Downloaded by guest on September 30, 2021 a fetvl oteesgasaepol nesod hsisehas issue This respond understood. poorly to are organisms signals allow these that However, to effectively mechanisms (14). the strategies flows, fluctuating, optimal chaotic stochastically by and advected are (13), chemoattractants when locomotion mechan- of fluid the ics 12), (11, pathways signalling chemical of context nadvrerneo pce vrarneo cls o low For scales. of organ- range in a chemotaxis as over such environments species isms viscous of number, range observed Reynolds and behavior diverse insects flying ubiquitous a or a in animals aquatic is in seen environment commonly their about information in people locating or landmines, (8–10). applications explosives, operations with search-and-rescue of swarms, detection robot the olfactory in will as distributed (7). tasks of such development search environment technologies, the for performing their consequences for important of have capacity awareness their collective Understanding fish a as information-processing with as such viewed systems, be entities can complex herds, context, animal or this schools In indi- the (3–6). at level absent entirely vidual behaviors complex aggregations, benefits display social fully often of the not which properties and does emergent effect (2) potential this the Grünbaum However, capture by illustrated. made clearly sociality was of an type of this multiple study quantitative to of A due effect group. a dampened within that is individuals is by environment (1), sampling the principle wrongs” in “many noise the inherent as known this, for expla- that nation One noted navigation. animal been assist has greatly can It behavior fields. collective magnetic chemical or gradients, advected thermal as cues, such on information based varying environment their spatiotemporally navigating means this Often survival. T cooperation | search olfactory | intelligence collective interactions. local from result can emergent, that the via intelligence capacity group-level cognitive higher of path- evolution alternative the an to present way illustrates not behavior environment This level. the individual of the and awareness at signal an chemical displays a of so source doing the in locate to able interac- conditions is group context-dependent local the these tions, the Through to experiencing. according are modified they collective are inter- which the social rules on simple action following based individuals is autonomous method of behavior our com- perform Instead, or calculations. field, flow without plex the task of properties this diffusive solve direction, the gradient to measure spatial able assess to is required intelligence that higher the strategy a present technology. we inspired biologically Here of design the ani- for of and behavior envi- understanding mal the these for in implications the effective important to are has ronments that distance strategies with search monotonically Defining decrease source. not do concen- that filamental trations changing dynamically heterogeneous, gener- com- high the of or to due ation a number nontrivial highly Reynolds is becomes task high the number, signal either Peclet by advecting chemical the characterized When is advected organisms. medium living an many facing of challenge mon source the Locating Torney Colin aggregates social in behavior search emergent to leads interaction Context-dependent w.nsog/ci/di/1.03/pnas.0907929106 / 10.1073 / doi / cgi / www.pnas.org dtdb io .Lvn rneo nvriy rneo,N,adapoe oebr3 09(eevdfrrve uy1,2009) 15, July review for (received 2009 3, November approved and NJ, Princeton, University, Princeton Levin, A. Simon by Edited 08544 NJ Princeton, University, Princeton Biology, Evolutionary and Ecology colo ahmtclSine n ope n dpieSsesLbrtr,Uiest olg uln efil,Dbi ,Iead and Ireland; 4, Dublin Belfield, Dublin, College University Laboratory, Systems Adaptive and Complex and Sciences Mathematical of School h s fceia inl yognssi re ogain to order in organisms by signals chemical of use The ihtecalneo oaigtersucsrqie o their for required resources the locating faced of constantly challenge are the with organisms world, natural the hroughout shrci coli Escherichia a,b,1 otnNeufeld Zoltan , a ensuidetnieyi the in extensively studied been has a n anD Couzin D. 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Here a been have on approach (16) based information-theoretic or algorithms (15) and previously, addressed been cinrlsbsdo nasrcino grgto tendencies aggregation of abstraction an on based rules action units. then these are of parameters further terms All in time. defined unit in domain the cross will hsatcei NSDrc Submission. 1 Direct PNAS a is article This interest. paper. of the conflict wrote no I.D.C. declare research; and authors Z.N., performed C.T., The C.T. and research; data; analyzed designed I.D.C. I.D.C. and Z.N., and C.T., Z.N., C.T., Contributions: Author owo orsodnesol eadesd -al [email protected]. 