Mathematical model of fish schooling behaviour in a set-net

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1 ICES Journal of Marne Scence, 61: 114e13 (004) do: /j.cesjms Mathematcal model of fsh schoolng behavour n a set-net Tsutomu Takag, Yutaka Mortom, Jyun Iwata, Hrosh Nakamne, and Nobuo Sannomya Takag, T., Mortom, Y., Iwata, J., Nakamne, H., and Sannomya, N Mathematcal model of fsh schoolng behavour n a set-net. e ICES Journal of Marne Scence, 61: 114e13. We nvestgated the valdty of a mathematcal model to descrbe fsh schoolng behavour towards a smple set-net model. We apply a model consdered to be an autonomous decentralzed system and based on Newton s equaton of moton. It ncludes the parameter M, whch ndcates the quantty of nformaton exchange (.e. the number of neghbours that affect an ndvdual s behavour) and strongly affects fsh school sze and schoolng behavour n an enclosed space. To evaluate the model, smulatons of fsh schoolng behavour n a set-net model consstng of a leadng fence and a box-shaped trap smlar to a prmtve type of set-net were compared wth expermentally observed behavour of btterlng and mackerel, wth a focus on M. A small M nduces mproper behavour because there s low cooperaton among fsh n a school. On the other hand, f M s too large, mproper smulaton results of ndvduals n deadlock states n the trap are obtaned as a result of excessve nformaton exchange among the fsh. The results suggest that the mathematcal model can descrbe the behavour n a set-net model adequately when M s greater than and less than 10. Ó 004 Internatonal Councl for the Exploraton of the Sea. Publshed by Elsever Ltd. All rghts reserved. Keywords: fsh schoolng behavour, mathematcal model, quantty of nformaton exchange, set-net. Receved 15 March 003; accepted 1 May 004. T. Takag and J. Iwata: Department of Fsheres, Knk Unversty, Naka-mach Nara, Japan. Y. Mortom and H. Nakamne: Kyoto Unversty of Educaton, Fushm-ku, Kyoto, Japan. N. Sannomya: Kyoto Insttute of Technology, Sakyo-ku, Kyoto, Japan. Correspondence to T. Takag: tel: C , fax: C ; e-mal: takag@nara.knda.ac.jp. Introducton The behavour of an ndvdual fsh swmmng alone can be descrbed as not well ordered, whereas a school of fsh conssts of ndvduals that move n concert. Models that smulate the behavour of a school of fsh are useful for explorng how ndvduals affect overall school behavour and what cooperatve behavour s nvolved. Aok (198) developed a stochastc model of a school of fsh that was based on an ndvdual fsh model, and the behavour of each ndvdual affected that of ts neghbours. In the model, the orentaton and velocty of ndvduals were defned by the probablty densty functon of a normal dstrbuton and Gamma dstrbutons, respectvely. Smulatons usng the model showed that attracton to neghbours and parallel orentaton behavour are mportant factors n the organzaton of a fsh school. Aok s model was the bass for other stochastc models proposed subsequently (Huth and Wssel, 199, 1994; Reuter and Brecklng, 1994; Inada and Kawach, 00). Huth and Wssel (199) developed a fsh school model based on Aok s model that refned ndvduals orentaton decson rule, and the coheson and polarzaton smulated by ther model agreed wth expermental results (Huth and Wssel, 1994). Inada and Kawach (00) proposed a stochastc fsh school behavour model that could smulate the response to a predator, and demonstrated that the order and flexblty of a fsh school are affected by the number of fsh consdered neghbours and the randomness of each ndvdual. These smulatons of fsh school behavour have provded a better understandng of the bass of the behavour mechansm n fsh schoolng and valuable nformaton for fsheres scence. The smulaton results of Huse et al. (00), usng bods model (ndvdual-based) (Reynolds, 1987), demonstrated that the mgraton patterns of Norwegan sprng-spawnng herrng change n accordance wth the number of ndvduals wth an assgned locaton. Ths s nterestng because t suggests that dfferences n recrutment numbers may affect herrng mgraton patterns. Fsh school behavour can be smulated not just by these stochastc models but also by physcal mathematcal models that can smulate fsh school behavour determnstcally /$30.00 Ó 004 Internatonal Councl for the Exploraton of the Sea. Publshed by Elsever Ltd. All rghts reserved.

