PLos one OPEN 3 ACCESS Freely available online Quantifying the Interplay between Environmental and Social Effects on Aggregated-Fish Dynamics M anuela C ap ello1*, Mare Soria1, Pascal C otel1, Jean-Louis D en eu b ou rg2, Laurent D agorn3 1 UMR EME, Institut d e Recherche pour le D éveloppem ent (IRD), Saint Denis, La Réunion, France, 2 USE, Unit of Social Ecology, Université Libre d e Bruxelles (ULB), Bruxelles, Belgium, 3 UMR EME, Institut d e Recherche pour le D éveloppem ent (IRD), Victoria, Seychelles Abstract Dem onstrating and quantifying the respective roles o f social interactions and external stim uli governing fish dynamics is key to understanding fish spatial distribu tion. If seminal studies have con tribu te d to o u r understanding o f fish spatial organization in schools, little experim ental inform ation is available on fish in the ir natural environm ent, where aggregations often occur in the presence o f spatial heterogeneities. Here, we applied novel m odeling approaches coupled to accurate acoustic tracking fo r studying the dynamics o f a gro up o f gregarious fish in a heterogeneous environm ent. To this purpose, we acoustically tracked w ith subm eter resolution the positions o f tw elve small pelagic fish (Selar crum enophthalm us) in the presence o f an anchored floating object, con stituting a p o in t o f attraction for several fish species. We constructed a fieldbased m odel for aggregated-fish dynamics, deriving effective interactions for both social and external stim uli from experiments. We tuned the m odel parameters th a t best fit the experim ental data and quantified the im portance o f social interactions in the aggregation, providing an explanation for the spatial structure o f fish aggregations found around floating objects. Our results can be generalized to oth er gregarious species and contexts as long as it is possible to observe the fine-scale m ovem ents o f a subset o f individuals. C itation : Capello M, Soria M, Cotel P, D eneubourg J-L, Dagorn L (2011) Quantifying the Interplay betw een Environmental and Social Effects on Aggregated-Fish Dynamics. PLoS ONE 6(12): e28109. doi:10.1371/journal.pone.0028109 Editor: Gonzalo G. d e Polavieja, Cajal Institute, Consejo Superior d e Investigaciones Científicas, Spain Received Septem ber 6, 2011; A ccepted November 1, 2011; Published December 12, 2011 C opyrigh t: © 2011 Capello e t al. This is an open-access article distributed under th e terms of th e Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided th e original author and source are credited. Funding: This work was supported by th e Regional Council o f Reunion Island, the French Ministry of Overseas, th e 'Run Sea Science' (Theme - Capacity Building, contract 229968) and th e 'MADE: Mitigating adverse ecological impacts o f open ocean fisheries' (Theme 2 Food, Agriculture, Fisheries and Biotechnology, contract 210496) FP7 European projects. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of th e manuscript. C om peting Interests: The authors have declared th a t no com peting interests exist. * E-mail: manuela.capello@ird.fr distributions. O n one h a n d , m icroscale m odels [9-1 4 ], w hich consider individuals e m b ed d ed in a n hom ogeneous environm ent, can explain the observed schooling a n d m illing p h e n o m e n a b u t can n o t be used to m ake predictions on the spatial distribution o f the different fish species due to the difficulty in estim ating m odel p a ram ete rs experim entally. O n the o th er h a n d , m acroscoscale m odels [15] capable o f in co rp o ra tin g the response o f fish populations to e nvironm ental gradients for different species do not take into account behavioral features th a t could play a crucial role in the fish spatial distribution. In this study, w e w orked at an interm ediate scale, deriving a field-based m odel for fish dynam ics th a t could in co rp o rate a t the sam e tim e the basic ingredients for fish response to social stim uli a n d e nvironm ental heterogeneities. R em arkably, this m odeling a p p ro ac h can be applied to a large variety o f p h e n o m e n a w henever the spatial distribution o f individuals results from the m u tu al response to environm ental a n d social interactions. W e used the case o f a group o f fish in the presence o f a floating object, know n in the literature as a Fish A ggregation D evice (FAD) [16,17]. FA Ds can be artificial or na tu ra l floating structures, eith er drifting or a n ch o red . T h e y have b e en m assively deployed by com m ercial fisheries since the eighties because they constitute a po in t o f attrac tio n for m an y fish species. H ow ever, the reason fish aggregate a ro u n d FA Ds is still unknow n. U n d e rstan d in g aggregated-fish beh av io r is beco m in g m ore urgent, due to the large a n d continually grow ing exploitation o f FADs. R ecently, concerns th a t FA Ds m ay act as ecological traps for fish have b e en voiced [18], suggesting th a t the retain in g ch ara cte r o f Introduction D espite the social a n d econom ic im p o rtan ce o f fisheries, q u antitative tools capable o f pred ictin g fish distribution a n d its variations w ith respect to e nvironm ental changes a n d h u m an activities are still missing. T h e m ain a pproaches th a t are currently used in fisheries m an ag em en t require going b e yond the study o f isolated targ et fish species a n d d e m a n d taking into account intraa n d inter-specific interactions, b ehavioral factors as well as responses to the e nvironm ent [1,2]. M o re generally, they raise fu n d am en tal questions o n anim al o rganization in a n a tu ra l environm ent. D e m o n stratin g a n d quantifying the respective roles o f external factors a n d social influences is key to u n d e rstan d in g the spatial distribution