A simulation study on self-organization in the behavior of a heterogeneous fish school

A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

Copyright ':c:' 1999 IF AC ] 4th Triennial Vv' orld Congress..

14th World Congress ofIFAC

L-5a-04-3 Beijing~

P.R. China

A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEHAVIOR OF A HETEROGENEOUS FISH SCHOOL

Yajie Tian*, Nobuo Sannotniya'" and Ko Matuda**

*

Kyoto Institute of Technology !vlatsugasaki, Sakyo - ku, Kyoto 606 - 8585, Ja.pan phone: +81-75-724-7447 fax: +81-75-724-7400 e-mail: [email protected](Yajie Tian) e-mail [email protected]*ac.jp(XobuoSannomiya)

** Tokyo

llniversity of Fisheries Konan, }Jinato - ku, Tokyo 108 - 8477, Japan phone: +81-3-5463-0471 fax: +81-3-5463-0399

Abstract: A fish school is considered as a typic.al example of autonomous decentralized system (ADS) or a self-organizing system existing in nature, because it often shov~rs a high degree of coordinative behavior in the absence of a leader. A roathelnatical model was proposed in our earlier paper to describe the behavior of a homogeneous fish school which consists of individuals with almost the same character. In this paper, another model is proposed to describe the behavior of a heterogeneous £sh school, in Vv~hich there exist individuals \vith different characters. Since the behavior of a fish school varies \vith environmental variations, simulations are carried out by setting a box-shaped trap in a behavior space as an obstacle. The self-organization of the school and the variations of moving pattern with changing the quantity of information exchange among individuals are discussed. Copyright © 1999 IFAC

Key

Words

Autonomous decentralized systems, Self-organization, Simulation

1. INTRODUCTION Autonomous decentralized system (ADS) is a concept extracted fronl self-organizing biological systems. Recently, the concept of ADS is applied to the design and operation of large-scale and complex engineering systems. The ADS organizes the action of each subsystem without any supervisor to adapt itself to environmental variations and to cooperate \vith other subsystems. For this purpose a functional order of the entire system is re-

Behavior,

l\.{odeling,

quired to be generated (Ita, 1991). A fish school is considered as an ADS or a selforganizing system from the results of not only observation but also simulation. It is knovln that each individual has an ability as a subsystem to swiIn forward freely and sirnultaneously to adjust its own speed and direction to rnatch those of other individuals in the schooL A high degree of coordinative school behavior can be generated in the absence of a leader or external stimuli in the school (Sannomiya 7 et al., 1993; Sannomiya

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

and Doustari, 1996~ Sannomiya and Nakamine, 1998). As a self-organizing system: each fish can establish its position with its eyes and lateral lines and simultaneously can measure the speed of other fish in a schooL The correlation bet"v-een the velocity of a particular fish and those of other fish in a school is not strong (Partridge, 1982). T'he strong correlations are observed between the velocity of the individual and the average velocity of the entire school (Sannomiya and Tian, 1996; Tian and Sannomiya, 1997). When all individuals of a school keep their speeds and directions to be same as the average of the school, a special order of the school is established (Sannomiya and ~:akamine, 1998).

i\ lllathenlatical lnodel was proposed in our earlier paper (Sannomiya, et al., 1993) to describe the behavior of a homogeneous fish school. In this paper, another model is proposed for describing the behavior of a heterogeneous fish school which includes individuals with different characters. A water tank experiment was carried out to obtain an experimental data of a heterogeneous school (Liang, et al., 1995). The model parameters are estimated by using the time series data sampled {rolD that experimental data. In order to study and analyze the behavior of suc.h a heterogeneous fish school, various simulations are carried out by setting a box-shaped trap in the tank. The different moving patterns are observed from the simulation results by changing the quantity of information exchange in the schooL Further it is found that a fixed cooperation and flexible adaptability to environmental variations are generated when the quantity of information exchange is large and a low adaptability to environmental variations is generated when the quantity of information exchange is reduced to half.

