Hydrodynamic interaction between two copepods: a numerical study · 2005. 3. 9. · 4CURRENT...

I N T RO D U C T I O N

What is the nature of hydrodynamic interaction betweentwo copepods? Many copepod species can generatefeeding currents by moving their mouthparts (Alcaraz etal., 1980; Koehl and Strickler, 1981; Strickler, 1982, 1985;Paffenhöfer et al., 1982). Copepods can also swim or sinkthrough the water column (Tiselius and Jonsson, 1990;Bundy and Paffenhöfer, 1996). The three-dimensionalmovement of a copepod’s body and/or appendagesresults in the flow field around the copepod. When twocopepods are in close proximity or approaching eachother, the velocity field generated by the movement of one

copepod is transmitted through the fluid medium andaffects the flow field around, as well as the hydrodynamicforce and torque on the other copepod.

Studying the hydrodynamic interaction between twocopepods will benefit our understanding of several keyissues when copepods are in close proximity. First, arethere any energetic advantages for copepods in denseswarms or in close proximity to each other? If there arebenefits available to members, then it is advantageous tomaintain membership in a group. For example, it has beenhypothesized and tested that individuals in a fish school orbird flock can gain hydrodynamic advantages and there-fore save energy for swimming or flying by adopting

© Oxford University Press 2002

Hydrodynamic interaction between twocopepods: a numerical studyHOUSHUO JIANG1,4, THOMAS R. OSBORN1,3 AND CHARLES MENEVEAU2,3

1DEPARTMENT OF EARTH AND PLANETARY SCIENCES, 2DEPARTMENT OF MECHANICAL ENGINEERING, 3CENTER FOR ENVIRONMENTAL AND APPLIED

FLUID MECHANICS, THE JOHNS HOPKINS UNIVERSITY, BALTIMORE, MD , USA

4CURRENT ADDRESS: MS #, DEPARTMENT OF APPLIED OCEAN PHYSICS AND ENGINEERING, WOODS HOLE OCEANOGRAPHIC INSTITUTION, WOODS

HOLE, MA , USA

CORRESPONDING AUTHOR: E-MAIL: [email protected]

Numerical simulations were carried out to compute the flow field around two tethered, stationary or

swimming model-copepods with varied separation distances between them and for different relative

body positions and orientations. Based on each simulated flow field, the power expended by each

copepod in generating the flow field and volumetric flux through the capture area of each copepod were

calculated. The geometry of the flow field around each copepod was visualized by tracking fluid

particles to construct stream tubes. The hydrodynamic force on each copepod was calculated. Also,

velocity magnitudes and deformation rates were calculated along a line just above the antennules of

each copepod. All the results were compared to the counterpart results for a solitary copepod (sta-

tionary or swimming) to evaluate the hydrodynamic interaction between the two copepods. The calcu-

lations of the power and volumetric flux show that no energetic benefits are available for two copepods

in close proximity. The results of the stream tube and force calculations show that when two cope-

pods are in close proximity, the hydrodynamic interaction between them distorts the geometry of the

flow field around each copepod and changes the hydrodynamic force on each copepod. Two beneficial

roles of the hydrodynamic interactions are suggested for copepod swarms: (1) to maintain the integrity

of the swarms and (2) to separate the swarming members with large nearest neighbour distances

(usually more than five body lengths). To prevent strong hydrodynamic interactions, copepods in

swarms have to avoid positions of strong interactions, such as those directly above or below their

neighbours. The results of the velocity magnitudes and deformation rates demonstrate that the hydro-

dynamic interaction between two copepods generates the hydrodynamic signals detectable by the setae

on each copepod’s antennules. Based on the threshold of Yen et al. (1992), the results show that the

detection distance between two copepods of comparable size is about two to five body lengths. Cope-

pods may employ a simple form of pattern recognition to detect the distance, speed and direction of

an approaching copepod of comparable size.

JOURNAL OF PLANKTON RESEARCH VOLUME NUMBER PAGES ‒

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appropriate positions relative to their immediate neigh-bours in order to benefit from vortices shed by the neigh-bours (Breder, 1965; Lissaman and Schollenberger, 1970;Weihs, 1973). In contrast to fish schools or bird flocks,there is no indication of a preferred nearest-neighbour-bearing (NNB) in crustacean aggregations (O’Brien,1989); instead individuals in aggregations are at randomangles to their nearest neighbours. This suggests a lack ofan energetic advantage to individuals in crustacean aggre-gations. However, the studied species in O’Brien’sresearch were euphausiid and mysid species and little isknown about copepod swarms.