a n d organization o f anim als in their e n viron­ m ent. In the last decade, several studies have show n how social interactions govern the dynam ics o f anim al groups, such as fish schools, b ird flocks, sheep herds o r aggregations o f insects [3-9]. H ow ever, in a n a tu ra l environm ent, anim al aggregations often occur in the presence o f e nvironm ental heterogeneities, constitut­ ing a p o in t o f a ttrac tio n for feeding, sheltering o r o th er behaviors. T his d em ands the creation o f dedicated analytical a n d m odeling tools capable o f taking into account interactions, b o th w ith the o th er individuals a n d w ith a heterogeneous environm ent. T h e re is substantial evidence th a t social beh av io r is im p o rta n t for m an y fish species, yet w ithin the existing e x perim ental a n d m odeling approaches, it is difficult to quantify the respective roles played by fish social interactions a n d external stim uli in th eir spatial PLoS ONE I w w w .p lo s o n e .o rg 1 D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics FA Ds m ay alter th e biological characteristics o f fish populations associated w ith them , like m igration, grow th, condition factors, p re d atio n a n d n a tu ra l m ortality. T h e validity o f these scenarios strongly depends on the type o f m echanism s leading to aggregation, a n d this requires furth er investigation. T o this end, precise inform ation o n the range a n d structure o f fish aggregations is needed. T o date, several acoustic surveys have b e en done in o rd e r to ch aracterize fish aggregations a ro u n d FA D s [19-21]. H ow ever, no in form ation was available so far on aggregated-fish dynam ics at subm eter scales, w here fish b e h av io r could be studied in detail. 0.2 0.15 => 0.1 o c cr 1— U- 0.05 •3 -2 1 0 1 2 3 0 [radians] In this p ap er, we in tro d u ced a new m odeling a p p ro ach , coupled to accu rate acoustic tracking m easurem ents, capable o f quantify­ ing the driving forces leading to aggregation. W e considered an obligate schooling small pelagic fish species, Selar crumenophthalmus, in the p roxim ity o f a FA D located in Saint P aul’s Bay at R eu n io n Island (W est In d ian O cean) [22]. T h e FA D was a 12-m b o a t fixed at 17 m d e p th by five anchors to p rev en t any m ovem ent. Tw elve fish w ere tagged w ith H T I ™ acoustic pingers (H ydroacoustic T echnology Inc., Seattle, USA) a n d released next to the FA D, along w ith o th er n on-tagged individuals. T h e 3D tracking o f each tagged fish (one position every second w ith sub-m eter resolution) was possible w ithin a radius o f a pproxim ately 50 m from the FA D w ith the use o f a n H T I ™ acoustic detection system. E xperim ental d a ta collected d u rin g one h o u r w ere used to construct a m odel for aggregated-fish dynam ics, w hich took into acco u n t the possible interactions w ith the FA D a n d the o th er tagged fish, w hile all o th er factors (e.g., light, food a b u n d an c e, a n d currents) w ere considered constant d u rin g this p erio d o f observation. M odel p a ram ete rs w ere fine-tuned w ith experim ental data, w hich allow ed us to quantify the interplay b etw een social interactions a n d attrac tio n to the FA D a n d to gain insights o n the aggregation p h e n o m e n a. 0.1 0.05 0 0 0.5 1 1.5 V [m/s] Figure 1 . Analysis o f individual-fish dynam ics. (A) T u rn in g a n g le d is trib u tio n : e x p e rim e n ta l p o in ts (red) a n d fit (black) w ith th e W ra p p e d C au c h y D istrib u tio n <D(0) = — ’ sin h (p )— . ^ f¡t t ¡n g p a ra m e te r 2 k cosh(p) —cos(0) a K p — 0.16. (B) Individual s w im m in g s p e e d d is trib u tio n : e x p e rim e n ta l p o in ts (b lu e ) a n d fit (black) w ith th e G a m m a d is trib u tio n f ( x ) = r.vexp( — x /s ) w ith sca le p a r a m e te r 5 = 0 .1 6 a n d n o rm a liz a tio n c o n s ta n t c — 1.2. d o i:1 0 .1 3 7 1 /jo u rn a l.p o n e .0 0 2 8 1 0 9 .g 0 0 1 Results w here r is the radial distance am o n g fish pairs (Fig. 2B). T h e depletion o f g ( r ) for distances sm aller th a n 0.3 m indicated a zone o f repulsion. A t in term ed iate distances, the fish p air-correlation was constant a n d m axim um , revealing a zone o f com fort. F or distances larg er th a n 0.9 m , the pair-co rrelatio n decayed expo­ nentially, revealing th a t it was less p ro b a b le to find inter-individual distances in this range. B oth the speed a n d tu rn in g angle distribution, as well as the pair-co rrelatio n function, w ere in d ep e n d en t o f the radial distance from the FA D , see Fig.S4 in S upplem entary m aterial. E x p e rim e n ta l d a t a an aly sis T w o variables w ere considered to characterize the individual fish dynam ics: tu rn in g angle a n d sw im m ing speed in the ,vy plane. T h e turning-angle distribution was well described by a w ra p p ed C auchy distribution, w ith a sharp p e ak cen tered at zero (Fig. 1A). A t the tim e scale o f o u r observations, no evidence o f correlation a m o n g subsequent tu rn in g angles was found [23]. T h e speed distribution h a d a m ax im u m at approxim ately 0.16 + 0.03 m /s (Fig. IB). Based on the observation th a t the average fork length o f o u r tagged fish was 0.17 + 0.02 m , this finding was com patible w ith the w idely accepted kinetic rule o f 1 b o d y -len g th /se co n d . At h igher speeds, the distribution decayed exponentially. T h e re co rd e d fish trajectories in the ,vy plan e (see Fig.S2 in S upplem entary m aterial) d e m o n stra ted a radial sym m etry a ro u n d the FA D . T herefore, in o rd e r to analyze the fish spatial distribution, we calculated the tim e-averaged radial distribution P i(R ) for each fish / w ith respect to the FA D position, w here R is the radial distance from the FA D in the ,vv plan e (Fig. 