Editing Swarm behaviour (section)

==Models==
{{See also|Collective animal behaviour}}

In recent decades, scientists have turned to modeling swarm behaviour to gain a deeper understanding of the behaviour.

===Mathematical models===
[[File:Metric vs topological distance in schools of fish.png|right|thumb|400px|In the [[metric distance]] model of a [[fish school]] (left), the focal fish (yellow) pays attention to all fish within the small zone of repulsion (red), the zone of alignment (lighter red) and the larger zone of attraction (lightest red). In the [[topological space|topological distance]] model (right), the focal fish only pays attention to the six or seven closest fish (green), regardless of their distance.]]
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|image1=[http://www.red3d.com/cwr/boids/ Boids simulation]
|image2=[http://www.aridolan.com/ofiles/iFloys.html iFloys simulation]
|image3=[http://www.aridolan.com/ofiles/Efloys.aspx Efloys simulation]}}
Early studies of swarm behaviour employed mathematical models to simulate and understand the behaviour. The simplest mathematical models of animal swarms generally represent individual animals as following three rules:
* Move in the same direction as their neighbours
* Remain close to their neighbours
* Avoid collisions with their neighbours

The [[boids]] computer program, created by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, simulates swarm behaviour following the above rules.<ref name="Reynolds"/> Many subsequent and current models use variations on these rules, often implementing them by means of concentric "zones" around each animal. In the "zone of repulsion", very close to the animal, the focal animal will seek to distance itself from its neighbours to avoid collision. Slightly further away, in the "zone of alignment", the focal animal will seek to align its direction of motion with its neighbours. In the outermost "zone of attraction", which extends as far away from the focal animal as it is able to sense, the focal animal will seek to move towards a neighbour.

The shape of these zones will necessarily be affected by the sensory capabilities of a given animal. For example, the visual field of a bird does not extend behind its body. Fish rely on both vision and on [[hydrodynamic]] perceptions relayed through their [[lateral line]]s, while Antarctic [[krill]] rely both on vision and hydrodynamic signals relayed through [[antenna (biology)|antennae]].

However recent studies of starling flocks have shown that each bird modifies its position, relative to the six or seven animals directly surrounding it, no matter how close or how far away those animals are.<ref name="Ballerini et al">{{cite journal |doi=10.1073/pnas.0711437105 |vauthors=Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Lecomte V, Orlandi A, Parisi G, Procaccini A, Viale M, Zdravkovic V |title= Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=105 |issue=4 |pages=1232–7 |year=2008 |pmid=18227508 |pmc=2234121|arxiv= 0709.1916 |bibcode= 2008PNAS..105.1232B|doi-access=free }}</ref> Interactions between flocking starlings are thus based on a [[topological]], rather than a metric, rule. It remains to be seen whether this applies to other animals. Another recent study, based on an analysis of high-speed camera footage of flocks above Rome and assuming minimal behavioural rules, has convincingly simulated a number of aspects of flock behaviour.<ref>{{cite journal |vauthors=Hildenbrandt H, Carere C, Hemelrijk CK |year= 2010 |title= Self-organized aerial displays of thousands of starlings: a model |journal= Behavioral Ecology |volume= 21 |issue= 6 |pages= 1349–1359 |doi= 10.1093/beheco/arq149|doi-access= free |arxiv= 0908.2677 }}</ref><ref>{{cite journal |vauthors=Hemelrijk CK, Hildenbrandt H |year= 2011 |title= Some causes of the variable shape of flocks of birds |journal= PLOS ONE |volume= 6 |issue= 8 |page= e22479 |doi= 10.1371/journal.pone.0022479 |pmid=21829627 |pmc=3150374 |bibcode=2011PLoSO...622479H|doi-access= free }}</ref><ref>{{cite web |url=http://www.rug.nl/sciencelinx/exhibits/swarming/index|title=Zwermen en scholen - Swarming - Permanente expo - Bezoek onze expo's & workshops! - Science LinX - Rijksuniversiteit Groningen|date=10 November 2007}}</ref><ref>{{cite web |url=http://www.rug.nl/fmns-research/beso/_people/hemelrijk|title=Onderzoek aan de Faculteit Wiskunde en Natuurwetenschappen - Faculteit Wiskunde en Natuurwetenschappen - Over ons - Rijksuniversiteit Groningen|date=25 October 2012}}</ref>