2 Mathematcal model of fsh schoolng behavour n a set-net 115 Sannomya et al. (1990, 1993) and Nakamne and Sannomya (1995, 1998) developed an ndvdual-based model n whch ndvduals behavour s decded by certan external forces that have a defned functon. Snce there are no stochastc processes n ths model, the effect of each external force on the behavour of a school of fsh can be quanttatvely evaluated n more detal. In the model, the behavour of a school of fsh s treated as beng that of an autonomous decentralzed system, through a systems engneerng approach. Ths means that overall order emerges from each autonomous subsystem that consttutes the whole system, and consequently the system acheves ts objectves. Ths s smlar to the concept of ndvdualbased models. Snce such a model can nclude the behavour of ndvduals n response to obstacles such as walls and f such a model of fsh school behavour can descrbe actual fsh schoolng behavour n varous stuatons, t could be appled to problems n the feld of fshng technology, such as estmatng the capture process, or decdng the best type of fshng gear to use. Smulaton results derved from the model could thus provde suggestons for fshng operatons or fshng gear desgn, allowng the catch to be controlled more effectvely. Usng the most recent mathematcal model Nakamne and Sannomya (1995, 1998) n the present study, we smulated fsh schoolng behavour n a smple set-net smlar to a prmtve type of set-net. Smulaton results were evaluated by comparson wth the behavour of an actual school of fsh under expermental settngs. In ths model, there are certan parameters that need to be quantfed before runnng smulatons. The parameter representng the quantty of nformaton exchange (M),.e. the number of ndvduals that nfluence the behavour of each ndvdual, s one of the most mportant. The quantty of nformaton exchange has a strong effect on the sze of a school of fsh and on ts behavour, especally n enclosed spaces. Therefore, to accurately represent the behavour of an actual school of fsh, the optmum value for M must be decded. We examned the schoolng behavour of two speces of fsh e a freshwater fsh, btterlng (Rhodeus ocellatus ocellatus), and a saltwater fsh, chub mackerel (Scomber japoncus) e to compare expermental data wth the results of smulaton. The am of the study was to develop a sutable mathematcal model that could realstcally descrbe the behavour of a school of fsh n a set-net, wth partcular focus on the nformaton exchange parameter. respectvely. The motons of N f fsh n a school are then descrbed by the followng equatons: x _ Zv ; mv _ ZF 1CF CF 3 Z1; ; /; N f ð1þ where mv _ s the nertal force of an ndvdual. F 1,F, and F 3 are the forces defnng the moton of an ndvdual,, defned as the propulsve force, the force exerted by the walls and nets, and the force of nteractons among ndvduals, respectvely. These forces are explaned as follows. The propulsve force, F 1, s used to express the fact that an ndvdual fsh swms forward at ts own preferred speed: F 1 Z ÿ a 1 ðkv k ÿ a Þðkv k ÿ a 3 Þv ðþ where a!a3, v s swmmng speed and v _ s the acceleraton. When there are no other forces actng on the moton of the fsh, Equaton () descrbes the swmmng speed of an ndvdual,, and converges on a stable equlbrum pont (Fgure 1). The swmmng speed of an ndvdual,, converges on a or a 3 or 0, dependng on the ntal condton. Snce v _ s the dfferental coeffcent of frst order wth respect to tme, v ncreases when v _ s postve and decreases when v _ s negatve. The parameters a 1, a, and a3 are unknown, and are unque to each ndvdual. F expresses the envronmental effects of the force exerted by walls and nettng screens. As fsh swm, they do not collde wth obstacles, such as walls or nettng, but v v 0 a a 1 > 0 a 3 v Methods Mathematcal model of fsh schoolng behavour The moton of ndvduals s assumed to be restrcted to wthn a two-dmensonal space, and s expressed by usng Newton s equaton of moton. The poston and the velocty of an ndvdual are x Zðx 1 ; x Þ and v Zðv 1 ; v Þ, 0 a a 3 a 1 < 0 Fgure 1. Soluton trajectory of Equaton () n the v ev _ plane. v s swmmng speed and v _ s the acceleraton. v