2A). T his qu an tity gave inform ation on the probability to find a fish at a distance R from the FA D [24], R em arkably, all o f the tagged fish show ed the sam e rad ial distribution. Close to the FA D , th ere was a region (at a distance sm aller th a n 2 m) characterized by a high a n d constant P i{R ). T h ere afte r, P¡{R ) decreased exponentially u p to a distance o f a pproxim ately 10 m , w here the radial distribution was o f the o rd e r o f the constant distribution associated to a fish occupying hom ogeneously the detection area. T his scale sets the b o u n d a ry o f the zone o f aggregation. T h e interactions betw een tagged fish, as well as th eir variations in space, w ere investigated th ro u g h the tim e-averaged fish pair-correlation function g (r) [24], PLoS ONE I w w w .p lo s o n e .o rg >, M o d e l resu lts In o rd e r to gain quantitative insights into the fish response to b o th the FA D a n d the o th er fish, w e derived effective interactions from the experim ental quantities discussed above. E xpressing the average fish-radial distribution P (R ) a ro u n d the FA D as the exponential o f a B oltzm ann w eight P ( i ? ) ~ e x p [ — F f,id(í?)] [25], we o b tain e d the effective fish-FAD interaction V f a d ( R ) ~ log[P (R )]. F ro m the b ehavior o f the experim ental P (R ) this lead to the follow ing expression for the fish-FA D interaction: Vf ad ( R ) = \ f c o n st for R < R s, [ yR for R > R st f “ (!) w here R st is the stationarity radius, found a t 2 m , w hich corresponds to the region w ith constant probability to find the fish. B eyond this distance was the zone o f FA D attractio n , w here fish resp o n d ed to the presence o f the FA D th ro u g h a constant attractive force, w hose strength, a, was o b tain e d by com parison 2 D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics 0.1 0.01 fe 0.001 0.0001 1e-05 R [m] 0.04 0.035 0.03 0.025 I 0.1 0.01 0.001 0.0001 1e-05 1e-06 0^02 0.015 0.01 0.005 0 5 10 15 20 B 0 0 2 4 6 8 10 12 14 16 1820 r[m ] Figure 2. Analysis o f fish spatial d istrib ution around the FAD. (A) Radial d is trib u tio n a ro u n d th e FAD fo r e a c h o f th e 12 ta g g e d fish ( r e p r e s e n te d b y d if f e r e n t c o lo rs ) in s e m ilo g a r ith m ic s c a le . T h e h o riz o n ta l line in d ic a te s th e b e h a v io r o f P ( R ) fo r a fish h a v in g a h o m o g e n e o u s d is trib u tio n in a circle o f ra d iu s e q u a l to 3 0 m . Inset: m e a n v a lu e o v e r all fish. (B) Fish p a ir-c o rre la tio n fu n c tio n . Inset: th e s a m e in s e m ilo g a rith m ic scale. d o i:1 0 .1 3 7 1 /jo u rn a l.p o n e .0 0 2 8 1 0 9 .g 0 0 2 w ith experim ental d ata, as show n below . T his zone was assum ed to be m u ch larger th a n the o th er spatial scales governing fish dynam ics [26]. T h e definition o f its bo u n d aries w ent b eyond the scope a n d experim ental lim its o f this study. A nalogously, w e derived effective fish-fish interactions from the pair-co rrelatio n function, defining F/¡¿/¡(r)~log[g(r)] [25]. A c­ c o rding to the b e h av io r o f g(r), w e identified th ree m ain zones for the fish-fish interaction: a zone o f repulsion w ithin a radius r rep = 0-3 m, a zone o f com fort u p to rcomf = 0.9 m a n d a zone o f a ttrac tio n a t larg er distances. T his lead to the follow ing effective fish-fish interaction: P (R ) averaged over all fish (Fig. 3A). T h e pair-co rrelatio n function (Fig. 3B) calculated for the optim ized non-social system was different th a n the one found from the experim ental d ata, signaling th a t the observed fish aggregation a ro u n d the FA D was n o t purely a consequence o f the FA D attractio n . W e th e n adjusted the p a ram ete rs a a n d ß in E q .l a n d E q.2 to find the m in im u m o f the SS R for b o th the fish radial distribution a n d the p air-correlation function (see T ab le 1). Indeed, ad d in g fish-fish interactions resulted in agreem ent b etw een the m odel a n d experim ental d a ta for all quantities a n d show ed th a t the fish dynam ics a ro u n d the FA D was also the fingerprint o f a tru e fish-fish in teractio n (Fig. 3G a n d 3D). M oreover, starting from the optim ized m odel th a t best fit the experim ental d ata, we studied the system sensitivity to changes o f one m odel p a ra m e te r a t a tim e. First, we calculated the radial distribution a ro u n d the FA D for different values o f the individual fish speed, v, keeping constant the fish-fish a n d fish-FAD interactions. Sm all changes in v affected the zone o f aggregation significantly, w ith a n aggregation radius increasing w ith speed (Fig. 4A). N ext, we analyzed the role o f social interactions, keeping constant the FA D attractio n . W e o b tain e d th a t non-social fish should have a larger dispersion a ro u n d the FA D , w ith an aggregation radius o f a b o u t 60 m ra th e r th a n 10 m for social fish (Fig. 4B). Finally, we studied the fish-group dynam ics in the absence o f a FA D . T h e fish-group b a ric en te r p erfo rm ed a ra n d o m w alk a n d explored the env iro n m en t (Fig. 5A), w ith the group staying com pact. T his is clear from the com parison o f the fish paircorrelation in the p re sen c e /a b se n ce o f a FA D (Fig. 5B). D iscussion E xperim ental d a ta revealed the existence o f a sharp zone o f aggregation, w here Selar crumenophthalmus c o n cen trate a t small distances from the FA D . All tagged fish exhibited the sam e beh av io r a n d m ostly stayed w ithin 10 m from the FA D . W e could distinguish a stationarity region very close to the FA D ( < 2 m), w ith a constant a n d high p robability o f finding fish, as well as a N on-social 0 .0 2 5 Social 0 .0 2 5 mode l 0 .0 1 5 ~ m odel exp ,\ 0 .0 0 5 0 .0 0 5 V /M ir) = { c o n st ßr 0 for r < rrep for rrep < r < 