===Evolutionary models===
In order to gain insight into why animals evolve swarming behaviours, scientists have turned to evolutionary models that simulate populations of evolving animals. Typically these studies use a [[genetic algorithm]] to simulate [[evolution]] over many generations. These studies have investigated a number of hypotheses attempting to explain why animals evolve swarming behaviours, such as the [[selfish herd theory]]<ref>{{cite journal |last1=Yang |first1=W. |last2=Schmickl |first2=T. |date=2019 |title=Collective Motion as an Ultimate Effect in Crowded Selfish Herds |journal=Scientific Reports |volume=9 |issue=1 |pages=6618 | pmid=31036873 | doi=10.1038/s41598-019-43179-6|pmc=6488663 |bibcode=2019NatSR...9.6618Y | doi-access=free}}</ref><ref>{{cite book |vauthors=Olson RS, Knoester DB, Adami C |title= Proceedings of the 15th annual conference on Genetic and evolutionary computation |chapter= Critical interplay between density-dependent predation and evolution of the selfish herd |year= 2013 |chapter-url= https://dl.acm.org/citation.cfm?doid=2463372.2463394 |pages= 247–254 |doi= 10.1145/2463372.2463394 |isbn= 9781450319638 |series= Gecco '13|s2cid= 14414033 }}</ref><ref>{{cite journal |vauthors=Ward CR, Gobet F, Kendall G |year= 2001 |title= Evolving collective behavior in an artificial ecology |url= https://dl.acm.org/citation.cfm?id=569757 |journal= Artificial Life |volume= 7 |issue= 2 |pages= 191–209 |doi= 10.1162/106454601753139005 |pmid=11580880 |citeseerx= 10.1.1.108.3956|s2cid= 12133884 }}</ref><ref>{{cite journal |vauthors=Reluga TC, Viscido S |year= 2005 |title= Simulated evolution of selfish herd behavior |journal= Journal of Theoretical Biology |volume= 234 |issue= 2 |pages= 213–225 |doi= 10.1016/j.jtbi.2004.11.035 |pmid=15757680|bibcode= 2005JThBi.234..213R }}</ref><ref>{{cite journal |vauthors=Wood AJ, Ackland GJ |year= 2007 |title= Evolving the selfish herd: emergence of distinct aggregating strategies in an individual-based model |journal= Proc Biol Sci |volume= 274 |issue= 1618 |pages= 1637–1642 |doi= 10.1098/rspb.2007.0306 |pmid=17472913 |pmc=2169279}}</ref> the predator confusion effect,<ref>{{cite journal |vauthors=Olson RS, Hintze A, Dyer FC, Knoester DB, Adami C |year= 2013 |title= Predator confusion is sufficient to evolve swarming behaviour |journal= J. R. Soc. Interface |volume= 10 |issue= 85 |page=20130305 |doi= 10.1098/rsif.2013.0305 |pmid=23740485 |pmc=4043163}}</ref><ref>{{cite journal |vauthors=Demsar J, Hemelrijk CK, Hildenbrandt H, Bajec IL |year= 2015 |title= Simulating predator attacks on schools: Evolving composite tactics |journal= Ecological Modelling |volume= 304 |pages= 22–33 |doi=10.1016/j.ecolmodel.2015.02.018 |bibcode= 2015EcMod.304...22D |hdl= 11370/0bfcbb69-a101-4ec1-833a-df301e49d8ef |s2cid= 46988508 |url= https://pure.rug.nl/ws/files/85452752/Simulating_predator_attacks_on_schools_Evolving_composite_tactics.pdf|hdl-access= free }}</ref> the dilution effect,<ref>{{cite journal |author= Tosh CR |year= 2011 |title= Which conditions promote negative density dependent selection on prey aggregations? |journal= Journal of Theoretical Biology |volume= 281 |issue= 1 |pages= 24–30 |doi= 10.1016/j.jtbi.2011.04.014 |pmid=21540037 |bibcode= 2011JThBi.281...24T |url= https://hal.archives-ouvertes.fr/hal-00708525/file/PEER_stage2_10.1016%252Fj.jtbi.2011.04.014.pdf}}</ref><ref>{{cite journal |vauthors=Ioannou CC, Guttal V, Couzin ID |year= 2012 |title= Predatory Fish Select for Coordinated Collective Motion in Virtual Prey |journal= Science |volume= 337 |issue= 6099 |pages= 1212–1215 |doi= 10.1126/science.1218919 |bibcode= 2012Sci...337.1212I |pmid=22903520 |s2cid= 10203872 |url= http://nbn-resolving.de/urn:nbn:de:bsz:352-0-387618|doi-access= free }}</ref> the many eyes theory,<ref>{{cite journal |vauthors=Olson RS, Haley PB, Dyer FC, Adami C |year= 2015 |title= Exploring the evolution of a trade-off between vigilance and foraging in group-living organisms |journal= Royal Society Open Science |volume= 2 |issue= 9 |doi= 10.1098/rsos.150135 |page=150135 |pmid=26473039 |pmc=4593673 |arxiv=1408.1906 |bibcode=2015RSOS....250135O}}</ref> and the predator-prey survival pressure theory.<ref>{{cite journal |vauthors=Li JN, Li L, Zhao SY |year= 2023 |title= Predator–prey survival pressure is sufficient to evolve swarming behaviors |journal= New Journal of Physics |volume= 25 |issue= 9 |doi= 10.1088/1367-2630/acf33a |page= 092001|arxiv= 2308.12624 |bibcode= 2023NJPh...25i2001L }}</ref>

