Asynchronous Evolution: Emergence of Signal-Based Swarming

Olaf Witkowski and Takashi Ikegami

University of Tokyo, Japanolaf@sacral.c.u-tokyo.ac.jp

Abstract

Since Reynolds boids, swarming behavior has often been re-produced in artificial models, but the conditions leading toits emergence are still subject to research, with candidatesranging from obstacle avoidance to virtual leaders. In thispaper, we present a multi-agent model in which individualsdevelop swarming using only their ability to listen to eachothers signals. Our model uses an original asynchronous ge-netic algorithm to evolve a population of agents controlled byartificial neural networks, looking for an invisible resourcein a 3D environment. The results demonstrate that agentsuse the information exchanged between them via signalingto form temporary leader-follower relations allowing them toflock together.

IntroductionThe ability of fish schools, insect swarms or starling mur-murations (Figure 1) to shift shape as one and coordinatetheir motion in space has been studied extensively becauseof their implications for the evolution of social cognition,collective animal behavior and artificial life (Couzin 2009).

Figure 1: Starling murmuration1

Swarming is the phenomenon in which a large number ofindividuals organize into a coordinated motion. Using onlythe information at their disposition in the environment, theyare able to aggregate together, move en masse or migrate

towards a common direction.The movement itself may differ from species to species.

For example, fish and insects swarm in three dimensions,whereas herds of sheep move only in two dimensions. More-over, the collective motion can have quite diverse dynamics.While birds migrate in relatively ordered formations withconstant velocity, fish schools change directions by align-ing rapidly and keeping their distances, and insects swarmsmove in a messy and random-looking way (Budrene et al.1991, Czirok et al. 1997, Shimoyama et al. 1996).

Numerous evolutionary hypotheses have been proposedto explain swarming behavior across species. These includemore efficient mating, good environment for learning, com-bined search for food resources, and reducing risks of pre-dation (Zaera et al., 1996). Partridge and Pitcher (1979) alsomention energy saving in fish schools by reducing drag.

In an effort to test the multiple theories, the past decadescounted several experiments involving real animals, eitherinside an experimental setup (Partridge, 1982; Balleriniet al., 2008) or observed in their own ecological environ-ment (Parrish and Edelstein-Keshet, 1999). Those experi-ments present the inconvenience to be costly to reproduce.Furthermore, the colossal lapse of evolutionary time neededto evolve swarming, make it almost impossible to study theemergence of such behavior experimentally.

Computer modeling has recently provided researcherswith new, easier ways to test hypotheses on collective be-havior. As it is well known, simulating individuals on ma-chines offers easy modification of setup conditions and pa-rameters, tremendous data generation, full reproducibility ofevery experiment, and easier identification of the underlyingdynamics of complex phenomena.

From Reynolds’ boids to recent approachesIn a massively cited paper, Craig Reynolds (1987) intro-duces the boids model simulating 3D swarming of agentscalled boids controlled only by three simple rules:

• Alignment: move in the same direction as neighbours1Copyright Walter Baxter and licensed for reuse under this Cre-

ative Commons Licence.

• Cohesion: Remain close to neighbours

• Separation: Avoid collisions with neighbours

Various works have since then reproduced swarming be-havior, often by the means of an explicitly coded set of rules.For instance, Mataric (1992) proposes a generalization ofReynolds’ original model with an optimally weighted com-bination of six basic interaction primitives2. Hartman &Benes (2006) come up with yet another variant of the origi-nal model, by adding a complementary force to the align-ment rule, that they call change of leadership. Unfortu-nately, in spite of the insight this kind of approach bringsinto the dynamics of swarming, it shows little about the pres-sures leading to its emergence. Many other approaches arebased on informed agents or fixed leaders (Cucker & Huepe2008, Su et al. 2009, Yu et al. 2010).

For that reason, experimenters attempted to simulateswarming without a fixed set of rules, rather by incorporat-ing into each agent an artificial neural network brain thatcontrols its movements. The swarming behavior is evolvedby copy with mutations of the chromosomes encoding theneural network parameters. By comparing the impact of dif-ferent selective pressures, this type of methodology eventu-ally allows to analyze the evolutionary emergence of swarm-ing.

Tu and Terzopoulos (1994) have swarming emergefrom the application of artificial pressures consisting ofhunger, libido and fear. Other experimenters have studiedprey/predator systems to show the importance of sensorysystem and predator confusion in the evolution of swarm-ing in preys (Ward et al. 2001, Olson et al. 2013).