5 10 15 2 R [m] f com f > (2) for r > r comf w here the analytic form o f the short-range repulsion was assum ed linear for simplicity, a n d v is the individual fish speed (v = 0.17 m / s, following the rule o f 1 b o dy-length/second). T h e p a ram ete r, ß, setting the degree o f attrac tio n b etw een fish, was o b tain ed by com parison w ith experim ental d ata, as show n below . G iven these effective interactions, we m odeled the system follow ing a co rrelated ra n d o m walk dynam ics (CRW ) em b e d d ed in a force field [27,28], In o rd e r to evaluate the role o f the fish-FAD interaction in the m easu red quantities, we first c o m p a red results from the non-social m odel (i.e., no fish-fish interaction) w ith experim ents. T ak in g the expression o f the effective fish-FAD interaction in E q .l, the only free p a ra m e te r was the FA D a ttrac tio n strength, a. W e estim ated the value o f a th a t m inim ized the sum o f squared residuals (SSR) for the rad ial fish distribution PLoS ONE I w w w .p lo s o n e .o rg c \ ' —r r - 0 .0 1 5 k 0 .0 2 5 0 .0 4 3 0 .0 3 5 0 .0 3 ,* 0 .0 2 5 i 1 0 .0 1 5 0 .0 1 5 0 .0 0 5 0 .0 0 5 0 .0 3 5 mode l model e xp ™ . *■ ^ ■ V D , 10 15 20 r[m ] Figure 3. Com parison am ong o p tim ize d m odel and experi­ m ental results for the fish radial d istrib ution P(R) around the FAD and the p air correlation function g (r ). Left p a n e ls: n o n -so c ial fish m o d e l (A) radial d is trib u tio n a n d (B) p a ir-c o rre la tio n fu n c tio n . R ight p a n e ls: social fish m o d e l (C) radial d is trib u tio n a n d (D) p a ir-c o rre la tio n fu n c tio n . d o i:1 0 .1 3 7 1 /jo u rn a l.p o n e .0 0 2 8 1 0 9 .g 0 0 3 D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics 80 T ab le 1 . Model parameters. M o d el P aram eter Sym bol Number o f fish N 12 12 Individual speed [m/s] V 0.17 0.17 W rapped Cauchy distribution param eter P 0.16 0.16 FAD Stationarity radius [m] Rst 2 2 FAD attracting potential strength * a 0.014 0.0015 Fish-fish repulsion radius [m] rrep - 0.3 Fish-fish comfort radius [m] r conf B 0.9 Fish-fish attracting potential strength * ß N on-social - 60 Social > 20 0 0 larg e r-ran g e d zone (2—10 m) w ith a n exponentially decaying p robability. T h is level o f spatial accuracy, as well as in form ation o n th e specific shape o f th e zone o f aggregation, c a n n o t b e re ac h ed v=0.13 v=0.15 v=0.17 .19 0.01 0.001 40 60 I m/s m/s m/s m m/s 0.1 0.01 0.001 0.0001 16-05 1e-06 1e-07 1e-08 1e-09 0 5 w ith sta n d ard acoustic techniques [19-21], revealing th a t this experim ental a p p ro ac h is ideally suited for u n d e rstan d in g the beh av io r o f fish aggregated to FA Ds a t a fine scale. In o rd e r to identify the factors th a t shaped the zone o f aggregation, we constructed the sim plest m odel o f fish dynam ics th a t w ould take the m ain ingredients into consideration, w here the fish-FAD a n d the fish-fish interaction was d educed from the spatial distribution o f the tagged fish. A lthough the attrac tio n o f the FA D alone allow ed m odeling o f the fish aggregation to the object, the m atch in g b etw een experim ental a n d m odeled d a ta was only possible w hen taking into account the fish-fish interaction. T h e optim ized value o f the FA D attractio n , a, was m u ch sm aller th an the C R W individual speed p a ram ete r, v, indicating th a t the stochastic term was playing a m ajo r role. In o th er w ords, w h en fish stayed in the zone o f aggregation, th eir m ovem ents w ere d o m in a ted by the c o rre late d -ra n d o m w alk com ponent. T h e o ptim ized value o f the fish-fish attractio n , ß, was larger th a n a a n d was necessary in o rd e r to rep ro d u ce the ex p erim ental paircorrelation function. In this way, the m odel highlighted the im p o rta n t role o f social interactions in the distribution o f fish a ro u n d a FA D . Indeed, alth o u g h we only tagged som e individuals from a group, the tim e average o f o u r tagged-fish p air-correlation offered insights into the entire fish aggregation n e a r the FA D . T h e effective fish-fish interaction im plicitly took into account the presence o f o th er n o n-tagged fish in the system. T his represents a key im provem ent in the study o f social b eh av io r o f wild anim als in their environm ent, as an exhaustive observation o f all m em bers o f 5 10 15 20 25 30 35 40 R[m] NON SOCIAL SOCIAL 0.001 0.0001 ST le -0 5 20 m o d e l p re d ic tio n fo r th e (A) fis h -g ro u p b a ric e n te r (FAD re p re s e n te d w ith a b lu e circle) a n d (B) p a ir-c o rre la tio n in th e p re s e n c e /a b s e n c e o f a FAD. do i:1 0 .1 3 7 1 /jo u rn a l.p o n e .0 0 2 8 1 0 9 .g 0 0 5 0 £■ 15 Figure 5. Role o f FAD attraction on fish agg reg atio n. O p tim iz e d 1e-06 0.01 10 r[m] le -0 5 0.1 100 120 NO FAD FAD 0.0001 ST 80 X [m] Model param eters used to fit th e experimental data in Fig. 3. Stars indicate the free param eters estim ated through minimization o f th e SSR on the radial distribution function and th e pair correlation function. Third column, non-social fish model. Forth column, social fish model. doi:10.1371 /journal.pone.0028109.t001 0.1 20 0.003 1e-06 1e-07 1e-08 0 10 20 30 40 50 60 70 R[m] F igure 4. Role o f ind ividu al sw im m ing speed and social interaction on fish agg reg atio n. Radial d is trib u tio n a ro u n d th e FAD o b ta in e d fro m th e o p tim iz e d m o d e l (w ith fish-fish in te ra c tio n s) w h e n v a ry in g (A) th e in d iv id u al fish s w im m in g s p e e d v, (B) th e social in te ra c tio n p a r a m e te r ß (NON-SOCIAL c o rre s p o n d s to ß — 0 a n d SOCIAL in d ic a te s th e o p tim iz e d m o d e l). d o i:1 0 .1 3 7 1 /jo u rn a l.p o n e .0 0 2 8 1 0 9 .g 0 0 4 PLoS ONE I w w w .p lo s o n e .o rg 4 