Journal of Theoretical Biology 245 - Paulo R. Guimarães Jr.

ARTICLE IN PRESSJournal of Theoretical Biology 245 (2007) 784–789Letter to Editorwww.elsevier.com/locate/yjtbiInvestigating small fish schools: Selection of school—formation models by means of general linear models andnumerical simulations1. IntroductionThe aggregation of organisms is a ubiquitous biologicalphenomenon that occurs in a wide range of species, frombacteria to large vertebrates, and across a range oftemporal stability, from ephemeral reproductive assemblagesto obligatory schools or herds (Allee, 1933). Ingeneral, the processes that lead to the aggregation ofindividuals are driven by a complex interplay betweenhabitat characteristics and individual behaviour. Toillustrate the importance of individual behaviour, we notethat a number of studies have suggested that importantfeatures of aggregations, such as size, shape, and architecture,emerge as a consequence of simple behaviouralrules that govern the interactions among individuals(Parrish and Edelstein-Keshet, 1999; Niwa, 2005). Understandingaggregation therefore always entails identifyingthe behavioural rules that generate the observed patterns inanimal aggregations.Aggregation size, usually defined as the number ofindividuals that are part of an aggregation, has receivedconsiderable attention, especially in the characterizationand prediction of the observed variation in school size infish species (Bonabeau et al., 1999; Bonabeau and Dagorn,1995; Niwa, 1996, 1998, 2003, 2004). Most of these studieshave noted that the probability of finding a large schooldecays as flat-tailed distributions such as power-laws andpower-laws with an exponential truncation for large schoolsize (Bonabeau and Dagorn, 1995), although exponentialdistributions have also been recorded (see Niwa, 2003).Flat-tailed distributions are relevant because they characterizestatistical properties associated with differentecological systems such as mutualistic networks (Jordanoet al., 2003) and ecological communities (Hubbell, 2001).The characterization of school-size distributions aspower-laws, truncated power-laws or exponentials oftenfocuses on fish species such as the tuna (e.g. Niwa, 2003),for which schools are very large (of the order of severalhundred individuals), thereby allowing a precise statisticalcharacterization of school-size distribution. However,many fish species do not form large schools. On thecontrary, some fishes aggregate forming small schools, inwhich the number of individuals is seldom larger than a fewdozens. In such cases, the characterization of school-sizedistribution lacks statistical confidence and, as a consequence,the study of the organizing principles of schoolformation has been limited to fishes that aggregate in largeschools (e.g. Bonabeau and Dagorn, 1995; Niwa, 2003;Powers, 2004). Statistical procedures that allow the studyof fish schools with few individuals will allow theinvestigation of the organization of schools of many fishspecies and will be also important in a broader sense. Asrecently noted by Amaral et al. (2004) in the context ofweb-like systems, the characterization of the structure ofnatural systems containing small number of elements is oneof the main challenges in the study of complex systems (seealso Guimara˜es et al., 2005).Here we explore a possible solution for the characterizationof low dimensional systems using as reference systemthe Southwest Atlantic parrotfish Sparisoma axillare(Perciformes: Scaridae), which forms small schools atfeeding sites. We combine general linear models andsimulations to describe the functional form of school-sizedistribution and select between two recently proposedschool-formation models (Powers, 2004). We demonstratedthat the proposed characterization allows the selectionamong candidate models even for species in which schoolsizes are often small.2. Material and methods2.1. Reference systemThe study site consisted of sandy areas adjacent to therocky shore of the Praia da Conceic-a˜o, at Fernando deNoronha archipelago ð03 50 0 S; 32 25 0 WÞ, about 345 km offnortheast Brazil (see Carleton and Olson, 1999 fordescription of the archipelago), from May to June 2003.During the study, horizontal visibility was 20 m, whichallowed adequate observation of fish schools. The watertemperature was 27 C and the depth of the study siteranged 2–7 m. The observations were done by divingduring daylight hours, from 0900 to 1700 h in five nonconsecutivedays while snorkelling. No nocturnal observationswere made since parrotfishes are inactive at night(Starck and Davis, 1966).Parrotfishes are bottom-foragers, feeding mostly onalgae and dead coral (Bruggemann et al., 1994a, b). Theparrotfish S. axillare is endemic to the tropical and0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2006.12.001

