Editing Swarm behaviour (section)

===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}}
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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}}
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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
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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>