D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics a g roup is alm ost never possible. M oreover, o u r optim ized m odel shows th a t the fish-fish in teractio n alone, estim ated th ro u g h our field-based a p p ro ac h n e ar the FA D , c an ensure a stable group dynam ic, even in the absence o f a FA D . T his pred ictio n signals th a t the observed aggregation is a stable entity a n d could be the precu rso r o f schooling [17,22]. Finally, o u r m odel allow ed us to m ake predictions for different values o f the param eters controlling the aggregated-fish dynam ics. W ithin o u r schem e, the boundaries o f the zone o f aggregation a p p ea r to be very sensitive to b o th changes in the fish speed a n d social interaction. Indeed, social individuals w ould b e closer to the FA D th an non-social fish, im plying a n am plification o f the individual response to the FA D attractio n in a social group [29]. M oreover, fish characterized by a high speed w ould have a w ider zone o f aggregation, signaling possible correlations am o n g the fish sw im m ing speed (or size) a n d their spatial distribution a ro u n d FADs. T his provides a unique explanation for the structure o f fish aggregations a ro u n d floating objects, w here sm aller species (or sm aller individuals), characterized by a sm aller sw im m ing speed, are found closer to FA Ds th an larger fish [19-21]. T his ap p ro ac h can be used to study the fish dynam ics o f o th er species a n d pred ict the occurrence o f different shells o f fish concentrations a ro u n d the FA D. A fine-tuned analysis o f the strength o f interactions w ould be req u ired to b e tte r assess the aggregation radius o f each species. Flowever, by fixing the sam e ratio am o n g all quantities a n d simply scaling all the param eters governing the dynam ics o f a factor, F, our m odel predicts low er a n d u p p e r b ou n d s for the zone o f aggregation o f social a n d non-social fish a t lO x F m a n d 60 x F m , respectively. By using a value o f 5 for F, a scale th a t could correspond to the individual sw im m ing speed a n d size o f a tuna, w ould result in an aggregation radius betw een 50 m a n d 300 m . T his low er b o u n d is close to the center o f mass for tu n a aggregations found in recent acoustic experim ents [20,21], suggesting a potentially strong social effect for tu n a aggregated to FADs. transm issions p e r second) was prog ram m ed to be different for each tagged fish a n d ranged betw een 1.43 a n d 1.16 i _ 1. T hese settings w ere optim al for the duratio n o f o u r experience, w ith a theoretical period o f life tags o f six days. D a ta processing involved two steps. First, the acoustic record o f each tag on each o f the five hydrophones was m anually p roofed using H T I ™ M ark T ags Software to exclude acoustic noise. Second, files w ere processed in H T I ™ Acoustic T a g pro g ram to track acoustic echoes. T his p rocedure used a hyperbolic algorithm to solve for the transm itter 3D position. In addition, a tim e-stam p was calculated so th a t the transm itter was referenced in b o th space a n d tim e. T h e accuracy o f the position in the horizontal plane ranged from 0.1 to 0.3 m in the m onitoring netw ork. In the vertical direction, the accuracy was low er (around 1.0 m). W e m easured c u rre n t through the A an d eraa R M C 9 Self R ecording C u rre n t M eter, w hich was fixed u n d e r the b o a t (at 5 m eters depth) in order to record the horizontal cu rren t speed a n d direction. T h e m inim um -current p eriod recorded by the cu rren t m eter occurred w hen the direction o f the tidal cu rre n t reversed (Fig. S3 in Supplem entary Inform ation), betw een 13:00 a n d 14:00. Fish S p e c ie s a n d T a g g i n g p r o c e d u r e T h e fish species we studied was the big-eye scad (Selar crumenophthalmus). It is a small coastal pelagic fish com m on in the circum tropical area [31] a n d is a n obligate schooler (i.e., unable to survive outside a fish school [32,33]), w hich is know n to associate aro u n d FADs [34]. In R eunion Island, Saint Paul’s bay is the m ain a rea w here this species is caught. H ere, traditional beach seiners target shoals o f bigeye scads aggregated aro u n d anchored FADs near the shore. Forty fish w ere caught using h a n d lines in Saint P aul’s bay, transported in baskets a n d m aintained in tanks a t the A quarium of R eunion Island during ten days for acclim ation. T hey w ere fed and treated w ith a solution o f m ethylene blue (from the first day) and copper sulfate to kill bacteria a n d to prevent the proliferation o f fungi. M ost fish showed only superficial wounds, w hich w ere caused by fishing (hook) or handling. T h e tagging operation was carried out on the 1st o f M ay 2003. Twelve fish w ere anesthetized w ith a solution of clove off [35]. T h e acoustic tags w ere im planted gastrically by ingurgitation, a tagging technique well suited for short-term experim ents [36]. T h e fish were held for two subsequent days in tanks to ensure fish survival a n d tag retention. N o further m ortality was observed in either tagged or untagged fish during this period. All fish (tagged a n d non-tagged) w ere released on the 3rd o f M ay 2003 at 12:00 in the proxim ity o f the boat, anchored in the nearby o f the fishing location. Based on visual observation, we could estim ate that fish im m ediately form ed a small school. All fish stayed w ithin the zone o f detection until 19:00, w ith very few excursions outside the range of detection. T hree fish stayed a t night, leaving the zone the subsequent m orning, while the others left aro u n d 19:00. F our fish m ade short visits during the second a n d third day o f the experim ent. M aterials and M ethods Ethic S t a t e m e n t T h e fish e x perim ental protocols w ere p e rm itte