3 116 T. Takag et al. nstead swm along the obstacles, followng them closely. Consequently, the model assumes that these obstacles act as repulsve and attractve forces on ndvduals. These forces are represented by: F Zk C w X L lz1 f C wl Ckÿ w X L lz1 f ÿ wl f C wl Z d vl e C ÿd l l for v d C l O0 and d l!d C 0 otherwse f ÿ wl Z d vl e ÿ ÿd l for v l d l!0 and d ÿ l!d ÿ 0 otherwse ð3þ ð4þ ð5þ where L s the number of sdes to the wall. The unt vector e l refers to wall l and v l s the velocty component normal to wall l, gven by v l Z ÿ e l v. The quantty d l descrbes the dstance between an ndvdual,, and the wall, l. d C and d ÿ are the certan dstance that repulsve and propulsve forces nfluence, respectvely. k C w and k ÿ w are unknown parameters that are unque to each ndvdual. F 3 expresses the force exerted on each ndvdual by ts neghbours. The force F 3 s gven by F 3 Z X j NðÞ b ÿ x j ÿ x r j C X c ÿ v j ÿ v r j r j M j NðÞ 8 9 b ÿ k < b ÿk1 b a 1 r j Ck 1 b 0!r j!a 1 = r j Z : k b a 1!r j!a ; 0 r j Oa c ÿ kc 0!r r j Z j %d 0 r j Od ð6þ ð7þ ð8þ where r j s the dstance between ndvdual and ndvdual j, gven by r j Z xj ÿ x. The frst term on the rght-hand sde of Equaton (6) s the nteractve force that mantans a sutable dstance between neghborng ndvduals. The second term s the schoolng force that makes the velocty of each ÿ ndvdual ÿ unform. Fgure shows ÿ these functons for b rj and c rj. The graph of b rj ndcates that when the dstance between ndvduals s small, the force acts on ndvduals repulsvely, whereas when r j s large, t acts attractvely. k 1 b,k b,k c, a 1, and a and d are the unknown parameters. N() s the subset whose elements consst of the ndvduals exstng n the vcnty of ndvdual. Let M be the number of elements n N(), where M s the quantty of nformaton exchange and ndcates the number of ndvduals that have an effect on ndvdual when the nteractve and schoolng forces are estmated. Therefore, the smulaton results for schoolng behavour dffer as the quantty of b(rj) b k k 1 b c(rj) kc α1 α nformaton exchange s vared. For nstance, f M s three, the elements of the subset N() consst of three nearer neghbours around wthn the dstance a. The present study sought to estmate an approprate value for M n order to develop a model that reflected the behavour seen n an actual set-net trap. The model equatons can be found by substtutng unque parameters that are defned for each ndvdual by Equaton (1), whch s a second-order smultaneous ordnary dfferental equaton system. We can therefore conjecture the swmmng trajectory of each ndvdual from the ntal poston of the ndvdual and ts velocty, and by usng computatonal numercal calculatons performed on a PC. In the present study, we used the ffth or sxth order of the RungeeKutta method to solve these dfferental equatons. Model parameters To smulate behavour, we need to estmate the unknown parameters ncluded n the model. We used the parameters estmated for btterlng usng the tme-seres data obtaned from experments performed n prevous studes (Sannomya et al., 1993; Nakamne and Sannomya, 1995). Fve btterlng were placed n a 150! 100 cm rectangular water tank, and tme-seres data were acqured for the trajectores of the ndvduals. If d C,d ÿ, a, d, x, and v can be defned, lnear equatons can be obtaned for the other parameters a 1,a,a3,kC w,kÿ w,k1 b,k b,k c, and a 1. Thus, these parameters can be dentfed by usng a leastsquares method from the tme-seres data. The parameters for btterlng are lsted n Table 1. In order to estmate the unknown parameters for mackerel, fve ndvduals were placed n a water tank 90 cm n dameter wthout the trap. The moton of each fsh δ rj rj Fgure. Graphs of b ÿ rj and c ÿ rj.