Second, how much can the hydrodynamic interactionsbetween copepods affect the flow field around individualcopepods? A solitary copepod maintains a flow field witha certain geometry around its body. The flow field aroundthe copepod is ecologically important, since it brings foodto the copepod and transmits signals to the chemorecep-tors and mechanoreceptors on the copepod, benefitingthe copepod’s feeding and predator avoidance. When twocopepods are in close proximity, the flow field generatedby one copepod has to satisfy the boundary conditions onthe other body, and hence the flow field around eachcopepod is different. The influence due to the hydrody-namic interactions on an individual copepod’s flow fielddepends on the separation distance between the two cope-pods, their relative body positions and orientations, theirrelative swimming velocities and their respective move-ments of mouthparts and appendages. It is unclear towhat extent the flow-field differences resulting from thehydrodynamic interactions between copepods can affectthe ecological functions of the flow field around the indi-vidual copepods.

Third, how, and at what distance, can a copepod detectthe presence of nearby copepods of comparable size?This is an extremely important question. In the labora-tory, one frequently observes that when a copepod isapproached by another copepod, at a certain separationbetween the two, the former detects the latter and startsan escape reaction [from video tapes provided by Prof. J.R. Strickler; see also (Strickler, 1975)]. When formingswarms, copepods have to be aware of the presence ofothers in order to maintain the integrity of the swarmswhile avoiding physical contact. It is likely that in swarmscopepods use mechanoreception to detect conspecificsand avoid bumping into each other (Strickler, 1975). Formechanoreception, the hydrodynamic interactionsbetween copepods can generate disturbances that can bepotentially detected by the copepods and enable thedetection. Components of these fluid disturbancesinclude fluid velocity, deformation rate, vorticity andacceleration (Kiørboe and Visser, 1999). Previously, thedetection distances between copepods were examined in

terms of the detection distances between predators andprey (Lenz and Yen, 1993). However, Haury andYamazaki pointed out that the nearest-neighbour dis-tances (NNDs) between monospecific swarming indi-viduals are much greater than known reaction distances inpredator–prey interactions (Haury and Yamazaki, 1995)and further suggested that the copepods in aggregationsare always aware of the presence of others at muchgreater distances than those in predator-prey interactions.This is supported by some studies of laboratory-inducedcopepod swarms, which suggest that the fluid disturbancesurrounding individual moving copepods, rather than theexoskeleton, maintains a minimum separation distance inswarms (Leising and Yen, 1997). Even though the reactiondistances as reported by Haury and Yamazaki (Haury andYamazaki, 1995) may not be the detection distances (sincea copepod may detect a signal physiologically, but thesignal may not be strong enough to elicit an obviousbehavioural response), their study may still indicate thatthe detection distances between copepods and their preymay not necessarily represent the detection distancesbetween copepods of comparable size. This is under-standable. Since the body size of conspecific swarmingmembers is quite uniform (Ritz, 1994), the membersshould be mutually affected. Whereas, in the situation ofthe predator–prey interactions, because of the consider-able size differences, the hydrodynamic disturbancesgenerated by prey may not be powerful enough to affectthe hydrodynamic field set up by predators. Hence thedetection distances between copepods of comparable sizeare expected to be greater than those in copepod preda-tor–prey interactions.

To answer these questions, we need to quantify the flowfield around two copepods of comparable size—in closeproximity—and examine their hydrodynamic interaction.It is difficult experimentally to visualize and measure thehydrodynamic signals between copepods in close proxim-ity or within a copepod swarm. However, numerical simu-lation methods can be employed to attack this problem.Jiang et al. ( Jiang et al., 1999) developed a computationalfluid dynamics (CFD) method to simulate the feedingcurrent around a tethered copepod and demonstratedthat the direct numerical simulation of the feeding currentaround a copepod is possible. The authors used thenumerical method together with some basic concepts influid mechanics to describe and explain properties of thefeeding currents of copepods. Based on the copepod’smorphology and the movement of the feeding currentgenerating appendages of the copepod, a numericalfeeding current was produced and found to be com-parable to observations. It was shown that the morphol-ogy and the movement actually determine the geometryof the flow field. The geometry was visualized by tracking


06jiang (44w)(ds) 13/2/02 9:39 am 36

fluid particles to construct stream tubes in the calculatedflow field. The volumetric flux through the copepod’scapture area was introduced to study the effectiveness ofthe feeding current as a vehicle to draw algal particles tothe capture area and energetic efficiency examined. Theviscous-dissipation-rate field due to the feeding current,which also reflects the deformation of the flow field, wasshown to be a possible mechanosensory signal field,potentially detected by prey or predators of the copepod.In the present work, we will use a similar method to cal-culate the flow field around two copepods of identical sizein close proximity and similar data analysis methods toexamine the hydrodynamic interaction between the twocopepods.

