Selforganization of fish schools: an object-oriented ...

ELSEVIER

Ecological Modelling 75/76 (1994) 147-159

Selforganization of fish schools: an object-oriented model Hauke Reuter a,., Broder Breckling

b

~'Arbeitsgruppe Boden6kologie und Okosystemforschung, Unit,ersitiit Bremen, Postfach 33 04 40, D-28334 Bremen, Germany, /' Projektzentrum Okosystemforschung, Christian Albrechts Unit,ersith't Schauenburger Str. 112, D-24118 Kiel, Germany

Abstract

Simulation models help to understand how individual interactions can lead to selforganization of large, moving polarized schools of fish. To some extend the underlying mechanisms in these models are still hypothetical. This paper investigates and confirms previously published results and suggests an alternative model. Previous models of fish schools used the assumption that individuals modify their swimming direction and speed as a reaction to a small fixed number of other individuals swimming closest to their own swimming direction (front priority). They also used discrete transitions between different reaction types to their neighbours (see Huth and Wissel, this volume). These models are able to simulate school movement in a homogeneous environment but fail under certain heterogeneous conditions (e.g. when schools meet obstacles) and do not adequately simulate the formation process of schools. We present an alternative model, in which a fish's new speed and swimming direction is influenced by all visible neighbours. Each neighbour contributes a vector component weighted according to its distance. These components are aggregated to determine to what extent the different behavioural modes of searching, attraction, parallel orientation and repulsion contribute to a modification of an individual's movement in a particular situation. The resulting model reproduces schooling behaviour, organization of schools and has fewer artificial requirements concerning a biological interpretation of the underlying mechanisms. It shows how a comparison of different modelling approaches, empirical observation of inter-organismic interaction, and physiological research can stimulate each other in order to understand a complex, dynamic phenomenon. Key words': Fish school; Object-oriented model; Selforganization

* Corresponding author. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0120-R

148

H. Reuter, B. Breckling / Ecological Modelling 75/76 (1994) 147-159

I. Introduction

Many fish species form polarized schools with a high degree of synchronization of swimming speed and direction (Radakov, 1973; Partridge et al., 1980; Aoki et al., 1986). The main biological function of schooling seems to be anti-predator defense, as these fish detect predators much earlier, tend to feed longer under threat, and predators due to a "confusion effect" have difficulties in catching one fish from a tight school (Magurran, 1990). In addition, schooling fish find food much faster than single individuals (Pitcher et al., 1982) and due to a cooperative effect they are capable of following gradients (plankton densities, oxygen, etc.) contrary to solitary fish (Kils, 1986). Schools vary widely in form and structure, depending on species, season, part of life cycle, and activity. Common to nearly all schooling species is the egalitarian state of its members (for exceptions see Klimley, 1985). As it is difficult to analyse or synthesize the complex interaction patterns with traditional methods, in recent years computer models were used to evaluate hypotheses about fish schools (in particular Aoki, 1982; Huth and Wissel, 1992, 1994). In these models the calculation of the new orientation of an individual is done according to the coordinates of other individuals. In the model of Huth and Wissel those fish which are found closest to the swimming direction of the respective individual are selected (front priority), nearly always ignoring much nearer lateral neighbours. It is very difficult to specify the underlying biological mechanisms which can actually lead to this kind of individual behaviour. The reason to use front priority as a model assumption is not based on physiological considerations but is due to the fact that it provides better schooling behaviour than the selection of a limited number of nearest neighbours. The latter leads to the formation of subgroups and splitting of the school. Nevertheless it is known that the visual angle of fish eyes is often larger than 300 ° and the perception angle of the lateral line system is similarly high. A further aspect, which is difficult to bring into accordance with biological knowledge, is the definition of behavioural zones with discrete boundaries, where an individual switches e.g. from attraction to parallel orientation. While we are still lacking the possibility to reconstruct schooling behaviour on a strict neurophysiological base, the discussion of different mechanisms which give rise to schooling interaction is of interest. Therefore we reproduced the model of Huth and Wissel basing on the verbal description and introduce an alternative model. For both models we used an individual based approach employing the objectoriented language SIMULA under 386/ix on an Siemens PCD3TS Workstation with a mathematical coprocessor. The computations were carried out at the Centre for Local Nets at the University of Bremen. 2. The model of Huth and Wissei