4 Mathematcal model of fsh schoolng behavour n a set-net 117 Table 1. Model parameters for btterlng. a 1 (g s cm ÿ ) a (cm s ÿ1 ) a 3 (cm s ÿ1 ) k 1 b (g cm s ÿ ) k b (g cm s ÿ ) k c (g s ÿ1 ) a l (cm) k C w (g cm ÿ1 ) k ÿ w (g cm ÿ1 ) 1 ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ was lmted by the shallow water depth (10 cm) because the moton s n the two-dmensonal space. Vdeo mages of fsh school behavour were taken usng a dgtal vdeo camera.5 m above the tank, and tme-seres data for ther trajectores were acqured usng dgtal mage analyss. The xey poston of each ndvdual was recorded at 0.1-s ntervals over a perod of 00 s. The poston of each ndvdual was used as tme-seres data n the fsh school behavour model (Nakamne and Sannomya, 1995). The unknown parameters were dentfed by a least-squares method, and are lsted n Table. For both btterlng and mackerel, d C,d ÿ, a, and d were defned from the observaton data as 5 cm, 0 cm, 50 cm, and 50 cm. Water tank experments To evaluate the behavoural model, we compared actual fsh schoolng behavour n a water tank wth that of numercal smulatons. A set-net s a complcated structure that conssts of leader fence nets, a playground, bag nets, and other parts. In ths study, however, a smple model of a set net, consstng of a box-shaped trap and a leadng fence, was used to obtan the basc characterstcs of fsh schoolng behavour n a set-net. To compare the behavour of a school of fsh as determned by smulaton wth that derved from expermental data, two experments were carred out, usng btterlng and mackerel, as follows. In the experments wth btterlng, a box-shaped trap, 0 cm n length, was used as a bag net. Ths, together wth a leadng fence as a leader net, 60 cm n length, was set n a crcular water tank wth a dameter of 180 cm (Fgure 3a). The water depth was 10 cm to restrct the fsh to twodmensonal movement. Twenty ndvdual btterlngs, wth an average total length of 4.7 cm and an average body weght of.5 g, were used for the experment. Table. Model parameters for mackerel. We appled the same methodology as descrbed above n the experments wth mackerel. Fve mackerel, wth an average total length of 17.7 cm and an average body weght of 61.5 g, were used for the experment. In ths case, a boxshaped trap 50 cm n length was used, and two screen fences were attached at the nlet of the trap to serve as a flapper net (Fgure 3b). The screen was 5 cm n length, and the angle between the screen and the entrance lne was 60(. In both experments, the trajectory of each ndvdual was obtaned by vdeo mage analyss of dgtal vdeos of the schoolng behavours of the fsh, usng a PC. Tmeseres data of the xey coordnates of each ndvdual were recorded every 0.1 s, once the fsh had acclmatzed to the expermental arenas. Results Experment 1: btterlng Fgure 4 shows the trajectores of the school of btterlng durng the process of enterng and escapng from the trap. The school of btterlng tended to swm along the wall of the crcular tank, and rarely entered the trap. Any temporary rregular behavour on the part of the leadng ndvduals of the school caused the whole school to change course sgnfcantly, as ndvduals movements were coordnated wth each other. When the leader of the school encountered the leadng fence, the entre school swam along the fence and entered the trap. As the ndvdual fsh enterng the trap were slowmovng, they tended to reman n the trap and not escape mmedately. Although some ndvduals were near the ext and had the chance to escape, they dd not, because ther movements were coordnated wth those of the other fsh. Although the drecton of travel of each ndvdual was smlar before enterng the trap, once nsde no regular a 1 (g s cm ÿ ) a (cm s ÿ1 ) a 3 (cm s ÿ1 ) k 1 b (g cm s ÿ ) k b (g cm s ÿ ) k c (g s ÿ1 ) a l (cm) k C w (g cm ÿ1 ) k ÿ w (g cm ÿ1 ) 1 ÿ ÿ ÿ.7 ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ1. 5 ÿ ÿ ÿ34.4