===Agents===
{{Main|Agent-based model in biology}}
{{See also|Agent-based models|Intelligent agent|Autonomous agent|Quorum sensing}}

* {{cite book |last1= Mach |first1= Robert |last2= Schweitzer |first2= Frank |year= 2003 |chapter= Multi-Agent Model of Biological Swarming |citeseerx= 10.1.1.87.8022 |title= Advances In Artificial Life |volume= 2801 |pages= 810–820 |doi= 10.1007/978-3-540-39432-7_87 |series= [[Lecture Notes in Computer Science]] |isbn= 978-3-540-20057-4}}

===Self-organization===
[[File:Sort sol ved Ørnsø 2007.jpg|right|thumb|Flocking birds are an example of [[biological organisation|self-organization in biology]]]]
{{See also|Self-organization|Biological organisation}}

===Emergence===
{{Main|Emergence}}

The concept of emergence—that the properties and functions found at a hierarchical level are not present and are irrelevant at the lower levels–is often a basic principle behind [[self-organization|self-organizing systems]].<ref name="importance-ppt">{{cite web |url=http://www.authorstream.com/Presentation/wdorsey-88049-hierarchy-life-biology-education-ppt-powerpoint/ |access-date=6 October 2009 |title=Hierarchy of Life |date=14 September 2008 |archive-date=3 July 2016 |archive-url=https://web.archive.org/web/20160703121704/http://www.authorstream.com/Presentation/wdorsey-88049-hierarchy-life-biology-education-ppt-powerpoint/ |url-status=dead }}</ref> An example of [[biological organisation|self-organization in biology]] leading to emergence in the natural world occurs in ant colonies. The queen does not give direct orders and does not tell the ants what to do.{{citation needed|date=July 2018}} Instead, each ant reacts to stimuli in the form of chemical scents from larvae, other ants, intruders, food and buildup of waste, and leaves behind a chemical trail, which, in turn, provides a stimulus to other ants. Here each ant is an autonomous unit that reacts depending only on its local environment and the genetically encoded rules for its variety. Despite the lack of centralized decision making, ant colonies exhibit complex behaviours and have even been able to demonstrate the ability to solve geometric problems. For example, colonies routinely find the maximum distance from all colony entrances to dispose of dead bodies.

