in this study is characterized by its composition. In this structure, individuals positioned themselves around a central circle with a tight
subgroup of individuals above and under the circle. Fig. 4c and f both show this typical structure under two different views.
3.4.
Randomness
As no perfect move can be realised in nature, many models of animal grouping have incorporated a random factor in individual’s moving direction and/or speed (Reuter and Breckling, 1994; Huth and Wissel, 1994). We realised, therefore, two sets of simulations in order to test the effect of randomness on the group structure described just above. We added random factors of different intensities to the moving direction (Fig. 5a–c) or to the speed (Fig. 5d–f). As shown in Fig. 5, it is necessary to add a large random factor to make indistinct the structure obtained without it.
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Fig. 4 – Simulated group structures in a 3D environment using AARa (top) or AARe (down) functions with 100 individuals and 100% of influential neighbours. The alignment zone is (a) 150 units, (b) 50 units or (c) 10 units wide. The alignment intensity is (d) 0.5, (e) 0.05 or (f) 0.005.
4.
Discussion
Most models of animal grouping, published in the literature, are based on three distinct individual behaviours, attraction, alignment and repulsion. In these models, attraction operates at long distances, repulsion at short distances, and alignment usually within a wider range of distances. However, there is some variability in the functions used in the models, namely to the way of linking the strength of these behaviours to the distance between individuals. Different types of functions can actually be used (Fig. 1) to simulate animal grouping across a wide range of parameter values. Indeed, when starting with random locations and moving directions of individuals, our simulations rapidly lead to polarized groups, except under very specific conditions. The different functions, when used under comparable conditions, lead to groups of similar shapes (roughly spherical in 3D) and volumes, but which differ in their distributions of individuals within them (Fig. 2). The number of neighbours, considered by the individuals to update their moving directions and the strength of the alignment behaviour, was shown to have major influences on these distributions. The simulated groups appeared to be particularly heterogeneous when the number of neighbours was high (Fig. 3) and the alignment weak (Fig. 4). These heterogeneities took mainly the form of an artificial concentration of individu-
als in subgroups, along lines, or at the periphery of the groups. Importantly, such structures have never been observed in nature, and have no plausible biological existence. Our simulations enabled to determine the conditions under which such artificial structures would not occur. The conclusion of these results is that to perform a simulation with correctly polarized and homogeneous groups, models of animal grouping should follow three different constraints. (i) Every other parameters being held constant, functions which present a smooth transition from attraction to repulsion (see Fig. 1e), show less tendency to produce structures. (ii) Strong alignment intensity comparatively to that of repulsion and attraction (in particular within the transition zone between attraction and repulsion), leads to less sensibility to other parameters (like NN) and (iii) a low number of neighbours as well. This last dependence is due to the impossibility, with low neighbours numbers, to find any non-biological (never observed) structure that diminishes the constraints linked to the model rules, the only acceptable structures being homogeneous. The most visible structures in the distribution of individuals, within our simulated groups, are only obtained for parameter values that may not be biologically plausible. To pretend that a fish, for example, can interact with any member of a big school is surely exaggerated. However, via their lateral lines, fish are influenced by individuals that are not their direct neighbours. Formations of subgroups of individ-
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Fig. 5 – Simulated group structures in a 3D environment using AARe functions, with 100 individuals, 100% of influential neighbours and an alignment intensity value of 1. A random factor of (a) −1◦ to 1◦ , (b) −50◦ to 50◦ and (c) −100◦ to 100◦ was added to moving direction. A random factor of (d) −1 to 1 unit, (e) −2 to 2 units and (f) −5 to 5 units was added to speed. uals within groups, as seen for instance in Fig. 3, will not necessarily be visually perceptible in simulations when few influential neighbours are used. However, such structures will influence the indices used to characterise grouping dynam-
ics (e.g., Parrish et al., 2002), especially those at the individual level, such as the nearest neighbour distance (NND). Since the structures obtained in our simulations revealed concentrations of individuals rather than regular spacing, NND is
Fig. 6 – Nearest neighbour distance (NND) calculated for different values of the strength of the alignment behaviour, i.e., (a) intensity of the alignment behaviour using AARe (as percentage of the weight presented on Fig. 1) and (b) size of the alignment zone using AARa. The median value for each dataset is indicated by the black centre lines. The first and third quartiles are the edges of the box, which is known as the inter-quartile range (IQR). Dotted lines indicate the extreme (minimal and maximal) values. Mean values were used to assess the effect of the strength of alignment on the simulated group structures (Fig. 4).