In spite of many pressures hypothesized to produceswarming behavior, designed setups presented in the liter-ature are often complex and specific. Previous works typ-ically introduce models with very specific environments,where agents are given specialized sensors sensitive. Whilethey are bringing valuable results to the community, one maywonder about systems with a simpler design.

In addition, even when studies focus on fish or insects thatswarm in 3D (Ward et al. 2001) most keep their model in2D. While the swarming can be considered to be similar inmost cases, the mapping from 2D to 3D is found to be non-trivial (Sayama 2012). Indeed, the addition of a third degreeof freedom may enable agents to produce significantly dis-tinct and more complex behaviors.

Signaling agents in a resource finding taskThis paper studies the emergence of swarming in a popula-tion of agents using a basic signaling system, while perform-ing a simple resource gathering task.

2Namely, those primitives are collision avoidance, following,dispersion, aggregation, homing and flocking.

Simulated agents move around in a three dimensionalspace, looking for a vital but invisible food resource ran-domly distributed in the environment. The agents are emit-ting signals that can be perceived by other individuals’ sen-sors within a certain radius. Both agent’s motion and signal-ing are controlled by an artificial neural network embeddedin each agent, evolved over time by an asynchronous geneticalgorithm. Agents that consume enough food are enabled toreproduce, whereas those whose energy drops to zero areremoved from the simulation.

During the first phase, we observe that agents progres-sively coordinate into clustered formations, which are pre-served through the second phase. Such patterns do not ap-pear in control experiments having the simulation start di-rectly from the second phase, with the absence of resourcelocations. If at any point the signaling is switched off,the agents immediately stop swarming together. They startswarming again as soon as the communication is turnedback on. Furthermore, it is observed that simulationswith signaling lead to agents gathering very closely aroundfood patches, whereas the control simulations with silentedagents end up with them wandering around erratically.

The main contribution of this work is to demonstrate thatcollective motion can originate, without explicit central co-ordination, from the combination of a generic communica-tion system and a simple resource gathering task. A spe-cific genetic algorithm with an asynchronous reproductionscheme is developed and used to evolve the agents’ neuralcontrollers. In addition, the search for resource is shownto improve from the agents clustering, eventually leadingto the agents gathering closely around goal areas. An in-depth analysis shows increasing information transfer be-tween agents throughout the learning phase, and the devel-opment of leader/follower relations that eventually push theagents to organize into clustered formations.

The rest of the paper is organized as follows. The nextsection describes the details of our model. Then simulationsettings and results are discussed, before finally drawing aconclusion in the last section.

ModelAgents in a 3D worldWe simulate a group of agents moving around in a cubic,toroidal arena of 600×600×600. The agents rely on energyto survive. If at any point an agent’s energy drops to zero, itis immediately removed from the environment. The task forthe agents is to get as close as possible to a preset resourcespot. By getting close to one of those spots, agents can gainmore energy, allowing them to counterbalance the energylosses due to movement and signaling. An agent whose en-ergy drops to zero is removed from the simulation. In thisregard, the energy also represents each agent’s fitness, andin this paper both terms are used interchangeably.

The agent’s position is determined by three floating pointcoordinates between 0.0 and 600.0. Each agent is positionedrandomly at the start of the simulation, and then moves at afixed speed of 1 unit per iteration. The direction of motion isdecided by two motors controlling Euler angles ψ for pitch(i.e. elevation) and θ for yaw (i.e. heading).

Communication among agentsEvery agent is also provided with one communication ac-tuator capable of sending signals with intensities (signalsare encoded as floating point values ranging from 0.0 to1.0), and six communication sensors allowing it to de-tect signals produced by other agents up to a distanceof 100 from 6 directions, namely frontal (0, 1, 0), rear(0,−1, 0), left (1, 0, 0), right (−1, 0, 0), top (0, 0, 1) andbottom (0, 0,−1)). The communication sensors are imple-mented so that every source point in a 100-radius spherearound the agent is linked to one and only one of its sen-sors. The sensor whose direction is the closest to the sig-naling source receives one float value, equal to the sum ofevery signal emitted within range, divided by the distance,and normalized between 0.0 and 1.0.