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individuals in a school are described by

(1)

,,"here m is the mean mass of the individuals. Fit, F i2 and F i3 are the forces which cause the motion of individual ,i. They are expressed as follo\vs.

2~1

Fish Own Swimming Ability

Each individual in a school has an ability to swim forv{ard freely~ The character is given by

at

w here is the coefficient related to the s\vimming ability. In the absence of F i2 and F i3 , the following one-dimensional system for Vi is obtained by substituting (2) into (1).

In this case Eq.(3) shows that an individual ha..'i a character of swimming forward with a constant speed (called the characteristic veloci t y ). The parameters af and a~ (a; < a~) are the quantities related to the characteristic velocity. Figure 1 is an example of the phase plane of (3) with a! > o. I t is concluded from t he figure that in t he case of > 0 > the characteristic velocity is given by and in the case of 0 > > a7, the characteristic velocity is zero.

at

a;,

ar

ar

1;-z

2. MODEL

characteristic

/velOCity Each fish in a school has an ability to s\vim forward at its o\vn speed, and to measure the speeds of other individuals in the school ,vith its eyes and lateral lines. According to the information obtained from other individuals, each individual constantly adjusts its speed and direction to lllatch those of other individuals. A cooperative behavior of the school is generated based on the quantity of information exchanges and the interactions among individuals. This leads to establishment of the rnovernent nlodel to describe the behavior of a fish school. The nlotion of fish is assuuled to be restricted to a tVlo-diInensional Hpace. Let the position and the velocity of individual fish i be Xi and Vi, respectively, ",rhere Xi, Vi E R2. 'T'hen the motions of 1'11

----~IoP-----.. Vi

(a)

aI

Fig~l

>

0

and

a;

<

0

<

at

(b)

at

>

0 and

Solution trajectory of (3) in the

at < a~

Vi-Vi

<

0

plane

2.2 Interaction among Individuals

Each individual keeps itself in a school on a basis of the interactions among the near neighbors. By the information exchanges, it adjusts its speed and direction to rnatch those of the neighbors. The character is given by

5956

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

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to wall l, given by vn == -eT Vi. The quantity d iZ means the distance from individual i to wall t. (4)

In (7), the first and the second term are called the repulsive and the attractive force from the v..~all respectively. The parameter~ k+. and k-· are their W1coefficients. From (8) and (9)) the repulsive force acts on individual i \vhen it approaches the wall, i.e. vu > 0 and the attractive force acts on individual i ,vhen it goes away from the wall, i.e. Vi < o. d+ and d- are the critical distances related to the repulsive and the attractive force, respectively. W~

(5)

o< r 1 'j

T~j

>

:5 c

(8)

D

\\~here r ij == llx j - Xi 11. The first term is the interactive force to keep a proper distance between the neighboring individuals~ The second term is the schooling force to make the velocity of each individual uniform. In (4), N(i) is the subset of {I, 2, Ni} whose elements consist of the individual numbers existing in the neighborhood of individual i. M is the number of the members in N(i). Uil and Q2 are the positive constants related to the interactive force. 8 is the critical distance for the schooling force. kti( < 0) and kti( > 0) arc the coefficients of the interactive force. k c'i ( > 0) is the coefficient of the schooling force. P

••

,

2.3 Environ1nental Effect Each individual perceives environmental variations and can measure the distance to an obstacle with its eyes and lateral lines~ Then the individual adjusts its own action to adapt itself to environmental variations and keeps a cooperative behavior with neighbors. In this model, the walls of a water tank and a trap are the environmental effects to the movement of a fish school. When a fish school swims in the water tank, it often moves along the wall, stays at and goes out of the trap, but never strikes against them. Consequently, there are both an attractive and a repulsive actions for the environmental effect. The character is expressed by L