In the simulation model of Huth and Wissel (1992, 1994) the orientation and speed of each fish is determined through the orientation, speed and position of

tl. Reuter, B. Breckling / Ecological Modelling 75/76 (1994) 147-159

149

four neighbours. The behavioural modes, Searching, Attraction, Parallel Orientation and Repulsion, are fixed to discrete zones of distances in which the neighbour fish are found. The influence of the four considered neighbours which are selected as the nearest ones to the swimming direction of the fish in question is averaged. For further details see Huth and Wissel (1994). A re-programming using the description given by Huth and Wissel (1992) allows us to confirm their results. It also provides a base of comparison with our model. To reproduce the high alignment they observed, it was necessary to restrict the starting orientation of the fish in the beginning of a simulation run to +90 ° deviation to a given direction as described by Huth (1992). Otherwise larger numbers of individuals often do not aggregate to one school but move to two or more different directions in independent schools. Nevertheless the inner structure of a school shows good similarity with real fish schools. We can conclude that the assumed schooling mechanism is not in contradiction with empirical observation. The front priority approach implies that a fish would have to exclude about 80% of its visual perception (school of 30 fish) in order to act accordingly. This might be possible under the assumption that schooling is done strictly in a binocular mode and is different from other orientation capacities. But there is no biological support for this assumption. On the contrary, the spherical fish eye leads to an acute image all over the retina. Distance and size can be estimated in monocular visual angle (Douglas and Djamgoz, 1990) and there are brain nuclei for all parts of the visual field. There is more evidence that fish are not only capable of perceiving their whole surroundings (excluding the body shadow behind) but also use this information. Experiments analysing the internal structure of fish schools in general showed the largest correlation between a fish and its nearest neighbour (Partridge and Pitcher, 1980; Partridge, 1981), only slightly depending on the relative position of the neighbour fish. Preferred are those in front (10 °) and those alongside (90°). This suggests a discussion of additional approaches in order to understand the range of possible underlying mechanisms.

3.

An

alternative

model

The model we propose calculates speed and direction of an individual in relation to all other visible fish, which are weighted according to their distance. Thus each percieved neighbour contributes a small component to a change of the orientation vector for the next moving step. By this we can model a gradual shift between the different reaction modes of attraction, parallel orientation and repulsion.

150


Table 1 Values of parameters and abbreviations (length units: cm) Fishlength Number of fish Number of neighbour fish Orientation vector Turning angle Standard orientation Weight factors Side length of starting area Time step Standard deviation of time step of change of orientation of change of velocity of solitary motion Minimum velocity per time step Maximum velocity per time step Sight range Angle of blind area Distance of max parallel orientation Repulsion area Corrected average distance of neighbours

fl i j wa dwaij osd wfj sl d(t) sd(t) sd(wa) sd(u) sd(s) t~min L,max sr wd md ra rna

30 units

210 units 0.5 s 0.05 s 5° 1 unit/step 10° 10 units 26 units 500 units 60° 50 units md / 6

3.1. M o d e l a s s u m p t i o n s

T h e m a i n a s s u m p t i o n s for t h e s i m u l a t i o n m o d e l a r e b a s e d on an a b s t r a c t i o n of t h e biological b a c k g r o u n d as d e s c r i b e d by P a r t r i d g e a n d P i t c h e r (1980), P a r t r i d g e (1981) a n d A o k i et al. (1986): - A l l fish have i d e n t i c a l b e h a v i o u r a l characteristics; t h e r e a r e no leaders. - T h e velocity a n d o r i e n t a t i o n of e a c h fish is c a l c u l a t e d as a r e a c t i o n to velocity, o r i e n t a t i o n a n d p o s i t i o n of its n e i g h b o u r s , c o m p l e t e d by a m i n o r stochastic effect. - T h e r e a r e no e x t e r n a l stimuli. - M a i n b e h a v i o u r a i p a t t e r n s (Aoki, 1982) a r e - N o n - c o n s i d e r a t i o n , if a n e i g h b o u r ' s d i s t a n c e e x c e e d s visibility. - A t t r a c t i o n , if t h e d i s t a n c e b e t w e e n fish is large. P a r a l l e l o r i e n t a t i o n within t h e r a n g e of the p r e f e r r e d swimming distance. - R e p u l s i o n , if individuals get too close to e a c h other. U s i n g this b a c k g r o u n d , t h e following c a l c u l a t i o n is d o n e to d e t e r m i n e t h e individual behaviour: T h e influence o f all visible n e i g h b o u r s is w e i g h t e d r e c i p r o c a l to t h e i r distance. T h u s we o b t a i n weight factors wfi (for a b b r e v i a t i o n s a n d scaling units see T a b l e 1) for all n e i g h b o u r s , for e x a m p l e 20% for t h e n e a r e s t a n d less t h a n 1% for the most d i s t a n t visible n e i g h b o u r s . T o o b t a i n the s t a n d a r d o r i e n t a t i o n osd, that is t h e new o r i e n t a t i o n a fish w o u l d have if all n e i g h b o u r fish w e r e in the p r e f e r r e d s w i m m i n g d i s t a n c e to each o t h e r , we a d d the angles dwaij by which the fish i m u s t turn to -