5 118 T. Takag et al. (a) 0cm 180cm 60cm 0cm 50cm 50cm 60 50cm Fgure 3. Confguraton of a set-net model n a crcular water tank n the experments wth btterlng (a) and mackerel (b). behavour was observed on the part of the school. There were on average 17 s between the frst fsh enterng and the last fsh escapng. The majorty of the fsh n the school dd not change swmmng speed between enterng and escapng, but once 4/5 of the fsh had escaped, the remanng fsh chased after the leadng group by swmmng faster. Smulaton results were obtaned by reproducng the same condtons as those n the experments. Fve parameter sets, shown n Table 1, were allocated to the model s equatons for 0 ndvduals. The numercal calculatons were thus performed usng ntal condtons derved from the expermental data for the postons and veloctes of the fsh. The xey coordnates for ndvduals were estmated from the ntal states, every 0.1 s for 150 s. Smulaton results for 19 cases were obtaned by varyng the number of nformaton exchange, M, from 1 to 19. Fgure 5 shows the results of smulaton when M Z 7. The school of fsh swam along the leadng fence lengthwse and entered the trap; at ths pont, ndvduals encounterng the end of the trap could not swm forward, and turned towards the ext of the trap. The school of fsh then spread out n the trap. Indvduals near the ext escaped and others followed, untl fnally all (b) ndvduals escaped. When M Z 7, the smulaton results agreed well wth those of the experment; however, the tme from enterng to escapng was more than 30 s, whch was longer than the tme taken for the expermental result. The behavour was qute dfferent for dfferent values of M. In partcular, when M was a relatvely large or small number, the behavour dd not reflect that of the experment. Usng an napproprate number for M caused unrealstc smulaton results. Fgure 6 shows the case when M Z 1, wth 19 smulaton results after 100 s had elapsed. When M Z 1, a few ndvduals left the trap slowly, but the school spread out and consequently dd not move,.e. the system attaned a state of deadlock. M Z 19 also nduced deadlock and hgh densty n the school; as ndvduals near the ext had to cooperate wth those n the slow-movng area near the end of the trap, they could not escape. To compare the results of smulaton wth those of the experment quanttatvely, we estmated the centre pont of a fsh school, CP(xp,yp), and d ðnþ (Z1; ; /; N f and n s tme-step), whch s the dstance between each ndvdual and CP n each tme-step, as well as the tme remanng T (Z1; ; /; N f ) for each ndvdual n the trap. We estmated dðnþ, the average value of d ðnþ for all ndvduals n each tme-step. For quanttatve comparson, based on the sze of the school of fsh and the remanng tme n the trap, we defned two ndcators D and T. These values are gven as follows: DZ 1 n1 X n1 nz1 TZ 1 N f X N f dðnþ T Z1 ð9þ ð10þ Fgure 7 shows D and T for the expermental and smulaton results for values of M rangng from 1 to 19. Fgure 4. Trajectores of the school of 0 btterlng durng the process of enterng and escapng from the trap.

6 Mathematcal model of fsh schoolng behavour n a set-net 119 Note that DnFgure 7 ndcates the average of a tmeseres of dðnþ, from the ntal poston, to the tme at whch 1 s had elapsed after the centre poston of the school had left the trap. The T of 150 s n Fgure 7 ndcates that the school of fsh became deadlocked n the trap. In the smulaton results, all fsh enterng the trap became deadlocked when M Z 11, 1, 14, 16, 17, 18, or 19. When M Z 1, one ndvdual entered the trap, then others entered, and the system subsequently became deadlocked. When M Z, 3, and 4, some ndvduals escaped wthn 30 to 80 s, but the remanng ndvduals became deadlocked. Although all ndvduals could escape when MR5, a correlaton between the tme requred to escape and M was not well estmated. When M Z 7, the tme requred to escape was the shortest, and was close to the expermental result, D; ths gave the densty of the school of fsh as 6. cm, and was n agreement wth the expermental value of 6.7 cm. Fgure 5. The smulaton results for btterlng when M Z 7. Addtonally, we found some nterestng behavoural patterns n the smulaton results. When M was greater than 10, some ndvduals n the leadng group turned back toward the centre of the school soon after startng the smulaton, but ths behavour was not seen n the experment, suggestng that the school sze gven as the ntal condton was too small to be realstc. Therefore, the approprate value of M can also be evaluated from the behavour patterns seen n the smulaton results. Experment : mackerel Fgure 8 shows the trajectores of a school of fve mackerel. Ther behavour n the trap wth screen fences used as flapper nets was qute dfferent from that of the btterlng n the trap wthout a flapper screen. When ndvduals encountered the end of the trap, they turned toward the ext, but ther escape was prevented by the flapper screen Fgure 6. The smulaton results for btterlng n the case M Z 1 and 19 after 100 s had elapsed.