===Stigmergy===
{{Main|Stigmergy}}

A further key concept in the field of swarm intelligence is [[stigmergy]].<ref name="Parunak2003">Parunak, H. v D. (2003). [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.1990&rep=rep1&type=pdf "Making swarming happen"] In: Proceedings of Conference on Swarming and Network Enabled Command, Control, Communications, Computers, Intelligence, Surveillance and Reconnaissance (C4ISR), McLean, Virginia, USA, 3 January 2003.</ref><ref name=Marsh2007>{{cite journal |author1=Marsh L. |author2=Onof C. |year= 2008 |title= Stigmergic epistemology, stigmergic cognition |url= http://mpra.ub.uni-muenchen.de/10004/1/3z2fx4r7prqwob3vfdq.pdf |journal= Cognitive Systems Research |volume= 9 |issue= 1 |pages= 136–149 |doi= 10.1016/j.cogsys.2007.06.009|s2cid=23140721 }}</ref> Stigmergy is a mechanism of indirect coordination between agents or actions. The principle is that the trace left in the environment by an action stimulates the performance of a next action, by the same or a different agent. In that way, subsequent actions tend to reinforce and build on each other, leading to the spontaneous emergence of coherent, apparently systematic activity. Stigmergy is a form of self-organization. It produces complex, seemingly intelligent structures, without need for any planning, control, or even direct communication between the agents. As such it supports efficient collaboration between extremely simple agents, who lack any memory, intelligence or even awareness of each other.<ref name=Marsh2007/>

===Swarm intelligence===
{{Main|Swarm intelligence}}

[[Swarm intelligence]] is the [[collective behaviour]] of [[decentralization|decentralized]], [[self-organization|self-organized]] systems, natural or artificial. The concept is employed in work on [[artificial intelligence]]. The expression was introduced by [[Gerardo Beni]] and Jing Wang in 1989, in the context of [[cellular automaton|cellular robotic]] systems.<ref>Beni, G., Wang, J. Swarm Intelligence in Cellular Robotic Systems, Proceed. NATO Advanced Workshop on Robots and Biological Systems, Tuscany, Italy, June 26–30 (1989)</ref>

Swarm intelligence systems are typically made up of a population of simple [[agent-based model in biology|agents]] such as [[boids]] interacting locally with one another and with their environment. The agents follow very simple rules, and although there is no centralized control structure dictating how individual agents should behave, local, and to a certain degree random, interactions between such agents lead to the [[emergence]] of intelligent global behaviour, unknown to the individual agents.

Swarm intelligence research is multidisciplinary. It can be divided into natural swarm research studying biological systems and artificial swarm research studying human artefacts. There is also a scientific stream attempting to model the swarm systems themselves and understand their underlying mechanisms, and an engineering stream focused on applying the insights developed by the scientific stream to solve practical problems in other areas.<ref>{{cite journal |last1= Dorigo |first1= M |last2= Birattari |first2= M |year= 2007 |title= Swarm intelligence |journal= Scholarpedia |volume= 2 |issue= 9 |page= 1462 |doi= 10.4249/scholarpedia.1462 |bibcode= 2007SchpJ...2.1462D|doi-access= free }}</ref>

===Algorithms===
Swarm algorithms follow a Lagrangian approach or an [[Euler equations (fluid dynamics)|Eulerian]] approach.<ref name="Li et al">{{cite journal |last1=Li |first1= YX|last2= Lukeman |first2=R|last3=Edelstein-Keshet|first3=L |year= 2007 |title= Minimal mechanisms for school formation in self-propelled particles |url= http://www.math.ubc.ca/~keshet/Papers/YXL_Lukeman_LEK.pdf |journal= Physica D: Nonlinear Phenomena |volume= 237 |issue= 5 |pages= 699–720 |doi= 10.1016/j.physd.2007.10.009 |bibcode= 2008PhyD..237..699L}}</ref> The Eulerian approach views the swarm as a [[field (physics)|field]], working with the density of the swarm and deriving mean field properties. It is a hydrodynamic approach, and can be useful for modelling the overall dynamics of large swarms.<ref>Toner J and Tu Y (1995) "Long-range order in a two-dimensional xy model: how birds fly together" ''Physical Revue Letters,'' '''75''' (23)(1995), 4326–4329.</ref><ref>{{cite journal |vauthors=Topaz C, Bertozzi A |year= 2004 |title= Swarming patterns in a two-dimensional kinematic model for biological groups |journal= SIAM J Appl Math |volume= 65 |issue= 1 |pages= 152–174 |doi= 10.1137/S0036139903437424 |citeseerx= 10.1.1.88.3071 |bibcode= 2004APS..MAR.t9004T|s2cid= 18468679 }}</ref><ref>{{cite journal |vauthors=Topaz C, Bertozzi A, Lewis M |year= 2006 |title= A nonlocal continuum model for biological aggregation |journal= Bull Math Biol |volume= 68 |issue= 7 |pages= 1601–1623 |doi= 10.1007/s11538-006-9088-6 |pmid= 16858662 |arxiv= q-bio/0504001|s2cid= 14750061 }}</ref> However, most models work with the Lagrangian approach, which is an [[agent-based model]] following the individual agents (points or particles) that make up the swarm. Individual particle models can follow information on heading and spacing that is lost in the Eulerian approach.<ref name="Li et al"/><ref>{{cite book |last1= Carrillo |first1= J |last2= Fornasier |first2= M |last3= Toscani |first3= G |chapter= Particle, kinetic, and hydrodynamic models of swarming |series= Modeling and Simulation in Science, Engineering and Technology |year= 2010 |title= Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences |chapter-url= http://mate.unipv.it/~toscani/publi/swarming.pdf |volume= 3 |pages= 297–336 |doi= 10.1007/978-0-8176-4946-3_12 |isbn= 978-0-8176-4945-6 |citeseerx= 10.1.1.193.5047}}</ref>