F~3

== k;i

l=l

EXPERIMENTAL DATA AND MODEL PARAMETERS

In our earlier papers (Sannomiya, et al . , 1993; Tian and Sannomiya) 1997; Sannomiya and Nakamine, 1998), the experimental data were obtained from the water tank experiment by using Bitterling (Rhoudeus ocellatu5 acellatus) with 3 . . . . , 5cm in length, 2.58g in the average weight and a small rectangular water tank (150cmx lOOcm) . In this paper another data was obtained from the experiment carried out at Tokyo Lniversity of Fisheries (Liang, et al., 1995). The fish used in this experiment were Tilapia rnossambica Vw"hich are 20.7cm long and weigh 162.8g on average. The tank is a regular octagon and each side is 170cm long. The image of the fish behavior was recorded by using a video camera and a video tape recorder~ The video image was sampled at the intervals of 0.5 seconds. The position of each individual was calculated by using an image processing device. The number N j of individuals is five. The observation time T is 550 seconds.. The unknown parameters included in the model Vi-f"crc estimated by applying the least squares algorithm to the experimental data. Table 1 shows the result of the estimation. From this table, we kno"\v that > 0 (i =:: 1,2,·' ~,5). We have < a~ < 0 and thus the characteristic velocity of individual 1 is zero. Since < 0 < (i == 2,3,4,5), the characteristic velocities for the other individuals are Since there is an individual with zero characteristic velocity in five Tilapia, the school of TiIapia is called heterogeneous.

a1

at

aT

ar

at.

L

L f;;7J + k~i L f;;il

3~

(7)

l==l

(8) otherwise

(9) otherwise

¥t here L is the number of the v-lall sides of the tank and the trap. The unit vector Cl is floTIllal to wall l, and Vii is the velocity component of 'Vi nor-lllal

The phenomenon of Tilapia is different from that of Bitterling (Sannomiya and Nakamine~ 1998)~ In the case of Bitterlillg, five individuals swim together \vith almost the same speed and the model para.meter~ have sirnilar values among five Bitterling. Therefore, the school of Bitterling is called homogeneous. Since the parameter values depend on the body size of the individual, we convert the parameters to dimensionless values in order to compare them

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

Table 1

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Parameters estimates (Nf == 5)

a?k ci a1 a~ ktk~. 1. 2 1. 2 (g. sec)cm 2 ) (etu7sec) (CDn sec) (g·cnn/sec ) (g·cnalsec ) (g/sec)

7

1

O~O1489

-93.54

2 3 4

0.05499 0.07562 0.02958

5

0.03625

-48.51 -13.31 -91.10 -103.64 m

~

-1.45 10.04 13.59 14.88 14~43

162.8g,

8 =

-130~54

35.86

7.81

-416.35 -416.72 -427.53 -2079.83

28.72 37.24 34.70 143.70

35.54

Q2

==

Table 2 Dimensionless parameter values Tilapia Bitterling

aI

k bI

k b2

kc

k~

k;

0.011

0.254

0.021

0.146

0.862

0.179

0.037

0.132

0.082

0.276

2.505

0.646

between Bitterling and 'Tilapia. The averages of dimensionless values are

66.59 53.40 62.63

k-. (g~~c)

238.57

41.29 48.90 39.69 86.91 59.40

(g/sec)

20.79 30.92 27.74

252~74

23~81

24.99

d+ :;:: 30Cffi, d-

180cm,

kti

ail

(cm)

289.38 236.04 311.05

= 80cm

the behavior of a heterogeneous school with that of a hOIllogeneous school, a box-shaped trap and a leading wall are set in the water tank as shown in Fig.2. The box-shaped trap is a square with 80cm long in each side. It has three sides of \\raIls and one side of an entrance from which fish can enter and go out freely~ The various simulations are carried out by changing lv~(i) and M in (4). 170cm Bo~-shaped lrap

R~

(10)

",,'all

or waler tank

,/

Leading wall

VlIThere

D..t

::=

v

:,,~~.P .~ ... ~-\. [nitiaJ

is the average speed of individuals and

t.

0.5 is the sampling time interval.