7 10 T. Takag et al. T(s) D(cm) Exp. 1 fence. Indvduals often remaned n the area between the flapper screen fence and the wall of the trap; subsequently, a school would splt off, and a few ndvduals would escape. However, escapng ndvduals would then approach or enter the trap agan because other ndvduals n or near the trap affected ther behavour. Therefore, these Exp Fgure 7. D and T for the expermental and smulaton results for values of M rangng from 1 to 19. D ndcates the average of a tmeseres of dðnþ from the ntal poston to the tme at whch 1 s had elapsed after the centre poston of the school had left the trap expermental results showed that flapper screens dd affect the behavour of fsh that remaned n or near the trap. The parameter sets n Table were allocated for model equatons for fve ndvduals, and the numercal smulatons were carred out usng the ntal condtons gven by the expermental data. The xey coordnates for ndvduals Fgure 8. Trajectores of the school of fve mackerel durng the process of enterng and escapng from the trap wth flapper screen fences.

8 Mathematcal model of fsh schoolng behavour n a set-net 11 were calculated every 0.1 s for 50 s. Four sets of results were obtaned by changng the number of nformaton exchange, M, from 1 to 4. In the case of mackerel, ndvduals moved more actvely than btterlng, and splt off and joned together fludly. Enterng and escapng from the trap was often seen. Therefore, unlke the case for btterlng, t was dffcult to evaluate ths quanttatvely usng D and T; to evaluate the model, we therefore compared the smulaton results wth those of the experment. Fgure 9 shows the smulaton results when M Z 1. Here, ndvduals encountered the wall and turned back toward the ext of the trap. The escape of a few ndvduals, however, was prevented by the flapper screens (Fgure 9B). The ndvduals that could not escape repeatedly came nto contact wth the wall and the flapper screen, and remaned n the area between the two (Fgure 9C). These behavours were smlar to the expermental results. After a short tme, one ndvdual escaped but the others remaned n the trap (Fgure 9D); any ndvduals that escaped dd not enter the trap agan. When these ndvduals encountered each other closely, they dd not unte to form a school. Ths behavoural pattern was not seen n the experment. The smulaton results wth M Z 4 dffered consderably from those wth M Z 1(Fgure 10). The school of fsh entered the trap, turned back, and escaped after encounterng the wall and fence (Fgure 10B). However, the fsh that escaped frst were then agan affected by escapng ndvduals (Fgure 10C). These fsh then turned back towards the fsh near the ext and consequently entered the trap agan (Fgure 10D). An ndvdual outsde the trap, swmmng along the wall of the crcular tank, was also affected by other ndvduals, and subsequently swam towards the trap (Fgure 10F). These behavoural patterns were also seen n the experment. These characterstc behavours were obtaned more frequently when M Z 4 than when M Z 1,, or 3. As ndvduals affected each other more wth ncreasng M, the state of stable equlbrum was not mantaned. The smulaton results when M Z 4 were those most lke the expermental observatons. Dscusson Although the form of the functon n our ndvdual-based model s mportant for fsh school behavour because our model s a physcal mathematcal model that estmates ndvduals behavour determnstcally, organzed fsh school behavour also requres that ndvduals cooperate wth ther neghbours. Ths mportant fact can be demonstrated by the dfferences n behavours that result from changng M, the value for the quantty of nformaton exchange. Fsh school behavour n response to a set-net can be quanttatvely evaluated by usng the ndcators D and T. The results suggest that the value of M must be optmal to obtan realstc school behavour. If the value of M s too small, the smulated fsh schoolng wll be unrealstc and can result n a state of deadlock n the trap; Fgure 9. The smulaton results for mackerel when M Z 1.