A neural network inside each agentThe agent’s neural controller is implemented by an Elmanartificial neural network with 6 input neurons, encoding theactivation states of the corresponding 6 sensors, fully con-nected through a 10-neuron hidden layer to 3 output neu-rons controlling the two motors and the communication sig-nal emitted by the agent. The hidden layer is given a formof memory feedback from a 10-neuron context layer, con-taining the values of the hidden layer from the previous timestep.

All nodes in the neural network take values between 0.0and 1.0. All output values are also floating values between0.0 and 1.0, the motor outputs are then converted to anglesbetween −π to π. The activation state of internal neurons isupdated according to a sigmoid function.

An asynchronous reproduction schemeOur model differs from the usual genetic algorithmparadigm, in that it designs variation and selection in anasynchronous way. The reproduction takes place continu-ously throughout the simulation, creating overlapping gen-erations of agents. This allows for a more natural, continu-ous model, as no global clock is defined, that could bias orweaken the model.

Every new agent is born with an energy equal to 2.0. Inthe course of the simulation, each agent can gain or lose avariable amount of energy. At iteration t, the fitness func-tion fi for agent i is defined by fi(t) = r

di(t)where r

is the reward value and di is the agent’s distance to the

Figure 2: Architecture of the agent’s controller, composed of6 input neurons (I1 to I6) , 10 hidden neurons (H1 to H10), 10 context neurons (C1 to C10) and 3 output neurons (O1

to O3) .

goal. The reward value is controlled by the simulation suchthat the population remains between 100 and 500 agents.All the way through the simulation, the agents also spenda fixed amount of energy for movement (0.01 per itera-tion) and a variable amount of energy for signaling costs(0.001× signal intensity per iteration).

The weights of every connection in the neural network(apart from the links from hidden to context nodes, whichhave fixed weights) are encoded in a genotype and evolvedthrough generations of agents. Each weight is representedby a unique floating point value in the genotype vector, suchthat the size of the vector is simply equal to the total numberof connections in the neural network. The simulation uses agenetic algorithm with overlapping generations to evolve theweights of the neural networks. Whenever an agent accumu-lates 10.0 in energy, a replica of itself (with a 5% mutationin the genotype) is created and added to a random positionin the arena. The agent’s energy is decreased by 8.0 and thenew replica’s energy is set to 2.0. The choice for randominitial positions is to avoid biasing the proximity of agents,so that reproduction does not become a way for agents tocreate local clusters.

Indeed, a local reproduction scheme (i.e. giving birth tooffspring close to their parents), leads rapidly to an explo-sion in population size, as the agents that are close to theresource create many offspring that will be very fit too, thusable to replicate very fast as well. This is why the newbornoffspring is placed randomly in the environment.

For our genetic algorithm to be effective, the number ofagents must be maintained above a certain level. Also, the

computation power limits the population size. The fitnessallowed to the agents is therefore adjusted in order to main-tain an acceptable number of agents (between 100 and 1000)alive throughout the simulation. In addition, agents above acertain age (5000 time steps) are removed from the simula-tion, to keep the evolution from standing still.

ResultsEmergence of swarmingWe observe agents coordinate together in clustered groups.As shown in Figure 3 (top) the simulation goes through threedistinct phases. In the first one, agents wander in an appar-ently random way across the space. Then, during the secondphase, agents progressively cluster into a rapidly changingshape, reminiscent of animal flocks3. In the third phase, to-wards the end of the simulation, the flocks get closer andcloser to the goal4, forming a compact ball around it.

Figure 3: Visualization of the development of a typical runwith a single resource spot. The agents start off in a ran-dom motion (left), then progressively come to coordinate ina dynamic cluster (middle), and finally flock more and moreclosely to the goal (right).

Figure 4 shows more in detail the swarming behavior tak-ing place in the second phase. The agents coordinate in a dy-namic, quickly changing shape, continuously extending andcompressing, while each individual is executing fast pacedrotations on itself. Note that this fast looping seems to benecessary to the emergence of swarming, as all trials withslower rotation settings never achieved this kind of dynam-ics. One regularly notices some agents reaching the borderof a swarm cluster, leaving the group, and most of the timeending up coming back in the heart of the swarm.

In spite of the agents needing to pay a cost for signaling(cf. description of the model), the signal keeps an averagevalue between 0.2 and 0.5 during the whole experiment (inthe case with signalling activated).

It is also noted that a minimal rotation speed is necessaryfor the evolution of swarming. Indeed, it allows the agent to

3As mentioned in the introduction, swarming can take multipleforms depending on the situation and/or the species. In this case,the clustering resemble in some aspects mosquito or starling flock-ing.