H. Reuter, B. Breckling / Ecological Modelling 7 5 / 7 6

rp :

151

(1994) 1 4 7 - 1 5 9

repulsion po :

parallel orientation /

Z

a

:

attraction

300

350

100~

80~ 6O-40--

2O--

,/

md

50

100

g,~ teo

150

200

250

I 400

I 450 mq

I 500

Fig. 1. Function of behavioral patterns, m d : distance of maximal parallel orientation (preferred swimming distance); ma: corrected average distance of neighbours; pa: parallel orientation, for rna > m d : pa = l / ( l + ( ( m d - m a ) / ( 1 . 5 × md)2)); for m a < m d and m a > ra: pa = l / ( l + ( ( m d r n a ) / ( 1 . 5 x rod~6)3)); a: Attraction, for m a > md: a = 1 - ( 1 / ( 1 + ( ( m d - m a ) / ( l . 5 × rod)2))); rp: Repulsion, for m a < m d and m a > ra: rp = 1 - ( 1 / ( 1 + ( ( m d - m a ) / ( 1 . 5 × m d /6)3)).

get the same orientation as his neighbours j (parallel orientation). These turning angles are weighted with the respective factors (Eq. 1),

osd = Edwaij. wfs J

(1)

The procedure for finding the standard speed and an attraction point (using the vectors to the visible neighbour fish) is similar. Which type of behaviour the fish will follow depends on the density of the neighbour fish in sight range. The behavioural decision is characterized by a function consisting of the average distance of all visible neighbours (see Fig. 1). This measure (ma) has to be adjusted according to the number of visible fish. The basis for the adjustment is the ideal situation of a regular triangular packing. The closer ma is to the chosen preferred swimming distance (rod), the more parallel orientation dominates. Otherwise repulsion (or attraction) causes stronger effects. Thus the shift between the different behavioural patterns is a gradual one. Repulsion is modelled as an increase of stochastic change of direction. If one of the neighbours gets too close

H. Reuter, B. Breckling/ Ecological Modelfing 75/76 (1994) 147-159

152

(< rod~6), repulsion

is the only component. Then the new orientation is plus or minus 36 ° to the direction of the neighbour fish (waj), depending on which turning angle is smaller. Any change of direction and velocity ts overlayed with a stochastic aspect, normally distributed with a standard deviation of 5 ° for the direction and 1.0 length unit for speed per time step. 3.2. Simulation sets

To start a simulation run, fish individuals are placed with random position, velocity and orientation into a starting area. After each time step each individual surveys its surrounding and changes swimming orientation and speed according to orientation, speed and position of visible neighbours. For each fish time steps of an average length of 0.33 s ( _+standard deviation of 0.033) are calculated individually. Each simulation run covers at least 100 steps. The runs are repeated 20 times with different random seed. The number of fish is increased by 10 from 10 to 50 individuals for each simulation set. The schooling behaviour is calculated in two spatial dimensions. All parameter values are fixed in an order of magnitude to match biological knowledge. The time step of 0.33 s corresponds with experimental

90-

180 c E) co

80--

c]

=

polarization

,c~

=

expanse

160

5 70-

140

nearest neighbour distance

0

120

6 0 -, c

50-

1O 0

~ o Q) c

40-

-

5(

20 -

2C

30

/

J

60

40 o

10 ~

qrokf4

80

S

-

1(

20

I

I

I

I

I

10

20

30

40

50

number of fish

Fig. 2. Characterisation of the school structure (standard run). Polarization, expanse and nearest neighbour distance for increasing number of fish.