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ECOLOGY APPLIED MATHEMATICS 2 observed in biological systems. A range of theoretical models have RA(t) = (1 − Ci(t)) RA,max been introduced to describe the collective motion of animal groups R (t) = sin2(πC (t))R . [6] (see e.g., refs. 19–21); here we assume that the preferred direction O i O,max results from local alignment, repulsion, and attraction (5, 6, 22, 23). However, we assume the interaction zone over which an individ- Although the exact functional form of this relation is reason- ual responds to neighbors varies as a function of current local ably arbitrary, the key behavioral response they represent can be conditions. This approach follows a similar concept outlined in summarized as ref. 24, which demonstrated that simulated fish schools were able to track regions of improved abiotic conditions by modifying reac- • a sharp decrease in signal results in attraction only. An indi- tions to conspecifics and reducing speed when located in preferred vidual then moves toward the center of the group within its regions. maximum interaction zone; The direction of motion pi is modified on the basis of the posi- • a weakening signal results in moderate attraction and strong tion and velocity of neighboring individuals. A desired direction alignment with neighbors. It is this intermediate response that di is defined by three social interaction rules. In order to maintain allows net movement along the filament and prevents a low a minimum distance between neighbors, individuals move away polarity swarm from forming; from those within a repulsion zone ZR (this precludes any other • if an improving signal is being experienced, this is the optimal behavioral response) direction, interaction zones are reduced to zero, all neighbors are ignored, and current direction is maintained.  rj − ri di =− . [3] |rj − ri| Qualitatively similar results to those presented here were obtained j∈Z R by using different functions for each scaling factor provided the If no others are present within the repulsion zone, individuals dynamical responses listed were approximated. By following the type of rules outlined, an individual acting move toward those within an attraction zone ZA and seek to align themselves with their neighbors within an orientation zone Z within a group is able to effectively track a filament and locate O the source of the signal, be it a nutrient source, a potential mate,  r − r  or the location of a suitable habitat. Sequential snapshots of the d = j i + p [4] i | − | j, search behavior being performed are shown in Fig. 1 for 60 indi- ∈ rj ri ∈ j ZA j ZO viduals, and animations (Movies S1 and S2) are included in the SI Appendix. where the attraction zone is typically assumed to be larger than the This model is not aimed to be an accurate representation of alignment zone. Each individual turns toward di at a rate propor- the behavior of a particular species, and a range of other quali- tional to the difference between the current and desired direction tatively similar interactions could be employed by different types with a maximum angular velocity defined by a parameter γ.Ifno of animal groups and in various environments. For example, vary- neighbors are present within the interaction zone, the direction ing the turning rates or some weighting factors of the alignment, vector pi remains unchanged. attraction and actual direction as a function of some measure of confidence can lead to qualitatively similar results so long as there Context Dependence exists a local context dependence of the type we describe above. In order to couple environment to behavior and create an effective Evidence that animals can, and do, employ rules of this type search algorithm, each individual varies the size of the interaction comes from experiments involving schooling fish where observed zones of local alignment and attraction depending on the current group-level behavior has been accurately modeled by implement- concentration value experienced. It should be noted that this is ing variable, context-dependent interaction ranges. In ref. 26, it not a directional measure, no gradient is detected, and flow veloc- was shown that fish appear to dynamically alter the range of ity is ignored; therefore at the level of a single individual, a search interactions with other individuals according to locally perceived strategy does not exist. stimuli, such as increasing this range when alarm odor [indicative The kernel of the model search behavior lies in the expansion of predation risk (27)] is detected. Experimental studies of fish and contraction of the attraction and orientation zones depend- have also demonstrated that individuals will tend to restrict their ing on the local concentration experienced by an individual. The schooling tendency when they can gather reliable direct informa- radius of each zone is a function of a normalized individual concen- tion from the environment, such as the location of resources, but increase their tendency to group with others when this information tration parameter Ci(t) ∈[0, 1] that measures the confidence in the actual direction of motion based on the recently encountered is perceived to be unreliable or is scarce (28). Potential examples signal, defined as are not limited to aquatic animals, and a similar mechanism has been postulated as an explanation for the active recruitment of C(r , t) others to food patches by the cliff swallow Hirundo pyrrhonota = i [5] Ci(t) −τ/α . (29). This hypothesis is supported by the increased frequency of max[C(ri(t − τ), t − τ)e ] 0<τ