4Even though results with one goal are presented in the paper,same behaviors are obtained in the case of two or more resourcespots.

Figure 4: Visualization of the swarming behavior occurringin the second phase of the simulation.

react faster to the environment, as each turn making one sen-sor face a particular direction allows a reaction to the signalscoming from that direction. The faster the rotation, the morethe information gathered by the agent about its environmentis balanced for every direction.

Neighborhood analysisWe choose to measure swarming behavior in agents by look-ing at the average number of neighbors within a radius of100 distance around each agent. Figure 5 shows the evolu-tion of the average number of neighbors, over 10 differentruns, respectively with signaling turned on and off. A muchhigher value is reached around time step 105 in the signalingcase, while the value remains for the silent control.

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Figure 5: Comparison of the average number of neighbors(average over 10 runs, with 106 iterations) in the case sig-naling is turned on versus off

We also want to measure the influence of each agent on itsneighborhood. To do so, the inward average transfer entropy

on agent’s velocities is calculated5 between each neighborwithin a distance of 100 and the agent itself. We will re-fer to this measure as inward neighborhood transfer entropy(NTE). This can be considered a measure of how much theagents are “following” their neighborhood at a given timestep. The values rapidly take off on the regular simulation(with signaling switched on), while they remain low for thesilent control, as we can see for example on Figure 6.

Figure 6: Plot of the average inward neighborhood transferentropy for signaling on (red curve) and off (blue curve)

Similarly, we can calculate the outward neighborhoodtransfer entropy (i.e. the average transfer entropy from anagent to its neighbors). We may look at the evolution ofthis value through the simulation, in an attempt to capturethe apparition of local leaders in the swarm clusters. Eventhough the notion of leadership is hard to define, the study ofthe flow of information is essential in the study of swarms.The single individuals’ outward NTE shows a succession ofbursts coming everytime from different agents, as illustratedon Figure 7. This frequent switching of the origin of in-formation flow can be interpreted as a continual change ofleadership in the swarm. The agents tend to follow a smallnumber of agents, but this subset of leaders is not fixed overtime.

On the upper graph in Figure 8, between iteration 105 and2× 105, we see the average distance to the goal drop to val-ues oscillating between roughly 50 and 300, that is the bestagents reach 50 units away from the goal, while other agentsremain about 300 units away. On the control experimentgraph (Figure 8, bottom), we observe that the distance to thegoal remains around 400.

Swarming, allowed by the signaling behavior, allowsagents to stick close to each other. That ability allows fora winning strategy in the case when some agents already aresuccessful at remaining close to a resource area. Swarmingmay also help agents find goals in the fact that they consti-tute an efficient searching pattern. Whilst an agent alone issubject to basic dynamics making it drift away, a bunch of

5The calculations are analogous to Wibral et al. (2013).

Figure 7: Plot of the individual outward neighborhood trans-fer entropy. Each color corresponds to a distinct agent.

agents is more able to stick to a goal area once it finds it,since natural selection will increase the density of surviv-ing agents around a those areas. In the control experimentswithout signaling, it is observed that the agents, unable toform swarms, do not manage to gather around the goal inthe same way as when the signaling is active.

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Figure 8: Average distance of agents to the goal with sig-naling (top) and a control run with signaling switched off(bottom)

Controller responseAfter simulation, we test neural networks of each swarm-ing agent and qualitatively compare them to non-swarmingones’. We observed that characteristic shapes for the curveobtained with swarming agents presented a similarity (seeFigure 9, top), and differed from the patterns of non-swarming agents (see Figure 9, bottom) which were alsomore diverse. In swarming individuals’ neural networks,

patterns were observed leading to higher motor output re-sponses in the case of higher signal inputs. This is char-acteristic almost every swarming individuals, whereas non-swarming agents present a wide range of response func-tions. A higher motor response may allow the agent to slowdown its course accross the map by executing quick rota-tions around itself, therefore keeping its position nearly un-changed. If this behavior is adopted in the case where thesignal is high, that is in presence of signaling agents, theagent is able to remain close to them.

Figure 9: Plots of evolved agents’ motor responses to a rangeof value in input and context neurons. The three axes repre-sent signal input average values (right horizontal axis), con-text unit average level (left horizontal axis), and average mo-tor responses (vertical axis). The top two graphs correspondto the neural controllers of swarming agents, and the bottomones correspond to non-swarming ones’.