N U M E R I C A L M E T H O D

The present work is a direct application of the numericalmethod of Jiang et al. ( Jiang et al., 1999). Since the flowfield around two copepods in close proximity is dependenton the separation distance between the two copepods,their relative body positions and orientations, their relativeswimming velocities and their respective movements ofmouthparts and appendages, some adaptations to thesimulation method are needed to include these com-ponents. Specifically, the body shapes of two identical

model-copepods are portrayed by using the body-fitted coordinates. The separation distance between the twocopepods is varied for four relative body position andorientation scenarios: ventral-to-ventral, dorsal-to-dorsal,horizontally side-by-side and one-above-the-other. Suit-able outer boundary conditions are adopted to simulatethe swimming of the copepods. Then, the flow fieldaround the two copepods is computed with varied separ-ation distances between them for the four relative bodyposition and orientation scenarios. In the following wedescribe the numerical method in detail.

Model-copepod

The body shape of the ‘model-copepod’ consists of aprosome, a urosome and two antennules, which isdesigned after the external morphology of a species ofcoastal water copepods. The dimensions of the bodyparts are shown in Figure 1, which are in the range of atypical adult female copepod. The body shape is intro-duced into the numerical simulation using the curvilinearbody-fitted coordinates. Since the beating movement ofcopepods’ mouthparts plays a key role in shaping thegeometry of the flow field in the vicinity of the mouth-parts, it is crucial to include the effect of the beatingmovement into the numerical simulation in order toreproduce a realistic flow field around copepods’ body.

H. JIANG, T. R. OSBORN AND C. MENEVEAU HYDRODYNAMIC INTERACTION BETWEEN COPEPODS

3.2 mm

3.2

mm

0.6 mm

0.3

mm

0.4 mm

1.3 mm

capture area

capture areacapture area

(a)(b)

(c)

Fig. 1. Body shape, dimensions of body parts and definition of the capture area for the model-copepod. (a) Lateral view; (b) ventral view; (c)anterior view. In (c), the solid-lined ellipse circumscribing the capture area is the base of the stream tubes in Figures 4–9, and through the dash-lined rectangular around the capture area, volumetric flux is calculated.

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17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

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K1.3

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Fig. 2. Illustration of the distribution of forces representing the effect of the beating movement of the cephalic appendages of the model-copepod,on a J-plane. The distribution consists of eight J-planes: J = 22–25 and J = 28–31. On each J-plane, forces are applied on 29 finite volume cells asshown in this figure. For simplicity, all forces are applied along the negative z-direction. The magnitude of each force applied on K = 21, 22 is 5.7� 10–11 N, on K = 23, 24 is 1.14 � 10–10 N, on K = 25, 26 is 1.71 � 10–10 N, on K = 27, 28 is 2.28 � 10–10 N, and on K = 29, 30 and 31 is 2.85� 10–10 N. The total force is 4.8 � 10–8 N. The maximum velocity magnitude generated from this distribution is 4.5 mm s–1.

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However, it is difficult to use the curvilinear body-fittedco-ordinates to depict the morphology of the mouthpartsin detail and it is also difficult to simulate the complexmovement of the mouthparts. According to the analysisin Jiang et al. ( Jiang et al., 1999), the effect of the beatingmovement of the mouthparts can be represented byapplying distributed forces to the finite volume cells ven-trally adjacent to the main body of the model-copepod.For the present work, the force magnitudes and theirspatial distribution relative to the model-copepod areillustrated in Figure 2. A total force of 4.8 � 10–8 N isapplied and generates a flow field with the maximumvelocity magnitude of 4.5 mm s–1, which is comparableto some observational data (Strickler, 1982; Tiselius andJonsson, 1990; Bundy and Paffenhöfer, 1996).