====Ant colony optimization====
{{Main|Ant colony optimization algorithm}}
{{External media
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|image1=[https://www.youtube.com/watch?v=oBhv4pKksgU Swarmanoid robots find shortest path over double bridge]<ref>{{cite web|url=http://www.swarmanoid.org/swarmanoid_simulation.php#|archive-url=https://web.archive.org/web/20070705115810/http://www.swarmanoid.org/swarmanoid_simulation.php|url-status=dead|archive-date=5 July 2007|title=Swarmanoid project}}</ref>}}

Ant colony optimization is a widely used algorithm which was inspired by the behaviours of ants, and has been effective solving [[discrete optimization]] problems related to swarming.<ref>[http://iridia.ulb.ac.be/~mdorigo/ACO/ACO.html Ant colony optimization] Retrieved 15 December 2010.</ref> The algorithm was initially proposed by [[Marco Dorigo]] in 1992,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, Elsevier Publishing, 134–142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italie, 1992.</ref> and has since been diversified to solve a wider class of numerical problems. Species that have multiple queens may have a queen leaving the nest along with some workers to found a colony at a new site, a process akin to [[swarming (honey bee)|swarming in honeybees]].<ref name=HolldoblerWilsonAnts2>Hölldobler & Wilson (1990), pp. 143–179</ref><ref name="Dorigo99">{{cite book |first1=M.|last1=DORIGO|first2=G.|last2=DI CARO|first3= L. M.|last3= GAMBERELLA|year=1999|title= Ant Algorithms for Discrete Optimization, Artificial Life|publisher= MIT Press}}</ref>
*Ants are behaviourally unsophisticated; collectively they perform complex tasks. Ants have highly developed sophisticated sign-based communication.
*Ants communicate using pheromones; trails are laid that can be followed by other ants.
*Routing problem ants drop different pheromones used to compute the "shortest" path from source to destination(s).
* {{cite journal |last1= Rauch |first1= EM |last2= Millonas |first2= MM |last3= Chialvo |first3= DR |year= 1995 |title= Pattern formation and functionality in swarm models |journal= Physics Letters A |volume= 207 |issue= 3–4 |page= 185 |arxiv=adap-org/9507003 |doi=10.1016/0375-9601(95)00624-c |bibcode=1995PhLA..207..185R|s2cid= 120567147 }}

====Self-propelled particles====
{{Main|Self-propelled particles}}
{{External media
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|video1=[http://phet.colorado.edu/sims/self-driven-particle-model/self-driven-particle-model_en.jar SPP model interactive simulation]<ref>[http://www.colorado.edu/physics/pion/srr/particles/ Self driven particle model] {{webarchive|url=https://web.archive.org/web/20121014155808/http://www.colorado.edu/physics/pion/srr/particles/ |date=2012-10-14}} Interactive simulations, 2005, University of Colorado. Retrieved 10 April 2011.</ref><br/>– needs Java
}}