H. Reuter, B. Breckling/ Ecological Modelling 75/76 (1994) 147-159

153

data of Partridge (1980, 1981) and Partridge and Pitcher (1980) concerning reaction times of fish. Together with a speed of 10-26 units ( = cm) per time step this leads to a cruising speed of 30-78 c m / s which covers the range from normal to high speed swimming for a fish of the assumed size (30 cm). Reducing speed or time step to investigate discretisation effects of the model did not affect simulation results qualitatively. To quantify the school movement the following variables are used: - The Polarization is a measure for the parallel alignment of the school. Values are between 0 ° for an entire parallel school and 90 ° for random orientation. - The Expanse quantifies the compactness of the school. It is calculated as the average distance of each fish to the schools centre of mass. - The Nearest Neighbour Distance is the average distance of each fish to its nearest neighbour.

4. Results Regardless of how many fish the school consists, we find that the model performs the main characteristics of real fish schools. The polarization (Fig. 2)

2

Fig. 3. Orientation vectors for a school of 30 fish. The vectors are displayed every 15 time steps. The length of an arrow is equivalent to the length of a fish.

154


stays high (about 10°), the expanse shows the predicted rise and the nearest neighbour distance gets less with increasing number of school members which is also known from real schools (Partridge and Pitcher, 1980), though this effect is a little too distinct in the model compared to reality. Fig. 3 shows the high polarization over many time steps. Further tests with our model reveal that it does not only keep schools together but also leads to the formation of successively merging schools if the fish are spread randomly over a very large area (Fig. 4 A - D ) . This is a realistic behaviour, it can be observed that smaller schools tend to aggregate to bigger ones (the upper limit of individuals observable in real schools is much higher than the number of individuals we dealt with in the model). Using the described model, the four nearest fish contribute to about 40% to the determination of a fish's movement (school consisting of 30 fish). The contribution of the most distant visible fish declines to less than 1%. In case the weighting function is changed to the reciprocal of the square of distance we obtain an increase of the weight of the four nearest individuals to about 75%. These schools also show high polarization and good cohesion when having more than 20 members (Fig. 5) 4.1. Schooling as a part of the behavioural repertoire

For biological reasons it has to be assumed that schooling behaviour is integrated into a larger complex of behaviour. Therefore it seemed interesting to investigate how the school formation is affected if individual fish intermediately switch to other locomotory priorities. An approach which can be investigated easily is to simulate feeding behaviour as an uptake of close but randomly distributed particles. This can be done through the introduction of a movement consisting only of a normal distributed step deviating slightly from the given orientation without considering other fish in the school (only the repulsion mechanism works). It is interesting to see, that if the percentage of time steps performing solitary motions stays below 50% only gradual changes in polarization or expanse can be seen. At 75% the schools polarization gets considerably higher but still the school does not disperse (Fig. 6). We see that the maintenance of a polarized school does not require the full attention of the involved individuals.

5. Discussion

Both the models of Huth and Wissel (1992, 1994) and the one presented here show some corresponding results and perform characteristics of real fish schools. Differences occur when embedding the schooling behaviour in more complex

Fig. 4. Formation of schools, traces of the individuals. The size of the starting area is 3000x3000 units, equal to the size of the displayed swimming area. The sight range of a fish is 500 units. Fish reaching a boundary are reflected, md = 60 and sd(wa) = 15. (A) after 50 time steps; (B) after 100 time steps; (C) after 200 time steps; (D) after 400 time steps.

11. Reuter, B. Breckling / Ecological Modelling 75/76 (1994) 147-159

155

Y

A v~r

f -L

~r k4


156

C v~r

Fishlength Siqhtronge

l(d.

D vpr

I~t

Fig. 4 (continued).

H. Reuter, B. Breckfing / Ecological Modelling 75 / 76 (1994) 147-159

180

90

c 0 '~

157

122]

=

polarization

~.