Table 2 shows the comparison of the dimensionless parameter values between Bitterling and Tilapia. It is observed from Table 2 that all parameters of Bitterling except are larger than those of Tilapia. This means that the repulsive force among individuals is large for Tilapia and is small for Bitterling~ Therefore, the school of Tialpia is called repulsive and the school of Bitterling is called cooperative~

ki

J position p

...'11 I1L .. _ . . . . . . . . . . p ... •

Fig~2

\Vater tank model used in the simulation

We define that individual i keeps an order "vhen it sv.rims at the same direction as the nearest individual j, i.e. the angle between Vi and Vj is less than 0.1 rad as sho"vn in Fig.3. ]\lg is defined as the number of individuals keeping an order.

e

4. SIMULATION RESULTS In this paper, simulations are carried out by using the model mentioned above with the parameters shoVtoTn in Table 1. The number lV-I of the individuals is assumed to be twenty. Then Eq.(l) is a system of nonlinear differential equations \vith eigh ty variables. The school consists of five grou ps \vith each group having four individuals with the same parameter values. We focus our attention to the behavior of a fish school in the process of escaping from the trap, which was investigated for a homogeneous school ill our earlier paper (Sannomiya and Nakamine~ 1998). In order to compare

Fig.3 State of keeping an order

Simulations are carried out by changing jvl from 3 to 19~ The results of two cases such as !vI ::;;;;: 19 and 10 are shown as typical examples to express

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ISBN: 0 08 043248 4

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A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

20 is established and col1apsed easily during the simulation time~ This tendency is different frolll that of a homogeneous school (Sannomiya and Nakaminc, 1998). For a homogeneous school, because there is almost no difference in individuality among twenty individuals, the specific order established in the '\vhole school is difficult to be broken. Further, because all individuals keep thenlselves in an order, the flexibility and the adaptability of the individuals are reduced. Thus, the school is difficult to escape from the trap_ On the other hand, for a heterogeneous school, because there are individuals v.,rith zero characteristic velocit~.y· in the school and the repulsive force among individuals is strong, an order is difficult to be kept in the school. But when M == 19 all individuals join the information exchange. Consequently, the cooperation is increased and the diversity is reduced in the school. At the same times, high adaptability to the environmental variations yields so that the school escapes from and enters the trap rreely~

(a) Trajectories of tlventy individ uals

o

so

100

150

200

250

300

350

400

450

5ClO

t(sec)

(b) T'ime variation of N g Iv,' 30

4.2 The Case for l\f

25

~

10

20 15

IO

::; 50

100

150

200

250

300

350

400

450 500

r(sec)

(C) Speeds of twenty individuals Fig.4 Simulation result for

J.Vf

=

19

the different moving patterns of the fish school.

4~1

The Case far .J.7\rf

We consider the case where ltr-f = 10. This means that each individual exchanges information with a half of individuals in the school. Figure 5 shows the simulation result for this case. It is observed from (a) of Fig.5 that when t == 28sec~, nineteen individuals enter the trap and stay there until t =: 385sec.~ It is also found from (b) of Fig.5 that an order of .". Tg > 10 is difficult to be established. Even though it tends to be established, it is broken quickly. Thus the school spends long time to escape from the trap (t > 385sec.).

== 19

The siluulation result for NI == 19 is ShOVlll in Fig.4. In this case each individual exchanges inforluatiol1 with all the other individuals in a school. Figure (a) sho\vs the trajectory of the school during 230 seconds, Cb) shows the tirrle variation of school order 1Vg and (c) shows the speeds of twenty individuals during 500 seconds. \-,Then t == 23sec., all individuals enter the trap and when t == 76sec., they escape from the trap (sec (a). After that, all individuals sV\rim for\\~ard \vith almost the same speed (see (c), even though four individuals \vith zero characteristic velocity are included in the school. This results frOIn the self-organization and the interactions among all individuals. Sometimes, the environment makes the school separate, but the individuals s1vim together quickly under a certain condition. This indicates that the school has a high degree of self-organization and adaptability to the environnlent variations. We kno,v from (b) of Fig.4 that an order of 1"-[9

=:::

Since there exist four individuals \vith zero characteristic velocity in the school and the repulsive force among individuals is large, the school is difficult to reach unanimity. But if an overwhelrning majority of individuals in the school have a mechanisrn to d€terrnine a COillrllon behavior, a group decision can be nlade easily and then the adaptability to environmental variations is realized, as the case for 1i.f == 19. On the other hand, v.rhen /1,.1 is reduced, the state of school becomes confused for a sudden environmental variation~ Because there is a high system diversity in the school, the cooperation is difficult to realize. Consequently, the school spends long tiIne to escape frolll the trap, as the case for ,1\.1 := 10.