Phylogeny

To study the heterogeneity of the population, the phyloge-netic tree is visualized (Figure 10). At the center of the graphis the root of the tree, which corresponds to time zero of thesimulation, from which start the 200 initial branches. Asthose branches progress outward, they create ramificationsthat represent the descendance of each agent. The time stepscale is preserved, and the segment drawn below serves asa reference for 105 iterations. Every fork corresponds to anewborn agent6. Therefore, every “fork burst” correspondsto a period of high fitness for the concerned agents.

On Figure 11, one can observe another phylogenetic tree,represented horizontally in order to compare it to the average

6The parent forks counterclockwise, and the newborn forksclockwise.

number of neighbors throughout the simulation. The neigh-borhood becomes denser around iteration 400k, showing ahigher portion of swarming agents. This leads to a firstlystrong selection of the agents able to swarm together overthe other individuals, a selection that is soon relaxed due tothe signaling pattern being largely spread, resulting in a het-erogeneous population, as we can see on the upper plot, withnumerous branches towards the end of the simulation.

The phylogenetic tree shows some heterogeneity, and theaverage number of neighbors is a measure of swarming inthe population. The swarming takes off around iteration400k, where there seems to be a genetic drift, but the sig-naling helps agents form and keep swarming.

Figure 10: Phylogenetic tree of agents created during a run.The center corresponds to the start of the simulation. Eachbranch represents an agent, and every fork corresponds to areproduction process.

DiscussionIn our simulation, agents progressively evolve the ability toflock through communication to perform a foraging task.We observe a dynamical swarming behavior, including cou-pling/decoupling phases between agents, allowed by theonly interaction at their disposal, that is signaling. Eventu-ally, agents come to react to their neighbors’ signals, whichis the only information they can use to improve their for-aging. This can lead them to either head towards or moveaway from each other. While moving away from each otherhas no special effect, moving towards each other, on the con-trary, leads to swarming. Flocking with each other may leadagents to slow down their pace, which for some of them maykeep them closer to a food resource. This creates a benefi-cial feedback loop, since the fitness brought to the agentswill allow them to reproduce faster, and eventually multiplythis type of behavior within the total population.

Figure 11: Top: average number of neighbors during a singlerun. Bottom: agents phylogeny for the same run. The rootsare on the left, and each bifurcation represents a newbornagent.

In this scenario, agents do not need extremely complexlearning to swarm and eventually get more easily to theresource, but rather rely on dynamics emerging from theircommunication system to create inertia and remain close togoal areas.

It should be noted that the simulated population has strongheterogeneity due to the asynchronous reproduction schema,which can be visualized in the phylogenetic tree (Figure 10).Such heterogeneity may suppress swarming but the evolvedsignaling helps the population to form and keep swarming.The simulations do not exhibit strong selection pressuresto adopt specific behavior apart from the use of the signal-ing. Without high homogeneity in the population, the signal-ing alone allows for interaction dynamics sufficient to formswarms, which proves in turn to be beneficial to get extrafitness as it has been mentioned above.

The results presented in this paper can be compared tomany works in the literature. Ward et al. (2001) and Olsonet al. (2013) also show the emergence of swarming withoutexplicit fitness, though those are based on a predator-preymodel. The type of swarming obtained with simple pres-sures is usually similar to the one obtained in this study, thatpresents the advantage of being based on a very simple sys-tem based on resource finding and signaling/sensing. Mod-els such as Blondel et al. (2005), Cucker & Huepe (2008)and Su et al. (2009) achieve swarming behavior based on ex-plicit exchange of information from leaders. Our simulationimproves on this kind of research in the sense that agentsnaturally switch leadership and followership by exchanginginformation over a very limited channel of communication.

Finally, our results also show the advantage of swarming forresource finding (it’s only only through swarming, enabledby signaling behavior, that agents are able to reach and re-main around the goal areas), comparable to the advantagesof particle swarm optimizations (Kennedy et al. 1995), hereemerging in a model with a simplistic set of conditions.

ConclusionIn this work we have shown that swarming behavior canemerge from a communication system in a resource gather-ing task. We implemented a three-dimensional agent-basedmodel with an asynchronous evolution through mutation andselection. The results show that from decentralized leader-follower interactions, a population of agents can evolve col-lective motion, in turn improving its fitness by reaching in-visible target areas.