=

expanse

0

=

nearest neighbour distance

160

80

°a 7o

140

120

6O

5O

1O0

4O

80

30

60

2O

40 G

~

~

o

o

o 20

I0

Qrokf4

r

i-

T

T

]~

10

20

30

40 number

50 of

fish

Fig. 5. School structure data (neighboursweighted reciprocal to square distance). Polarization,expanse and nearest neighbour distance for increasing fish numbers. interactions especially concerning school aggregation. The model of Huth and Wissel does not reproduce aggregation when fish are set with random orientation into a starting area. This is a consequence of the orientation behaviour of their model fish, which determine their swimming direction according to the position of the individuals which have the smallest deviation angle to their own swimming direction (prolongation of body axis), regardless of distance (within sight range). The splitting of schools when reaching obstacles (e.g. boundaries) is due to the same effect of the formation of subgroups within the school. The investigation of schooling behaviour is an excellent example that the understanding of biological interaction of individuals requires a combination of field observation, theoretical considerations and laboratory investigations. The different hypotheses could stimulate neurophysiological experiments to learn more about signal processing in the fish brain and also encourage further analysis of fish school structure, to give proof to either assumption. Even if at present important questions remain open, it is possible to exclude some behavioural patterns through theoretical consideration and simulation experiments: Huth and Wissel (1992) showed that polarized schooling does not result if a fish orientates on just one neighbour. In addition considering a fixed number of nearest neighbours to calculate the new swimming direction and speed of a respective fish would not lead


158

:::'C °

90,if,

c o

g

-o

lj -

18o

160 Q. x 140

0 c 0 60

] 20 c

5o.

~

100

0 c

40 •

4C

80

30 -

3C

60

20 -

2C

40

10 -

1C

20

-~--I 0 - Qro kf44 . . . . . . . .

I I0

20

I

l 30

l

l 40

l

I 50

I

I 60

I

I

I I

70

80

percent

I

I

I

90 solitorv

I I O0 movement

Fig. 6. Increasing solitary movements. Polarization, expanse and nearest neighbour distance for increasing solitary movements.

to schooling with either model. Using only the four nearest neighbours to influence a fish leads to subgroup formation in which particular neighbours influence each other mutually enabling them to ignore the rest of the school and split it. Using a larger number of neighbour fish leads only to larger subgroups. At least two possibilities to prevent the stabilization of small subgroups are possible: to consider exclusively neighbours swimming directly in front (Huth and Wissel, 1992, 1994) or as suggested here to take into account all distant neighbours using a weighting function. We expect that future investigations on fish ethology will allow a deeper understanding of the mechanisms of individual action. The available simulation approaches allow an easy testing to confirm whether the proposed behaviour leads to the formation of the observed school type.

References Aoki, I., 1982. A simulation study on the schooling mechanisms in fish. Bull. Jpn. Soc. Sci. Fish., 48: 1081-1088. Aoki, I., Inagaki, T. and Long, L.V., 1986. Measurements of the 3-D-structure of free-swimming pelagic schools in natural environment. Bull. Jpn. Soc. Sci. Fish., 52: 2069-2078.


159

Douglas, R. and Djamgoz, M., 1990. The Visual System of Fish. Chapmann and Hall, London. Huth, A., 1992. Ein Simulationsmodell zur Erkl~irung der kooperativen Bewegung von polarisierten Fischschw~irmen. Dissertation, Universit~it Marburg. Huth, A. and Wissel, C., 1992. The simulation of the movement of fish schools. J. Theor. Biol., 156: 365-385. Huth, A. and Wissel, C., 1994. The simulation of fish schools in comparison with experimental data. Ecol. Modelling, 75/76: 135-146. Kils, U., 1986. Verhaltensphysiologische Untersuchungen an pelagischen Schw~irmen. Schwarmbildung als Strategie zur Orientierung an Umweltgradienten. Bedeutung der Schwarmbildung in der Aquakultur. Habilitationsschrift, Universit~it Kiel. Klimley, A.P., 1985. Schooling in Sphyrna lewini, a species with low risk of predation: a non-egalitarian state. Z. Tierpsychol., 70: 279-319. Magurran, A.E., 1990. The adaptive significance of schooling as an anti-predator defence in fish. Ann. Zool. Fenn., 27: 51-66. Partridge, B.L., 1980. The effect of school size on the structure and dynamics of minnow schools. Anim. Behav., 28: 68-77. Partridge, B.L., 1981. Internal dynamics and the interrelations of fish schools. J. Comp Physiol., 144: 313-325. Partridge, B.L. and Pitcher, T.J., 1980. The sensory basis of fish schools: relative roles of lateral line and vision. J. Comp. Physiol., 135: 315-325. Partridge, B.L., Pitcher, T.J., Cullen, J.M. and Wilson, J., 1980. The three dimensional structure of fish schools. Behav. Ecol. Sociobiol., 6: 277-288. Pitcher, T.J., Magurran, A.E. and Winfield, l.J., 1982. Fish in larger shoals find food faster. Behav. Ecol. Sociobiol., 10: 149-151. Radakov, D.V., 1973. Schooling in the Ecology of Fish. Halsted Press, New York.