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expected to decrease as the structures form. This hypothesis is confirmed by the result shown on Fig. 6 for two AAR functions. NND was calculated for different values regarding the strength of the alignment behaviour, i.e., size of the alignment zone (AARa) or intensity of the alignment behaviour (AARe). While the alignment behaviour gets weaker, structures start to form in the simulated groups (Fig. 4) and NND diminishes (Fig. 6). The fact of reducing the intensity of the alignment behaviour, by using AARe functions, appear to lead to a regular logarithmic decrease of the NND value (Fig. 6a). Thus, the reduction of the size of the alignment zone, by using AARa functions (Fig. 6b), induces a slower diminution in the values of NND. We purposely considered a simple grouping model so as to focus on the implications of using specific attraction–alignment–repulsion functions on the simulated group structures. Additional processes, taken into account in former models, may diminish these structures. For example, it is accepted that introducing stochasticity in the individual movements is somehow more biologically relevant. In nature, no individual can perform perfectly similar behaviour when similarly stimulated, and as a consequence errors occur. The stochasticity introduced in the model would translate this imperfection. As far as we could check (see Fig. 5), the use of a stochastic procedure to update individual moving directions similar to that used in former works (e.g., Huth and Wissel, 1993; Inada and Kawachi, 2002; Viscido et al., 2004) did not suppress spatial structures, though they were less visible. Other processes that may suppress, diminish or modify the simulated structures, still have to be explored. In our simulations, we regularly observed lines of individuals (e.g., Fig. 3h). We believe that this is due to the definition of the repulsion and attraction forces: when individuals are distributed along lines, the repulsion force associated to neighbours on the right side (oriented to the left) is largely compensated by the one associated to neighbours on the left side (oriented to the right). The same can be claimed concerning the attraction behaviour as the orientation of the force is opposed. The symmetry existing in the way attraction and repulsion forces are defined, one pointing in the opposite direction of the other, is probably largely responsible for the formation of structures observed. For example, we could observe that the structure identified when the alignment influence is weak (Fig. 4c and f) has its origin in that symmetry. Indeed, when the attraction and repulsion behaviours dominate, each individual will tend to be positioned in a location where the two tendencies are cancelling each other. Looking at members within the tight subgroups of the structure allowed us to separate the other members of the group in three categories. The ones belonging to the same subgroups are strongly repulsive for the individual considered, the ones belonging to the other subgroup are attractive, and, finally, group members of the circle are tangentially positioned at the attraction–repulsion limit, and have therefore no influence. Another way of breaking the symmetry within the group is to add a dead angle, which is biologically relevant, especially for fish. The fact that all individuals move at the same speed also probably participates to the rapid outcome of crystal-like structures. Such structures may occur less in models that are not built according to the assumption of constant speed, like those based on a Newto-
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nian approach (e.g., Viscido et al., 2004, 2005). However, some preliminary simulations that we have conducted with such a model induced the same structures under specific conditions. Finally, though individuals tend to join groups with members sharing similar characteristics with themselves (McRobert and Bradner, 1998; Hoare et al., 2000), individuals are not identical within groups. Individuals who display different behaviours, depending on internal parameters or external stimuli, surely influence the whole group dynamics (Romey, 1996; Huse et al., 2002; Couzin et al., 2002, 2005) and structure. One can argue that it is not necessarily relevant to simulate homogeneous and highly polarized groups. For example, Viscido et al. (2004) showed that the cohesion and polarity of small fish schools in experimental tanks was low, and that alignment had to be weak for simulations to reproduce the properties of the real schools. Within large pelagic fish schools, areas of low (e.g., vacuoles) or high (e.g., nucleus) ´ densities of individuals can be found (Freon et al., 1992; Gerlotto and Paramo, 2003; Paramo et al., 2003). Our results showed that such heterogeneities may be the consequences of local interactions among neighbours. However, this is more likely to occur in simulated fish schools when the number of neighbours is high, while in real schools it is believed that vacuoles, for example, result from local interactions involving few ´ neighbours (see Fig. 14A in Freon et al., 1992). Different species tend to have distinct grouping strategies as the number of individuals increases. For example, Barange et al. (1999) suggested that sardine schools become denser, whereas anchovy schools expand, when the number of fish increases. Nøttestad et al. (1996) also reported that herring schools sizes and densities ´ change much with their activity. Freon et al. (1992) studied the way fish school shape and internal structure changes when overpassed by a vessel. We therefore intend to use models of animal grouping to study the determinism of such behaviours.
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