d u n d e r the A q u ariu m o f R e u n io n Island anim al care certificates delivered by the F ren ch V eterin ary M edicine D irectorate. Protocols w ere carried o u t w ith the au th o rity o f the N atio n al V e terin ary School o f N antes (France) validating a certificate o f train in g in anim al experim entation a n d a degree in experim ental surgery o n fish. E x p e rim e n ta l s e t t i n g T h e experim ent was conducted in open field o f a shallow-water region (17 m depth in average) in the center o f S aint P aul’s Bay in R eu n io n Island (South W estern In d ian O cean). T h e H T I ™ Acoustic T a g T racking System (M odel 290) was com posed o f five hydrophones connected b y cables to the Acoustic T a g Receivers system em bedded on the boat. T hese hydrophones surrounded the b o a t in a square o f approxim ately 100 m eters p e r side (see F ig.S l in Supplem entary Inform ation). T h e hydrophones w ere arran g ed a t the surface a n d n ear the sea b e d to allow for optim al reception. T h e cables connecting the hydrophones to the b o a t reached the sea bo tto m straight below the boat, constituting a vertical subm erged structure w hose position was taken as o u r FA D position (Fig.S2 in Supplem entary Inform ation). T h e H T I ™ acoustic tags (M odel 795) w ere 7 m m diam eter, 17 m m length a n d 1.5 g w eight in the w ater. T his w eight was less th an 0.1% o f the m ean fish weight. W e were therefore confident th a t the tags did n o t affect the buoyancy o f the fish [30]. T h e in situ test led us to choose a pulse duratio n o f 4 msec. In order to discrim inate fish, the repetition rate (num ber of PLoS ONE I w w w .p lo s o n e .o rg M e t h o d s for d a t a an alysis E xperim ental d a ta analysis co n ce n tra ted on one h o u r, betw een 13:00 a n d 14:00 o f the first day, w hen the c u rre n t was negligible a n d all o f the tagged fish w ere present. T his allow ed us to collect good statistics, w ith a b o u t 3600 positions for each o f the twelve tagged fish. D u rin g the rest o f the day, o u r results still held, b u t we observed a shift in the position o f the fish baricen ter, due to nonnegligible c u rre n t effects. D u e to the system geom etry, w here the floating object was associated to a subm erged vertical structure reach in g the sea-bottom (see Fig.Sl), d a ta analysis focused on the x y plane, integrating over the vertical direction. T h is ap p ro ac h was supported b y previous acoustic survey m easurem ents [19-21] w ere the fish spatial distribution along z was n o t affected by the presence o f the FA D b u t ra th e r d e p en d e d on the fish species. T h e 5 D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics presence o f each fish i a ro u n d the FA D was analyzed th ro u g h the tim e-averaged radial distribution function: S ( R - R ,( t) ) P ,(R ) = 2nR dR w here the delta function selects the fish positions R ¡(t) a t tim e t with radial distance from the FA D w ithin the interval [R — d R ,R \ [37], a n d brackets denote the tim e average. T h e d en o m in ato r co rre­ sponds to the a rea o f the rin g o f radius R a n d w idth d R a ro u n d a FA D . W ith this d enom inator, the q uantity P ¡(R ) was norm alized: d V ( { x j,y j } ) J P i(R ) 2 n R d R = 1 Fy = T herefore, P ¡(R ) could be interpreted as the probability o f the presence o f fish i a t distance R from the FA D , p e r un it a rea .T h e strength o f o u r ap p ro ac h resided in the high n u m b e r o f sam pled points, w hich clearly allow ed us to speak in term s o f probabilities. W e chose d R = 0.3 m , w hich was com patible w ith the experim ental precision for fish detection in the x y plane. Because we h a d d etailed spatial inform ation concerning several fish, w e calculated the fish pair-correlation function (or pair-distribution function) g(r) am ong synchronous fish [24], T his gave us a n understan d in g o f fish interactions a n d their variations in space. W e took synchronicity intervals o f 1 second because this tim e fram e was sufficient to obtain a large n u m b e r o f synchronous fish. W e calculated the tim eaveraged fish pair-correlation function g(r), w hich in two dim ensions can be w ritten as: ^ ¿ ( r - r ÿ(0) g(r)= N (t) right h a n d side constituted the sta n d ard C R W dynam ics, w ith v b ein g a c o n stan t defining the individual sw im m ing speed. C o n cern in g the a n g u lar com ponent, a>t co rresp o n d ed to the fish o rien tatio n angle in o u r reference fram e a t tim e Í a n d 6 was a ra n d o m n u m b e r taken from a probability distribution ® (0) th a t sets the tu rn in g angle. T his p robability distribution, as well as the co n stan t for the individual sw im m ing speed, was taken from experim ents. In particular, v c o rresponded to the sta n d ard rule o f 1 body length p e r second a n d ® (0) follow ed a w ra p p ed C auchy distribution. T h e last term s in E q.3 constituted the determ inistic dV^Xuyd} p a rt o f the fish dynam ics, w here F x ( t ) = ---------- —-—-— 2 nrdr the X (y) co m p o n e n t of the determ inistic force associated w ith a poten tial V { { x j,y j} ) a t tim e i, d ep en d in g o n fish positions. W e considered additivity in the forces. O u r poten tial h a d the form: V(xi,yi) = Vfad (R í) + J 2 vM rv) v w here the first term was the FA D potential, a n d the second term set the fish-fish interaction, w ith R¡ = \ J ( x fad — Xi)2 + (yFAD — y i)1 bein g the radial distance am o n g the FA D a n d fish i a n d ry = \ J (xj — x¡)2 + (jy —y i f bein g the distance am o n g fish i a n d fish j . T h e analytic form s o f these potentials w ere derived from the experim ental radial distribution P (R ) a n d the p a ir correlation function g(r) [39]. T h e optim ized m odel p a ram ete rs are show n in T ab le 