9 1 T. Takag et al. cooperaton wth other fsh leads to schoolng behavour, but none of the fsh obtans suffcent nformaton to escape. By contrast, f M s too large, each ndvdual has to cooperate wth so many neghbours that when the ndvduals have varous orentatons, no one ndvdual can move, resultng n deadlock nsde the trap. Furthermore, outsde the trap, more fsh tend to school together than s the case expermentally. Thus, a model usng too large or too small a value for M cannot descrbe the actual behavour of a school of fsh. Our results suggest that M should be set between 5 and 10 n our model. In the case of btterlng, the behavour of the smulated school of fsh was smlar to that seen n the experment, when M Z 7, and also n accord wth quanttatve comparsons usng the ndcators n Fgure 7. However, the real school of fsh was always able to escape from the trap faster than suggested by the smulaton. Ths dfference mples that the quantty of nformaton exchange should not be a constant but rather should be treated as a varable value that depends on the stuaton. The smulaton showed that the school of fsh s lkely to become deadlocked n the trap f M s too large; ths suggests that when fsh are packed together, such as n a trap, changng M to a small value can avod deadlock and may permt prompt escape from the trap. In the experment usng mackerel, a few ndvduals often remaned by the flapper screens, and we observed that Fgure 10. The smulaton results for mackerel when M Z 4. these ndvduals outsde the trap attracted ndvduals nsde the trap and affected ther behavour. Ths behavour pattern can also be seen n the smulaton, regardless of the value of M. The smlarty between the smulaton and the experment supports the valdty of the model, and confrms that the force of nteracton among ndvduals n our model s mportant when there s a small number of ndvduals. However, after escapng from the trap, the ndvduals wll not form a school f M Z 1, and the behavour dffers from that observed n the experment or the results of the smulaton when M Z 4. Ths shows that M Z 1 s not approprate. Although the results of the mackerel experment cannot readly be compared wth that conducted wth btterlngs, because the former school conssted of only fve ndvduals, they do nevertheless support the suggeston that too small a value of M should not be selected to yeld realstc schoolng behavour. Our model needs to be further refned to address some basc nherent problems, but the expermental results do confrm that the concept of our model s vald. The present study focused on the quantty of nformaton exchange among ndvduals, but there reman other problems that need to be consdered n more detal. For nstance, the assumpton that the nfluence area n whch neghbours of ndvdual affect ndvdual s unform from 0( to 360( should be examned and verfed because the drectons of vsual and other sensory stmul are not unform.

10 Mathematcal model of fsh schoolng behavour n a set-net 13 Further nvestgaton of the relatonshp between the degree of nformaton exchange n a school and adaptve behavour n response to envronmental varables wll mprove the performance and accuracy of the model. Smulatons and expermental studes should be examned n more detal, and n greater numbers, to provde a bass for further practcal use of the model. Acknowledgements Ths paper owes much to the collaboraton of Professor Osamu Murata and to Shnj Yamamoto. We thank them for ther assstance and for provdng us wth the samples. Our thanks are extended to the staff and students of the Shrahama Fsheres Expermental Staton of Knk Unversty. We thank Katsuya Suzuk and Takash Shmzu, Graduate School of Fsheres Scence, Hokkado Unversty, for ther assstance n makng 3D computer graphcs. We also thank the edtors and anonymous referees for helpful comments and constructve crtcsm. Ths research was supported n part by a grant-n-ad for the 1st century COE program, Japan. References Aok, I A smulaton study on the schoolng mechansm n fsh. Bulletn of the Japanese Socety of Scentfc Fsheres, 48: 1081e1088. Huse, G., Ralsback, S., and Fernö, A. 00. Modelng changes n mgraton pattern of herrng collectve behavour and numercal domnaton. Journal of Fsh Bology, 60: 571e58. Huth, A., and Wssel, C The smulaton of the movement of fsh schools. Journal of Theoretcal Bology, 156: 365e385. Huth, A., and Wssel, C The smulaton of fsh schools n comparson wth expermental data. Ecologcal Modellng, 75/ 76: 135e145. Inada, Y., and Kawach, K. 00. Order and flexblty n the moton of fsh schools. Journal of Theoretcal Bology, 14: 371e387. Nakamne, H., and Sannomya, N Effect of autonomous decentralzaton mechansm n fsh behavour. Transactons of Insttute of Systems, Control and Informaton Engneers, 8: 350e356. Nakamne, H., and Sannomya, N A smulaton study on fsh behavour affected by obstacles. Proceedngs of the Amercan Control Conference, Phladelpha, USA, 4e6 June. Reuter, H., and Brecklng, B Self-organzaton of fsh schools: an object-orented model. Ecologcal Modellng, 75/76: 147e159. Reynolds, C. W Flocks, herds, and schools: a dstrbuted behavoural model. Computer Graphcs, 1: 5e34. Sannomya, N., Nakamne, H., and Iwasak, H A study on the valdty of a physcal model for fsh behavour. Transactons of Insttute of Systems, Control and Informaton Engneers, 3: 14e0. Sannomya, N., Shmada, A., and Nakamne, H Modelng of autonomous decentralzed mechansm n fsh behavour. Transactons of the Socety of Instrument and Control Engneers, 9: 11e19.