The concept of [[self-propelled particles]] (SPP) was introduced in 1995 by [[Tamás Vicsek]] ''et al.''<ref name="Vicsek1995">{{cite journal |vauthors= Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O |author-link= Vicsek T |year= 1995 |title= Novel type of phase transition in a system of self-driven particles |journal=[[Physical Review Letters]] |volume= 75 |issue= 6 |pages= 1226–1229 |doi= 10.1103/PhysRevLett.75.1226 |bibcode=1995PhRvL..75.1226V |arxiv= cond-mat/0611743 |pmid= 10060237|s2cid= 15918052 }}</ref> as a special case of the boids model introduced in 1986 by Reynolds.<ref name="Reynolds"/> An SPP swarm is modelled by a collection of particles that move with a constant speed and respond to random perturbations by adopting at each time increment the average direction of motion of the other particles in their local neighbourhood.<ref>{{cite journal |vauthors=Czirók A, Vicsek T |year= 2006 |title= Collective behavior of interacting self-propelled particles |journal= Physica A |volume= 281 |issue= 1–4 |pages= 17–29 |doi= 10.1016/S0378-4371(00)00013-3 |arxiv= cond-mat/0611742 |bibcode= 2000PhyA..281...17C|s2cid= 14211016 }}</ref>

Simulations demonstrate that a suitable "nearest neighbour rule" eventually results in all the particles swarming together, or moving in the same direction. This emerges, even though there is no centralized coordination, and even though the neighbours for each particle constantly change over time.<ref name="Vicsek1995"/> SPP models predict that swarming animals share certain properties at the group level, regardless of the type of animals in the swarm.<ref name="Buhl et al">{{cite journal |vauthors= Buhl J, ((Sumpter DJT)), Couzin D, Hale JJ, Despland E, Miller ER, Simpson SJ |display-authors= etal |year= 2006 |title= From disorder to order in marching locusts |url= http://webscript.princeton.edu/~icouzin/website/wp-content/plugins/bib2html/data/papers/buhl06.pdf |journal= Science |volume= 312 |issue= 5778 |pages= 1402–1406 |doi= 10.1126/science.1125142 |pmid= 16741126 |bibcode= 2006Sci...312.1402B |s2cid= 359329 |access-date= 2011-04-13 |archive-url= https://web.archive.org/web/20110929220754/http://webscript.princeton.edu/~icouzin/website/wp-content/plugins/bib2html/data/papers/buhl06.pdf |archive-date= 2011-09-29 |url-status= dead}}</ref> Swarming systems give rise to [[emergent behaviour]]s which occur at many different scales, some of which are both universal and robust. It has become a challenge in theoretical physics to find minimal statistical models that capture these behaviours.<ref>{{cite journal |vauthors= Toner J, Tu Y, Ramaswamy S |year= 2005 |title= Hydrodynamics and phases of flocks |url= http://eprints.iisc.ernet.in/3397/1/A89.pdf |journal= Annals of Physics |volume= 318 |issue= 1 |pages= 170–244 |bibcode= 2005AnPhy.318..170T |doi= 10.1016/j.aop.2005.04.011 |access-date= 13 April 2011 |archive-date= 18 July 2011 |archive-url= https://web.archive.org/web/20110718172510/http://eprints.iisc.ernet.in/3397/1/A89.pdf |url-status= dead }}</ref><ref name="Bertin et al">{{cite journal |last1= Bertin |first1= E |last2= Droz |last3= Grégoire |first3= G |year= 2009 |title= Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis |arxiv= 0907.4688 |journal= J. Phys. A |volume= 42 |issue= 44 |page= 445001 |doi= 10.1088/1751-8113/42/44/445001 |bibcode= 2009JPhA...42R5001B|s2cid= 17686543 }}</ref>