5. CONCLUSION Various cases of sim111ations have been carried out for describing the behavior of a heterogeneous fish

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A SIMULATION STUDY ON SELF-ORGANIZATION IN THE BEH...

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case of a large-scale system, centralization and concentrated management lead to many serious problems such as deterioration of system reliability and flexibility. Therefore, the idea of ADS is offered as an alternative, but the procedure for realizing the ADS is not completed yet. The simulation results mentioned above are considered to be effective for this purpose. Further investigation is continuing to that end~ and the result vlill be reported in a separate paper. This work is partly supported by the "Research for the Future" Program JSPS-RFTF97IOOI02 of the Japan Society for the Promotion of Science. (a) Trajectories of tVlenty individllals

6. REFERENCES

o

SO

100

150

200

250

300

350

400

450

Ito,M.(1991). Autonomous decentralized systems, Proc~ of 35th . .4 nnual Conf. of ISCIE, pp.916 (in Japanese)~ Liang.Z, T.Tokai, K~1llatuda and H. Kanehiro(1995). Propulsive mechanism in fish behavior model {or Tilapia, A'ippan Suisan Gakkaishi, 61; pp~375­ 379 (in Japanese). Partridge,B.L.(1982). The structure and function of fish schools, Sciecetific American, pp.90-99.

;500

r(sec)

Cb) Time variation of 1\/g

15

Sannomiya,N., A.Shimada, and

10 :5

50

J 00

] 50

200

250

300

3:'50

400

450 5t..)(]

r(sec)

(c) Speeds of twenty individuals

Fig.5 Simulation result for M

==

10

school by changing the quantity of information exchange_ The self-orga.nization and adaptation to the environmental variations have been analyzed ",~ith reference to the simulation results. The behaviors of a heterogeneous school have been cornpared with those of a homogeneous schooL For a homogeneous school, ,,,hen ]v.! ~ 19, a specific order is established easily and kept in the whole schooL But becau~e all individuals keep themselves in an order, the adaptability of individuals is reduced. Thus the school is difficult to escape from the trap. On the other hand for a heterogeneous school, when i.U ~ 19, an order is difficult to be kept in the school because of the difference in individuality among twenty individuals. But the school has a high degree of adaptability to the environmental variations. Therefore, the school enters and escapes from the trap freely.

H~Nakamine(1993)~

1fodeIing of autonomous decentralized m~chanism in fish behavior, Trans. Soc. In8trum. Control Eng., 29 pp.211-219 (in Japanese). Sannomiya~N. and l\·1.A.Doustari(1996). il simulation study on autonomous decentralized mechanism in fish beha.vior model, Intern. J OUT. Systems SC'ience~ 27, pp. 1001-1007. SannomiyaJN. and Y.J.Tian(1996). An aggregated model that describes random behavior of fish school '-\'"ith many individuals, Proc. of 13th IF AC tf,T arid Congr., J" pp.71-76. Sannomiya)N. and H.Nakamine(1998). -'"~ simulation study on cooperative behavior in a fish school, Proc. of 3Td J·ntern. Symp. on Artificial Life and Robotics, 1, pp~17-22. Tian~Y.J and N.Sannomiya(1997). Simulation and analysis of the behavior of a fish school ~vith many individuals by using an aggregated model, Intern. Jour. System.s Science~ 28, pp. 357-364.

t

For controlling a system, a hierarchical structure is

constructed to supervise and regulate all parts and subsystems of the system. All informations Illust be concentrated and anaJyzed in order to produce coruulands needed for the subsystems working nnder the supervisory of the entire system. In the

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