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The the towards pointing orienta- half-circle initial individuals the around random of distributed a Numerical tion with number filament performed. chemical given the is a on positioned with search for performed the potential were the as simulations ignore fracture and present simulation to individuals each of populations number of the beginning source. the as the the size of at locating group effects successfully to the here of refer probability first We the investigate the on we of size role model, group the the understand in and parameters behavior this quantify further To Effects Group-Size search group-level signal a different for evidence for mechanism. provide size also indi- group would the optimum properties beyond this exists would Examining functionality drops vidual. higher-level rate success a which that below suc- and indicate size threshold a group minimum of understood, function a fully nonmonotonic has a are both is interactions which the rate cess of that guarantee nature no exact are model the the of predictions the reproduce that experiments some Although of individuals. result between a is interactions this nontrivial that princi- hypothesized wrongs” be therefore “many can the it by and ple, expected be with would improve as linearly (31) not size group does chemoat- fish the schooling of that capability Furthermore, tractive demonstrates here. evidence discuss experimental we to type response existing there the in that of deficiencies expect stimuli predictable environmental we that and dynamically, characteristic such range be means) will their alter other cannot or organisms they by pharmacological, used genetic, modalities the (through modify systems. can example experimentalists these If in mechanism context-dependent to a stages strate different (30). at distances different employed at be conspecifics to could respond vision and sensitivity 1. 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ECOLOGY APPLIED MATHEMATICS but the group is able to maintain its position centered on the plume for long periods. In the other extreme, high values of α lead to fluctuations in concentration having a larger impact on success rates due to a weakening responsiveness to the decay of the filament perpen- dicular to the desired direction. Effectively, individuals that have experienced a high-concentration parcel are subsequently less able to accurately respond to the horizontal profile of the fila- ment within the time frame defined by the decay rate. The negative impact of longer memory is a result of intermittent fluctuations in the signal and is therefore a weak effect, meaning that once the required memory is reached, the success rate declines slowly as it is increased. To examine the intermediate regime, the value of α was varied and success rates recorded for different group sizes. These results are shown in Fig. 3A, from which it is clear that optimum memory length is dependent on group size. Although the global optimum value corresponds to the value required to detect the edges of the filament while also being able to forget high-concentration patches, different group sizes do not all share this same optimum. The reason for this can be found in the tradeoff between explo- ration and exploitation of the signal where exploration means the effective sampling of the signal and exploitation corresponds to

Fig. 2. Effect of group size and group area on probability of success. (A) Probability of success as a function of group size. Lines represent vary- ing repulsion zone size. Green dash, 1 × 10−3; red solid, 2 × 10−3; blue dot, 3 × 10−3; black dot-dash, 4 × 10−3; purple dot-small dash, 5 × 10−3.(B) Prob- ability of success as a function of group area. Green plus sign, 1 × 10−3; red x, 2×10−3; blue asterisk, 3×10−3; black square (outline), 4×10−3; purple square −3 (solid), 5×10 .(Inset) Interaction zone sizes as a function of Ci (t). Blue dash, attraction zone; red solid, orientation zone; black dash, repulsion zone.

represents an individual level parameter that can be tuned through evolution or experience to affect the success of the group. In the flow regime we are considering, the filamental structure of the advected chemical signal means that the largest gradient is transverse to the desired direction. The memory parameter there- fore controls how responsive an individual is to a reduction in signal strength as it moves towards the edges of the filament. To place the time scale of the memory in the context of the proper- ties of the signal, we approximate the memory parameter required to effectively trace the edges of the filament. By measuring the average width of the filament and assuming its profile follows a Gaussian distribution, we calculate the time taken to traverse one standard deviation travelling at 45◦ to the parallel direction as ≈0.0125. As the role of memory is to allow an individual to determine if conditions are improving or deteriorating, this value defines a memory length for which it can be assumed that an indi- vidual is able to detect that it is moving toward the edges of the filament. Fig. 3. Effects of group size and memory length. (A) Probability of suc- Decay rates which are greater than ≈0.0125 lead to less respon- cess as a function of group size for various memory lengths. Green dash, α = 25 × 10−3; red solid, α = 12.5 × 10−3; blue dot, α = 2.5 × 10−3; black dot- siveness and more binary behavior; individuals ignore others when dash, α = 1.25 × 10−3; purple dash-dot, α = 0.5 × 10−3.(B) Group polarity as experiencing any concentration value and then attract when the fil- a function of group size. Legend as in A.(Inset) Mean nearest-neighbor dis- ament is lost. In the limiting case of no memory, this corresponds tance. Legend as in A. The black thin-dashed line represents repulsion radius. to a low-polarization swarm as seen in groups of mosquitoes or These values represent ensemble average values, time averaged within and midges. In this region of parameter space there is no net motion, across multiple simulation instances.