====Particle swarm optimization====
{{Main|Particle swarm optimization}}
[[Particle swarm optimization]] is another algorithm widely used to solve problems related to swarms. It was developed in 1995 by [[James Kennedy (social psychologist)|Kennedy]] and [[Russell C. Eberhart|Eberhart]] and was first aimed at [[computer simulation|simulating]] the social behaviour and choreography of bird flocks and fish schools.<ref name=kennedy95particle>
{{cite conference
|last1=Kennedy
|first1=J.
|last2=Eberhart
|first2=R.
|title=Particle Swarm Optimization
|book-title=Proceedings of IEEE International Conference on Neural Networks
|year=1995
|volume=IV
|pages=1942–1948
}}</ref><ref name=kennedy97particle>
{{cite conference
|last1=Kennedy
|first1=J.
|title=The particle swarm: social adaptation of knowledge
|book-title=Proceedings of IEEE International Conference on Evolutionary Computation
|year=1997
|pages=303–308
}}</ref> The algorithm was simplified and it was observed to be performing optimization. The system initially seeds a population with random solutions. It then searches in the [[candidate solution|problem space]] through successive generations using [[stochastic optimization]] to find the best solutions. The solutions it finds are called [[Point particle|particles]]. Each particle stores its position as well as the best solution it has achieved so far. The particle swarm optimizer tracks the [[maxima and minima|best local value]] obtained so far by any particle in the local neighbourhood. The remaining particles then move through the problem space following the lead of the optimum particles. At each time iteration, the particle swarm optimiser accelerates each particle toward its optimum locations according to simple [[mathematical formulae|mathematical rules]]. In a related approach, Shvalb et al. (2024) introduced a statistical-physics-based framework for controlling large-scale multi-robot systems. By modeling robots as particles within a statistical ensemble, the study leverages macroscopic parameters—such as density and flow fields—to guide collective behavior without the need for individual identification or direct communication between agents. This method enables scalable and robust control of robot swarms, drawing conceptual parallels to particle swarm optimization by utilizing global information to influence local agent dynamics.<ref name=shvalb2024>
{{cite journal
 |last1=Shvalb
 |first1=Nir
 |last2=Hacohen
 |first2=Shlomi
 |last3=Medina
 |first3=Oded
 |title=Statistical Robotics: Controlling Multi Robot Systems using Statistical-physics
 |journal=IEEE Access
 |year=2024
 |volume=12
 |pages=134739–134753
 |doi=10.1109/ACCESS.2024.3406599
 |bibcode=2024IEEEA..12m4739S
 |url=https://www.researchgate.net/publication/380947211
 |access-date=2025-05-13
}}
</ref> 
Particle swarm optimization has been applied in many areas. It has few parameters to adjust, and a version that works well for a specific applications can also work well with minor modifications across a range of related applications.<ref>Hu X [http://www.swarmintelligence.org/tutorials.php Particle swarm optimization: Tutorial]. Retrieved 15 December 2010.</ref> A book by Kennedy and Eberhart describes some philosophical aspects of particle swarm optimization applications and swarm intelligence.<ref name=kennedy01swarm>
{{cite book
|title=Swarm Intelligence
|last1=Kennedy
|first1=J.
|last2=Eberhart
|first2=R.C.
|year=2001
|publisher=Morgan Kaufmann
|isbn=978-1-55860-595-4
}}</ref> An extensive survey of applications is made by Poli.<ref name=poli07analysis>{{cite journal
|last=Poli
|first=R.
|url=http://cswww.essex.ac.uk/technical-reports/2007/tr-csm469.pdf
|title=An analysis of publications on particle swarm optimisation applications
|journal=Technical Report CSM-469
|year=2007
|access-date=15 December 2010
|archive-date=16 July 2011
|archive-url=https://web.archive.org/web/20110716231935/http://cswww.essex.ac.uk/technical-reports/2007/tr-csm469.pdf
|url-status=dead
}}</ref><ref name=poli08analysis>
{{cite journal
|last=Poli
|first=R.
|url=http://downloads.hindawi.com/journals/jaea/2008/685175.pdf
|title=Analysis of the publications on the applications of particle swarm optimisation
|journal=Journal of Artificial Evolution and Applications
|year=2008
|volume=2008
|pages=1–10
|doi=10.1155/2008/685175
|doi-access=free
}}</ref>

====Altruism====
Researchers in Switzerland have developed an algorithm based on [[Hamilton's rule]] of kin selection. The algorithm shows how [[altruism in animals|altruism]] in a [[swarm]] of entities can, over time, evolve and result in more effective swarm behaviour.<ref>[http://genevalunch.com/blog/2011/05/04/altruism-helps-swarming-robots-fly-better-study-shows/ Altruism helps swarming robots fly better] {{Webarchive|url=https://web.archive.org/web/20120915092222/http://genevalunch.com/blog/2011/05/04/altruism-helps-swarming-robots-fly-better-study-shows/ |date=2012-09-15}} ''genevalunch.com'', 4 May 2011.</ref><ref>{{cite journal |last1= Waibel |first1= M |last2= Floreano |first2= D |last3= Keller |first3= L |year= 2011 |title= A quantitative test of Hamilton's rule for the evolution of altruism |journal= PLOS Biology |volume= 9 |issue= 5 |page= 1000615 |doi= 10.1371/journal.pbio.1000615 |pmid=21559320 |pmc=3086867 |doi-access= free }}</ref>