ARTICLE IN PRESSLetter to Editor / Journal of Theoretical Biology 245 (2007) 784–789 785subtropical Southwestern Atlantic (Moura et al., 2001).Initial phase individuals of S. axillare forage alone or inschools (Bonaldo et al., 2006). At the studied site, the initialphase individuals foraged along the rocky shore and onsand flats. In sand flats, S. axillare gather to forage on thesparse rocks and form near motionless schools (Bonaldoand Krajewski, unpublished results). We assessed theparrotfishes distribution at Praia da Conceic-a˜o with 16transect lines (5 m 100 m) parallel to the rocky shore,where we registered all the numbers of initial phaseindividual S. axillare in each group. The good observationconditions and the spatial homogeneity of sand flatssuggest that our sample is representative. During ourfieldwork, the schools observed were often small, thelargest having up to 20 fish.2.2. Characterizing the school-size distributionThe number of schools recorded for each school sizeformed the school-size distribution. In these analyses,solitary fish were recorded as 1-fish schools. We used theformalism of general linear models (McCulloch and Searle,2000) to assess whether the observed school-size distributionof S. axillare could be characterized as an exponentialor a power-law. Initially, we recall that both exponentials,y / e b 1k , and power-laws, y / k b 2, can be written as linearmodels, respectively, as logðyÞ ¼a 1 þ b 1 k þ e and logðyÞ ¼a 2 þ b 2 log k þ e, where a 1 and a 2 are the fitted normalizationconstants, b 1 , and b 2 are the fitted slope constants,k is the school-size, y is the number of recorded schoolswith a given school-size (kÞ, and e is the error. If theobserved school-size distribution is described by anexponential, we would expect b 1 k to be significant, whereasif a power-law adequately describes the data set, we wouldexpect b 2 log k to be significant. If both b 1 k and b 2 log kare statistically significant then school-size distribution canbe described by either an exponential or a power-law. Ifneither term is significant, the observed distribution cannotbe described by either an exponential or a power-law.2.3. School-formation modelsRecently, different types of splitting/coalescing modelswere proposed to explain school-size distributions (Bonabeauet al., 1999; Bonabeau and Dagorn, 1995; Niwa 1996,1998, 2003). However, the assumptions of splitting andcoalescing do not adequately describe the natural history ofS. axillare because these fishes aggregate at feeding siteswhere they form small groups that may remain nearlymotionless and seldom coalesce with other schools or splitinto new schools (Bonaldo, unpublished results). Singleindividuals may join schools, suggesting that schools mayincrease in size during the feeding period (morning toafternoon). In fact, larger schools were recorded atafternoon (Bonaldo et al., 2006). The two models usedhere are adapted versions of the recently proposed modelsinspired in complex networks (Powers, 2004). The twomodels used here assume that: (1) at t ¼ 0 there was asingle 1-fish school in the sand flats; (2) at each timeinterval, a fish moved from the rocky shore to the sandflats; (3) this new individual in the sand flats had a constantprobability, 1 p, of joining a school already present, orremained alone, creating a new 1-fish school with probabilityp; (4) if an individual joined a school already in thearea, it could choose among any of the fish schoolspresent—this assumption is supported by the spatialhomogeneity of sand flats; (5) individuals did not leavethe schools until the end of the simulation, and (6)the simulation ends when the number of fishes in thesand flats is equal to the observed number of fishes in thesand flats. The only difference between the two models washow a fish chooses among the schools already present: thefirst model, referred to as the random (R) model, is basedon the assumption that, if a fish joined a previouslyestablished school, the probability of it choosing the ithschool isP ¼ 1 N . (1)In this equation, N is the number of previously establishedschools and P is independent of school size. In contrast,the second model, referred to as the preferential attachment(PA) model, is based on the assumption that, if thefish joined a school, the probability of its choosing amongone of the already existing schools is not uniform. Rather,fish have a tendency to preferentially join larger schools.The tendency of large schools to be more attractive tosolitary individuals can be modelled using a conceptderived from network theory, the PA rule (Baraba´si andAlbert, 1999), in which the probability P that a fishjoins school i depends on school size, k i , in a linearfunctional formPðk i Þ¼k iPj k . (2)jIn this equation, P j k jis the sum of the sizes of all schoolsalready present.The few differences between our models and those ofPowers (2004) are related to particulars of the naturalhistory of S. axillare. These differences include theassumptions that: (i) there is no recruitment or deathduring simulation. This assumption is reasonable becauseschools of S. axillare are temporary since they form atfeeding sites (rocks on sand flats) and (ii) only 1-fishschools aggregate to previous schools. This assumption isbased on naturalistic observations that indicate thatschools seldom aggregate, but solitary fish usually joinschools and remain in the school while feeding (Bonaldo,unpublished results). It is also important to notice that thePA model is similar to the well-known Simon model ofgrowing aggregations, which has been used to studyabiotic, biotic and social systems (see Albert and Baraba´si,2002 for further discussion).