1. Supporting Information w here N ( t ) is the n u m b e r o f co p lan ar pairs detected in the tem poral interval [ í , í + l í ] a n d A /(0 is the p lan a r distance am ong synchronous fish i a n d j . C o planarity was established w hen two fish w ere w ithin 1 m in the z-direction, w hich was com patible with o u r experim ental accuracy in the vertical direction. T h e delta function selects fish pairs a t p lan a r distance in the range [r — d r,r\, w ith dr = 0.3 m , a n d brackets d enote the tim e average. F ig u re S I T h e H T I ™ e x perim ental setting. T h e boat, w ith the cables u n d e rn ea th , represents the floating object or ‘F A D ’. (TIF) M o d e l d efin itio n F ig u re S3 C u rre n t speed (red line) in c m / s a n d c u rre n t angle (blue line), w ith respect to the N o rth (East = 90; W est = 270) re co rd e d d u rin g the experim ent. (TIF) L ag ran g ian dynam ics [37,38] are c h aracterized by the use o f stochastic differential equations th a t describe the evolution o f the positions o f each individual in tim e. T h e system evolves u n d e r the effect o f b o th determ inistic forces a n d a ra n d o m com ponent. Flere, we used a v a ria n t o f the L ag ran g ian dynam ics, the co rrelated ra n d o m w alk (CRW ) [27,28], in the presence o f a force field. In the C R W m odel, a n anim al m akes discrete steps, w ith tu rn in g angles sam pled from a given p robability distribution. A t each step, the tu rn in g angle is in d ep e n d en t o f the previous one. Flere, in ad dition to isolated-fish C R W dynam ics, fish m ovem ent was influenced determ inistically. T h e tim e evolution o f the position o f fish i in the plane follow ed the equations: X i(t + A í) = x,-(i) + v e o s(cot + 0 )A t + A tFx (t) y¡(t + A í) = y¡(t) + v sinfuy + 0)A í + A tFy (t) (3) w here x,-(i) a n d y ¡(i) are the x a n d y co m p o n e n t o f the position o f fish i a t tim e Í, a n d A í was the tim e step. T h e first term s on the PLoS ONE I w w w .p lo s o n e .o rg F ig u re S2 T rajectories o f the tracked fish in the x y plane a ro u n d the FA D from 13:00 to 14:00 D ifferent colors indicate different fish a n d black p o in t indicates the FA D position. (TIF) F ig u re S4 Sw im m ing speed distribution (A), tu rn in g angle distribution (B) a n d pair-co rrelatio n function (C), calculated at different radial distances from the FA D . (TIF) A ck n ow led gm en ts W e thank G. Potin, M . T aquet, L. Bigot, P. Durville, M . Tim ko, G. Fritsh for their help in fishing an d tagging operations an d P. Fréon, F. Becca, O. Zagordi and G. Nicolis for interesting discussions. Author C ontributions C o n c eiv ed a n d d esig n ed the ex p erim en ts: M S . P e rfo rm e d the experim ents: M S PC LD. A nalyzed the data: M C M S. C o n trib u ted reag en ts/m aterials/an a ly sis tools: J-L D LD . W rote the paper: LD M C MS J-L D . D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9 M o d e lin g A g g re g a te d -F is h D yn a m ics R eferences 1. P ik itc h F , S a n to ra C . B a b c o c k F , B a k u n A , B o n fil R , e t al. (2004) E co sy stem b a s e d fish e ry m a n - a g e m e n t. S c ien c e 305: 3 4 6 —347. 2. F r é o n P. M is u n d O (1999) D y n a m ic s o f p e la g ic fish d istrib u tio n a n d b e h a v io u r: effects o n fish eries a n d sto ck assessm en t. B lackw ell S c ien c e , O x fo rd . 34 8 p. 3. P a rris h J , H a m n e r W (1997) A n im a l g ro u p s in th re e d im e n sio n s. C a m b rid g e U n iv . P ress, C a m - b rid g e . 33 6 p. 4. K r a u s e J , R u x to n G (2002) L iv in g in g ro u p s. O x fo rd U n iv e rsity P ress, U S A . 2 24 21. 22. 23. P5. S u m p te r D (2010) C o lle c tiv e A n im a l B e h a v io r P rin c e to n U n iv e rsity Press. P rin c e to n U n iv e rsity Press. 31 2 p. 6. P a rris h J , E d e lste in -K e s h e t L (1999) C o m p le x ity , p a tte rn , a n d e v o lu tio n a ry tra d e -o ffs in a n im a l a g g re g a tio n . S c ien c e 284: 9 9 —101. 7. C a v a g n a A , C im a re lli A , G ia rd in a I, P a risi G , S a n ta g a ti R , e t al. (2010) S cale­ 24. b io sc ie n ce s 214: 3 2 -3 7 . C h a n d le r D {1987) I n tr o d u c tio n to m o d e r n sta tistic a l m e c h a n ic s O x fo rd U n iv e rsity Press. 28 8 p. 26. G ir a r d C , D a g o r n L, T a q u e t M , A u m e e ru d d y R , P e ig n o n C , e t al. {2007) H o m in g ab ilities o f d o lp h in fis h { c o ry p h a e n a h ip p u ru s) d isp la c e d f ro m fish a g g re g a tin g d ev ices {fads) d e te r m in e d u s in g u ltra so n ic tele m e try . A q u a tic L iving R e s o u rc e s 20: 3 13—321. 27. K a re iv a P , S h ig e sa d a N {1983) A n a ly z in g in se c t m o v e m e n t as a c o rre la te d r a n d o m w alk. O e c o lo g ia 56: 234—238. 28. M c C u llo c h C , C a in M {1989) A n a ly z in g d isc re te m o v e m e n t d a ta as a c o rre la te d r a n d o m w alk. E co lo g y , p p 38 3 —388. 29. S u m p te r D , K r a u s e J , J a m e s R , C o u z in I, W a r d A {2008) C o n se n su s de c isio n m a k in g b y fish. C u r r e n t B iology 18: 1 7 7 3 -1 7 7 7 . 30. A lm e id a P , Q u in te lla B, C o s ta M , M o o re A {2007) D e v e lo p m e n ts in Fish T e le m e try : P ro c e e d in g s o f th e 6 th C o n fe re n c e o n F ish T e le m e try H e ld in E u ro p e . S p rin g e r-V e rla g N e w Y o rk . 30 0 p. 31. R o o s D , R o u x O , C o n a n d F {2007) N o te s o n th e b io lo g y o f th e b ig ey e sc ad , selar c ru m e n o p h th a lm u s {caran g id ae) a ro u n d re u n io n isla n d , s o u th w e st in d ia n o c e an . S c ien tia M a rin a 71: 1 3 7 -1 4 4 . 32. B re d e r C {1959) S tu d ies o n so cial g ro u p in g s in fishes. A m e r M u s N a t H is t 117: 3 9 7 —481. 