Our results represent an improvement on models usinghard-coded rules to simulate swarming behavior, as they areevolved from very simple conditions. Our model also doesnot rely on any explicit information from leaders, like previ-ously used in part of the literature (Cucker & Huepe 2008,Su et al. 2009). It does not impose any explicit leader-follower relationship beforehand, letting simply the leader-follower dynamics emerge and self-organize. In spite of be-ing theoretical, the swarming model presented in this pa-per offers a simple, general approach to the emergence ofswarming behavior once approached via the boids rules.

ReferencesBallerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E.,

Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini,A., Viale, M., and Zdravkovic, V. (2008). Interaction rulinganimal collective behavior depends on topological rather thanmetric distance: Evidence from a field study. Proceedings ofthe National Academy of Sciences, 105(4):1232–1237.

Blondel, V., Hendrickx, J. M., Olshevsky, A., and Tsitsiklis, J.(2005). Convergence in multiagent coordination, consensus,and flocking. In IEEE Conference on Decision and Control,volume 44, page 2996. IEEE; 1998.

Budrene, E. O., Berg, H. C., et al. (1991). Complex patterns formedby motile cells of escherichia coli. Nature, 349(6310):630–633.

Couzin, I. D. (2009). Collective cognition in animal groups. Trendsin cognitive sciences, 13(1):36–43.

Cucker, F. and Huepe, C. (2008). Flocking with informed agents.Mathematics in Action, 1(1):1–25.

Czirok, A., Barabasi, A.-L., and Vicsek, T. (1997). Collective mo-tion of self-propelled particles: Kinetic phase transition inone dimension. arXiv preprint cond-mat/9712154.

Huth, A. and Wissel, C. (1992). The simulation of the movement offish schools. Journal of theoretical biology, 156(3):365–385.

Kennedy, J., Eberhart, R., et al. (1995). Particle swarm optimiza-tion. In Proceedings of IEEE international conference on

neural networks, volume 4, pages 1942–1948. Perth, Aus-tralia.

Mataric, M. J. (1992). Integration of representation into goal-driven behavior-based robots. Robotics and Automation,IEEE Transactions on, 8(3):304–312.

Olson, R. S., Hintze, A., Dyer, F. C., Knoester, D. B., andAdami, C. (2013). Predator confusion is sufficient to evolveswarming behaviour. Journal of The Royal Society Interface,10(85):20130305.

Parrish, J. K. and Edelstein-Keshet, L. (1999). Complexity, pattern,and evolutionary trade-offs in animal aggregation. Science,284(5411):99–101.

Partridge, B. L. (1982). The structure and function of fish schools.Scientific american, 246(6):114–123.

Reynolds, C. W. (1987). Flocks, herds and schools: A distributedbehavioral model. In ACM SIGGRAPH Computer Graphics,volume 21, pages 25–34. ACM.

Sayama, H. (2012). Morphologies of self-organizing swarms in 3dswarm chemistry. In Proceedings of the fourteenth interna-tional conference on Genetic and evolutionary computationconference, pages 577–584. ACM.

Shimoyama, N., Sugawara, K., Mizuguchi, T., Hayakawa, Y., andSano, M. (1996). Collective motion in a system of motileelements. Physical Review Letters, 76(20):3870.

Su, H., Wang, X., and Lin, Z. (2009). Flocking of multi-agentswith a virtual leader. Automatic Control, IEEE Transactionson, 54(2):293–307.

Tu, X. and Terzopoulos, D. (1994). Artificial fishes: Physics, loco-motion, perception, behavior. In Proceedings of the 21st an-nual conference on Computer graphics and interactive tech-niques, pages 43–50. ACM.

Ward, C. R., Gobet, F., and Kendall, G. (2001). Evolving collectivebehavior in an artificial ecology. Artificial life, 7(2):191–209.

Wibral, M., Pampu, N., Priesemann, V., Siebenhuhner, F., Seiwert,H., Lindner, M., Lizier, J. T., and Vicente, R. (2013). Mea-suring information-transfer delays. PloS one, 8(2):e55809.

Yu, W., Chen, G., and Cao, M. (2010). Distributed leader–follower flocking control for multi-agent dynamical systemswith time-varying velocities. Systems & Control Letters,59(9):543–552.

Zaera, N., Cliff, D., and Janet, B. (1996). Not) evolving collectivebehaviours in synthetic fish. In Proceedings of InternationalConference on the Simulation of Adaptive Behavior. Citeseer.