786ARTICLE IN PRESSLetter to Editor / Journal of Theoretical Biology 245 (2007) 784–7892.4. School-formation model selection and numericalsimulationsThe R and PA models differ in their asymptoticbehaviours: the R model generates exponential distributions,in which all schools have similar sizes, whereas thePA model generates power-law distributions, in which agreat proportion of individuals are part of a few largeschools and the majority of schools are very small(Baraba´si and Albert, 1999; Powers, 2004). In contrast, insystems with a small number of component elements, suchas that analysed here, characterization of the distributionmay not be so easy (Guimara˜es et al., 2005). Indeed, wefound that the school-size distributions generated by bothmodels at small-size situations can be described by powerlaws(results not shown). To circumvent this problem, weused the estimated slope constant, b i , of the significantgeneral linear model rather than the qualitative (‘‘exponential’’or ‘‘power-law’’) description of the functionalform, to select between the two models of fish aggregation.The ability of the models to predict the observed value ofthe slope constant was examined by using simulations togenerate the empirical distribution of b i for each model(4000 simulations). We considered that the model reproducedthe field data if the observed value lies within the95% confidence interval of the empirical distribution.3. Results and discussionNeotropical parrotfish S. axillare at Fernando deNoronha archipelago often form small schools (often o20 individuals). We sampled 65 schools of S. axillare,which corresponded to a total of 124 individuals ( ¯X SD,¯k ¼ 1:9 3:0 individuals per school). The value of ^p wasdirectly estimated from field data, simply by dividing thenumber of schools recorded minus one (N 1 ¼ 64Þ, bythe number of fish recorded minus one ðn 1 ¼ 123Þ,which yielded ^p ¼ 0:52. To remove first order bias of ^p weused jackknife procedures (Davison and Hinkley, 1997),and simulations are performed using the corrected value^p ¼ 0:50. The exponential function did not adequatelydescribe the observed school-size distribution of S. axillare(d:f : ¼ 6; R 2 ¼ 0:47; F ¼ 4:46; P ¼ 0:1Þ. In contrast, thepower law distribution generated a good description ofa0.20.180.160.14Frequency0.120.10.080.060.040.02bFrequency00.160.140.120.10.080.060.040.020-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7β 2-2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1 -0.8β 2Fig. 1. The empirical distribution of b 2 generated by: (a) the R model and (b) the PA model. The dashed lines indicate the lower and upper limits of the95% confidence interval. The black arrow indicates the observed value for Sparisoma axillare.