33. S o ria M , F r é o n P , C h a b a n e t P {2007) S c h o o lin g p ro p e rtie s o f a n o b lig a te a n d a fa c u lta tiv e fish species. J o u r n a l o f F ish B io lo g y 71: 1257—1269. 34. T a q u e t M , S a n c h o G , D a g o r n L , G a e r tn e r J , I ta n o D , e t al. {2007) C h a r a c te riz in g fish c o m m u n itie s a sso c ia te d w ith d riftin g fish a g g re g a tin g devices {fads) in th e w e s te rn in d ia n o c e a n u sin g u n d e rw a te r v isu a l surveys. A q u a tic L iv in g R e s o u rc e s 20: 3 3 1 -3 4 1 . 35. D u rv ille P , C o lle t A {2001) C lo v e oil u se d as a n a n a e s th e tic w ith ju v e n ile tro p ic a l m a r in e fish. S P C L iv e R e e f F ish In fo r m a tio n B u lle tin o f M a rin e S c ien c e 9: 1 7 -1 9 . 36. M a rtin e lli T L , H a n s e l H C , Shively R S {1998) G ro w th a n d p h y sio lo g ic a l resp o n se s to su rg ic a l a n d g a stric ra d io tra n s m itte r im p la n ta tio n te c h n iq u e in s u b y e a rlin g c h in o o k sa lm o n { o n c o rh y n c h u s tsh aw y tsch a). H y d ro b io lo g ia 3 7 1 / 372: 7 9 -8 7 . 37. G r u n b a u m D , O k u b o A {1994) M o d e llin g so cial a n im a l a g g re g a tio n s. F ro n tie rs in th e o re tic a l Biology: L e c tu re n o te s in b io m a th e m a tic s 100: 2 9 6 —325. 38. N iw a H {1994) S e lf-o rg a n iz in g d y n a m ic m o d e l o f fish sch o o lin g . J T h e o r B iol 25. f re e c o rre la tio n s in s ta rlin g flocks. P ro c e e d in g s o f th e N a tio n a l A c a d e m y o f S ciences 107: 1 1 8 6 5 -1 1 8 7 0 . 8. L u k e m a n R , L i Y , E d e lste in -K e s h e t L (2010) In fe rrin g in d iv id u a l ru les fro m collective b e h a v io r. P ro c e e d in g s o f th e N a tio n a l A c a d e m y o f S cien ces 107: 9. 1 2 5 7 6 -1 2 5 8 0 . S c h e llin ck J , W h ite T (2011) A rev ie w o f a ttr a c tio n a n d re p u lsio n m o d els o f a g g re g a tio n : M e th o d s , fin d in g s a n d a d isc u ssio n o f m o d e l v a lid a tio n . E c o lo g ic a l 10. M o d e llin g 222: 1 8 9 7 -1 9 1 1 . C o u z in I, K r a u s e J , F ra n k s N , L e v in S (2005) E ffectiv e le a d e rsh ip a n d d e c isio n ­ m a k in g in a n im a l g ro u p s o n th e m o v e. N a tu r e 4 3 3 : 5 1 3 —51 6 . 11. 12. 13. 14. H e m e lrijk C K , H ild e n b r a n d t H (2008) S e lf-o rg a n iz ed s h a p e a n d f ro n ta l d e n sity o f f i s h schools. E th o lo g y 114: 2 4 5 -2 5 4 . H e m e lrijk C , K u n z H (2005) D e n s ity d istrib u tio n a n d size s o rtin g in fish schools: a n in d iv id u a l- b a s e d m o d el. B e h a v io ra l E c o lo g y 16: 178—187. L u k e m a n R , L i Y , E d e lste in -K e s h e t L (2009) A c o n c e p tu a l m o d e l fo r m illin g fo rm a tio n s in b io lo g ic al a g g re g a tes. B u lle tin o f m a th e m a tic a l b io lo g y 71: 3 5 2 -3 8 2 . K a tz Y , T u n s tro m K , I o a n n o u C , H u e p e C , C o u z in I (2011) In fe rrin g th e s tru c tu re a n d d y n a m ic s o f in te ra c tio n s in sc h o o lin g fish. P ro c e e d in g s o f th e N a tio n a l A c a d e m y o f Sciences;{in press). 15. T y le r J , R o s e K (1994) I n d iv id u a l v a ria b ility a n d sp a tia l h e te ro g e n e ity in fish p o p u la tio n m odels. R e v ie w s in F ish B iology a n d F ish eries 4: 9 1 —123. 16. D e m p s te r T , T a q u e t M (2004) F ish a g g re g a tio n d ev ice (fad) re s e a rc h : g a p s in c u r r e n t k n o w le d g e a n d f u tu re d ire c tio n s fo r ec o lo g ic al stu d ies. R e v ie w s in Fish 17. B iology a n d F ish eries 14: 2 1 -4 2 . F r é o n P , D a g o r n L (2000) R e v ie w o f fish a sso ciativ e b e h a v io u r: to w a rd a g e n e ra lis a tio n o f th e m e e tin g p o in t h y p o th esis. R e v ie w s in F ish B io lo g y a n d 18. F ish eries 10: 1 8 3 -2 0 7 . M a rs a c F , F o n te n e a u A , M é n a r d F {2001) D riftin g fad s u s e d in tu n a fisheries: a n ec o lo g ic al tra p ? A C T E S D E C O L L O Q U E S - I F R E M E R 28. 19. J o s s e E , D a g o r n L , B e r tr a n d A (2000) T y p o lo g y a n d b e h a v io u r o f tu n a a g g re g a tio n s a ro u n d fish a g g re g a tin g dev ices f ro m a c o u stic surveys in f re n c h p o ly n esia . A q u a tic L iv in g R e s o u rc e s 13: 183—192. 20. D o r a y M , J o s s e E , G e r v a in P , R e y n a l L , C h a n t r e l J (2 0 0 6 ) A c o u s tic c h a ra c te ris a tio n o f p e la g ic fish a g g re g a tio n s a ro u n d m o o re d fish a g g re g a tin g 39. d evices in m a r tin iq u e {lesser antilles). F ish eries re s e a rc h 82: 162—175. PLoS ONE I w w w .p lo s o n e .o rg M o re n o G , J o s s e E , B re h m e r P , N o tte s ta d L {2007) E c h o tra c e c la ssification a n d s p a tia l d istrib u - tio n o f p e la g ic fish a g g re g a tio n s a ro u n d d riftin g fish a g g re g a tin g d ev ices {dfad). A q u a tic L iv in g R e s o u rc e s 20: 3 4 3 —356. S o ria M , D a g o r n L , P o tin G , F r é o n P {2009) F irs t fie ld -b a se d e x p e rim e n t s u p p o rtin g th e m e e tin g p o in t h y p o th esis fo r sc h o o lin g in p e la g ic fish. A n im a l B e h a v io u r 78: 1 4 4 1 -1 4 4 6 . G a u tra is J , J o s t C , S o ria M , C a m p o A , M o ts c h S, e t al. {2009) A n a ly z in g fish m o v e m e n t as a p e rs is te n t t u rn in g w a lk e r. J o u r n a l o f m a th e m a tic a l b io lo g y 58: 4 2 9 —445. C a v a g n a A , C im a re lli A , G ia rd in a I, O r la n d i A , P a risi G , e t al. {2008) N e w sta tistic a l to o ls fo r a n a ly z in g th e s tru c tu re o f a n im a l g ro u p s. M a th e m a tic a l 7 171: 1 2 3 -1 3 6 . K irk w o o d J {1935) S ta tistica l m e c h a n ic s o f flu id m ix tu re s. T h e J o u r n a l o f C h e m ic a l P h y sics 3: 3 0 0 —313. D e c e m b e r 2011 | V o lu m e 6 | Issue 12 | e 2 8 1 0 9