22058 www.pnas.org / cgi / doi / 10.1073 / pnas.0907929106 Torney et al. 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(2005) Collective (2007) SA ID Levin Couzin NR, Franks 7. spatial J, and memory Krause Collective ID, (2002) N Couzin Franks G, Ruxton 6. R, James J, Krause I, Couzin 5. (2003) al. et S, Camazine 4. .PrihJ,EesenKse 19)Cmlxt,pten n vltoaytrade-offs evolutionary and pattern, Complexity, (1999) L environment. Edelstein–Keshet noisy JK, Parrish a in taxis for 3. strategy a as Schooling (1998) D navigation. Grünbaum group of advantage 2. The wrongs: Many (2004) A Simons 1. ntecs ftemdlpeetdhr,tecne ftefila- the of center the here, presented model the of case the In advection-dominated an study to chosen have we Although aigi nmlgop ntemove. the on groups animal in making groups. animal in sorting Princeton). naia aggregation. animal in 12:503–522. 19:453–454. Δr = Science efOgnzto nBooia Systems Biological in Self-Organization ho Biol Theor J p= N 1 284:99–101.  i=1 N Nature N 1 min 218:1–11.      j Nature  i=1 N 445:715. (|r p j (t i 433:513–516. (t ) ) −      , r i uo Robot Auton (t )|), PictnUi Press, Univ (Princeton rnsEo Evol Ecol Trends 20:183–184. which α, vlEcol Evol [7] [8] 5 akvk ,Sria 20)Ofcoysac thg enlsnumber. Reynolds high at search Olfactory (2002) optimality B the Shraiman and E, persistence Balkovsky Directional (2009) 15. PK Maini JP, in Armitage DV, Chemosensing Nicolau (2006) microorganisms. swimming 14. of NS hydrodynamics The Wingreen (2009) TR Y, Powers E, Lauga Meir M, 13. Skoge the of RG, model Endres stochastic extended JE, spatially A Keymer (2003) D 12. Bray SV, Aksenov TS, Shimizu 11. 0 ae ,MrioiA oda 20)Samrbtcoo oaiain Off-line localization: odor robotic Swarm (2003) R Goodman A, Martinoli A, Hayes 10. yuigascn-re ceewt oiin n eoiiso neighbors speed of Particle velocities step. and time positions updated per with were once positions scheme updated Particle semi- second-order 512. a a using of using mean constant by gridsize by Root a performed a in 0.31. was with imposed signal to method was chemical 0.6 Lagrangian set stochastic of the modes flow of a mean energy Advection a by direction. highest and 0.25 the space was of velocity Fourier squared scale in length the generated spectrum and energy was decaying exponentially an field with process Ornstein–Uhlenbeck flow isotropic The aeil n Methods of and absence Materials the in even effective be information. can global type illustrates model algo- this our genetic of applied as of mechanisms be optimization, domain swarm also the may particle in may e.g., paper problems rithms, but this search strategies abstract in search more outlined to robotic results to efficiently. the applied manufactured sense, be be traditional numbers can the large components as In robotic effectiveness com- simple cost given and of (any expendable), robustness is inherent ponent systems), autonomous time communication (in of absence the notably paradigm control, traditional centralized the of over advantages (35–37) many community have robotics they the are because in systems interest distributed considerable tasks of sophisticated These topics perform interactions. to local able simple are through they that shown has tems environment. the characteristics specific of process the to learning adapted or behavior optimal selection produce to evolutionary some be could by functions further para- response the The tuned of path. form exact optimal the the and the values of of meter selection track automatic losing the without and concentration, trajectories signal chemical different the of in is comparison factors fluctuations a these of of out result smoothing The samples. the consen- these a on makes based decision interactions, sus context-dependent the simultaneously area to larger much due a and, over sample to able Secondly, is filament. group the trajectories, the to diverging group signal; away exponentially the of anchor moved the interactions result collective and a on as dislodged signal position easily the maintaining from are individuals in lone provides although it stability the (34). required speed separate turning spatially the requires determine but accurately (33) to nature sensors in effec- observed be of is may and it edges mechanism tive counter-turning the lost, a to is through Responding signal filament relocate. the the to once lost. difficult as is highly ineffective, turns becomes signal random is the biased environments and if these sampling away in temporal moved on rapidly based strategy is A particle