ARTICLE IN PRESSLetter to Editor / Journal of Theoretical Biology 245 (2007) 784–789 787the school-size distribution of this species. The estimated The solution of the above equation isqk iqt ¼ m k iP N 1j¼1 k ¼ m k ij2mt ¼ k analytic prediction resulted from the small number of fishes.i2t . (3) These results reinforce the importance of the introducedapproach for the study of small complex systems.value of the slope constant was b 2 ¼ 1:21 and the linear model log y ¼ 2:99 1:21 log k was highly significant k i ¼ mt a(d:f : ¼ 6; R 2 ¼ 0:80; F ¼ 20:25, Po0:01Þ. Therefore, wet i(4)used b 2 in the procedure of model selection.in which a ¼ 1Although the schools are small, the organization of these2. The predicted exponent for the probabilityof a node showing k links derived from (4) is bschools is not random. The average b 2 predicted by the R2 ¼1=a 1 ¼ 3 (Albert and Baraba´si, 2002). In the PAmodel was lower than the observed b 2 ¼ 1:21 formodel for fish schools, the probability that the size of theS. axillare (Fig. 1a). In contrast, the empirical distributionschool i increases with time is slightly different from (3),generated by the PA model reproduced the observed b 2 depending on pfor S. axillare (Fig. 1b). Therefore, our analysis suggeststhat the model that assumes that fishes preferentially join qk ilarger schools reproduces the features of the school-size qt ¼ð1 pÞ k iP N 1j¼1distribution.¼ð1 pÞ k ijt . (5)We also note that the exponent generated by the Therefore, a ¼ð1 pÞ and the predicted exponent isnumerical simulations of the PA model is very different b 2 ¼½ 1=ð1 pÞŠ 1. In the case of the studied parrotfishes,from the analytic prediction of the original network model p ¼ 0:5 and, consequently, the above analytic approach(Baraba´si and Albert, 1999). In complex networks, PA predicts b 2 ¼ 3. Additional numerical simulations demonstratepredicts that the degree (kÞ of the node i varies with timefollowing the dynamical equation:that the analytic predictions are valid for large systems(Fig. 2). Therefore, the differences between the result ofnumerical simulations based on the reference system and theaSchool-size1 10 100 1000 100000.1Frequency0.010.001b0.00011School-size1 10 100 1000 100000.1Frequency0.010.0010.0001Fig. 2. (a) Agreement between numerical simulations and analytic predictions (lines) for large systems (n ¼ 10 000 fishes) for preferential attachmentmodel. School-size distributions were plotted as cumulative distributions to reduce finite-size fluctuations (Strogatz 2001): p ¼ 0:10 (circles), p ¼ 0:25(squares), and p ¼ 0:5 (triangles); (b) school-size distributions generated by numerical simulations using random attachment are plotted for comparisons.

ARTICLE IN PRESSLetter to Editor / Journal of Theoretical Biology 245 (2007) 784–789 789Niwa, H.S., 2004. Space-irrelevant scaling law for fish school sizes.J. Theor. Biol. 228, 347–357.Niwa, H.S., 2005. Power-law scaling in dimension-to-biomass relationshipof fish schools. J. Theor. Biol. 235, 419–430.Parrish, J.K., Edelstein-Keshet, L., 1999. Complexity, pattern, andevolutionary trade-offs in animal aggregation. Science 284,99–101.Powers, J.E., 2004. Recruitment as an evolving random process ofaggregation and mortality. Fish. Bull. 102, 349–365.Starck, A.S., Davis, W.P., 1966. Night habits of fishes of Alligator Reef,Florida. Ichthyologica. 38, 315–356.Strogatz, S.H., 2001. Exploring complex networks. Nature 410,268–276.Paulo. R. Guimara˜es Jr.Departamento de Física da Matéria Condensada,Instituto de Física Gleb Wataghin,UNICAMP, 13083-970, Campinas, SP, BrazilandIntegrative Ecology Group, Estación Biológica de Doñana,CSIC, Apdo. 1056, E–41080 Sevilla, SpainE-mail address: prguima@ifi.unicamp.brRoberta M. BonaldoJoa˜o P. KrajewskiPrograma de Pós-Graduac-ão em Ecologia,Instituto de Biologia, UNICAMP, 13083–970,Campinas, SP, BrazilE-mail addresses: robertabonaldo@yahoo.com(R.M. Bonaldo),jpaulokra@yahoo.com.br (J.P. Krajewski)Paulo Guimara˜esDaitan Labs, Galleria Office,Bloco 4, cj 444, Campinas, SP, BrazilE-mail address: prguima.pm@gmail.comAluı´sio PinheiroDepartamento de Estatística, Instituto de Matemática,Estatística e Ciência da Computac-ão, UNICAMP,13083–970, Campinas, SP, BrazilE-mail address: pinheiro@ime.unicamp.brJoseph Powers2147 Energy, Coast and the Environment Building,School of Coast and Environment,Louisiana State University, Baton Rouge,LA 70803, USAE-mail address: jepowers@lsu.eduSe´rgio Furtado dos ReisDepartamento de Parasitologia, Instituto de Biologia,UNICAMP, 13083–970, Campinas, SP, BrazilE-mail address: sfreis@unicamp.br