a as tageous, rs etefrHg-n Computing. High-End for the by Centre provided Irish were N00014-09-1-1074. support Award and facilities Research computational Naval Foundation High-performance of Science Office National and by sup- PHY-0848755 Scholar supported Award Searle were was by N. work supported Z. was This C. and 08-SPP-201. D. T. Award I. C. Ireland; Foundation discussions. Science useful by for ported Ioannou Christos and Bazazi, details ACKNOWLEDGMENTS. further For source. the the from see 0.8 cen- methods, of ball distance numerical a a of in at located distance filament initially a the were on within Individuals tered location. reaching source individuals the of of 0.025 number of recording and trials 1,000 (radians), u trcinradius attraction mum .MrusL ue ,d leaA(06 atcesambsdofcoygie search. guided olfactory swarm-based Particle (2006) A Almeia de U, Nunes L, Marques 9. h eetsuyo nmlgop scmlxaatv sys- adaptive complex as groups animal of study recent The Firstly factors. two in lies therefore sociality of advantage The cdSiUSA Sci Acad chemotaxis. run-and-tumble of 72:096601. Phys, receptors. 1791. two-state of regimes Two coli: escherichia pathway. signalling chemotaxis bacterial piiainadvldto ihra robots. real with validation and optimization Robot Auton dt = 99:12589–12593. .02,mxmmoinainradius orientation maximum 0.00025, 20:277–287. 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ECOLOGY APPLIED MATHEMATICS 16. Vergassola M, Villermaux E, Shraiman BI (2007) Infotaxis as a strategy for searching 28. Van Bergen Y, Coolen I, Laland K (2004) Nine-spined sticklebacks exploit the most without gradients. Nature 445:406–409. reliable source when public and private information conflict. Proc R Soc London Ser 17. Torney C, Neufeld Z (2008) Phototactic clustering of swimming microorganisms in a B 271:957–962. turbulent velocity field. Phys Rev Lett 101:078105. 29. Brown C, Brown M, Shaffer M (1991) Food-sharing signals among socially foraging 18. Martí AC, Sancho JM, Sagués F, Careta A (1997) Langevin approach to generate cliff swallows. Anim Behav 42:551–564. synthetic turbulent flows. Phys Fluids 9:1078–1084. 30. Partridge B, Pitcher T (1980) The sensory basis of fish schools: Relative roles of lateral 19. Vicsek T, Czirok A, Ben–Jacob E, Cohen I, Shochet O (1995) Novel type of phase line and vision. J Comp Physiol A 135:315–325. transition in a system of self-driven particles. Phys Rev Lett 75:1226–1229. 31. Steele C, Scarfe A, Owens D (1991) Effects of group size on the responsiveness of 20. Gueron S, Levin S, Rubenstein D (1996) The dynamics of herds: From individuals to zebrafish, Brachydanio rerio (hamilton buchanan), to alanine, a chemical attractant. aggregations. J Theor Biol 182:85–98. J Fish Biol 38:553–564. 21. Mogilner A, Edelstein–Keshet L (1999) A non-local model for a swarm. J Math Biol 32. Berg H (1993) Random Walks in Biology (Princeton Univ Press, Princeton). 38:534–570. 33. Grasso FW, Basil JA (2002) How lobsters, crayfishes, and crabs locate sources of odor: 22. Aoki I (1982) A simulation study on the schooling mechanism in fish. Bull Jpn Soc Sci current perspectives and future directions. Curr Opin Neurobiol 12:721–727. Fish 48:1081–1088. 23. Huth A, Wissel C (1992) The simulation of the movement of fish schools. J Theor Biol 34. Grasso FW, Consi TR, Mountain DC, Atema J (2000) Biomimetic robot lobster performs 156:365–385. chemo-orientation in turbulence using a pair of spatially separated sensors: Progress 24. Huth A, Wissel C (1994) The analysis of behaviour and the structure of fish schools by and challenges. Robot Auton Syst 30:115–131. means of computer simulations. Comments Theor Biol 3:169–201. 35. Bonabeau E, Dorigo M, Theraulaz G (1999) Swarm intelligence: From natural to 25. Segall J, Block S, Berg H (1986) Temporal comparisons in bacterial chemotaxis. Proc artificial systems (Oxford Univ Press, New York). Natl Acad Sci USA 83:8987–8991. 36. Desai J, Ostrowski J, Kumar V (2001) Modeling and control of formations of 26. Hoare D, Couzin I, Godin JG, Krause J (2004) Context-dependent group size choice in nonholonomic mobile robots. IEEE Trans Robotic Autom 17:905–908. fish. Anim Behav 67:155–164. 37. Ogren P, Fiorelli E, Leonard N (2004) Cooperative control of mobile sensor networks: 27. Kats LB, Dill L (1998) The scent of death: Chemosensory assessment of predation risk Adaptive gradient climbing in a distributed environment. IEEE Trans Automat Contr by prey animals. Ecoscience 5:361–394. 49:1292–1302.

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