Relative body position and orientation

In the present work, we consider four relative bodyposition and orientation scenarios: ventral-to-ventral,dorsal-to-dorsal, side-by-side and one-above-the-other.The meshes for the four scenarios are obtained by modi-fying the mesh of a solitary model-copepod, i.e. the base-line mesh for the present work, as illustrated in Figure 2.Basically, the meshes for these two-copepod scenarioshave same grid-distribution around each model-copepod’s body and distribution of forces applied to thefinite volume cells as those in the baseline mesh; however,the intermediate grid-distribution is modified to changethe separation distance between the two copepods. Theseparation distance between the two copepods is definedas the distance between the nearest surfaces of the mainbodies. Six meshes are made for two ventral-to-ventralcopepods with the respective separation distancesbetween the two copepods of 1.06 mm, 1.41 mm,1.78 mm, 2.18 mm, 3.09 mm and 4.95 mm. The mesheswith the shortest and longest separation distancesbetween the two ventral-to-ventral copepods are shownin Figures 3a and 3b, respectively. For two dorsal-to-dorsal copepods, seven separation distances betweenthe two copepods are considered, i.e. 0.83 mm, 1.23 mm,1.86 mm, 2.33 mm, 2.84 mm, 3.43 mm and 4.10 mm.Figures 3c and 3d show respectively the grid distributionsbetween the two dorsal-to-dorsal copepods with theshortest and longest separation distances. The longestand shortest separation distances between two side-by-side copepods are 4.88 mm and 3.43 mm, respectively(Figures 3e and 3f ), between which one separation distance is considered (i.e. 4.10 mm). Five separation distances are chosen for the scenario of two one-above-the-other copepods. The shortest separation distance is1.16 mm and the longest is 4.18 mm (Figures 3g and 3h).The three intermediate separation-distances are 1.63mm, 2.14 mm and 3.40 mm.

Calculation of the flow field

For each mesh, the flow field around the two copepods iscomputed by using a finite-volume code, FLUENT™. Asymmetry boundary condition is used for the cases of twoventral-to-ventral, dorsal-to-dorsal, or side-by-side cope-pods, as the physical geometry of interest and expectedflow patterns have mirror symmetry. For the purpose ofcomparison, the flow field around a solitary copepod isalso computed. In order to simulate the constant swim-ming motion of the copepods, suitable velocity inletboundary conditions are applied on the boundaries of thecomputational domain while the frame of reference isfixed on one copepod’s body. For the numerical schemessee Jiang et al. ( Jiang et al., 1999).

N U M E R I C A L R E S U LT S

Power and volumetric flux

Two stationary copepods

The first category considered is two copepods in closeproximity, both of which remain stationary and create astrong flow with their mouthparts. From the computedthree-dimensional flow field, the power expended by eachcopepod in generating the flow field and volumetric fluxthrough each copepod’s capture area can be calculated foreach two-copepod pair considered in the present study.The power (W· ) is estimated using the formula:

W· = f vi ii

N

1$

=

! (1)

where the sum is taken over the N cells on which the dis-tributed force is applied, fi is the force applied on the ithcell and vi is the velocity at the centre of the ith cell. Thevolumetric flux (Q) is calculated as

Q dv AA

$= ## (2)

where A is the capture area and v is the flow velocity. Forsimplicity of the numerical integration, a rectangular areaas shown in Figure 1c is chosen. The area is 0.28 mm inthe x-direction and 0.6 mm in the y-direction, centred atthe point (0.22 mm, 0.0, 0.16 mm) for the baseline mesh.In Table I the calculated power and volumetric flux aretabulated with respect to the four relative body positionand orientation scenarios and separation distancesbetween the two copepods, and the power/volumetricflux is normalized by that for a solitary copepod. Theresults clearly show that the power expended by a copepodin every stationary two-copepod pair is almost equal to thepower expended by the solitary copepod and that the volumetric flux through the capture area of a copepod inevery stationary two-copepod pair is almost identical to


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the volumetric flux for the solitary copepod, insensitive tothe separation distance between the two copepods andtheir relative body positions and orientations.

Two swimming copepods

The second category considered is two copepods in closeproximity, both of which swim in the same fashion whilemoving their mouthparts to generate a strong flow. Whilecopepods exhibit various swimming behaviours, in the

present study we only consider the behaviour of swim-ming dorsal-ward to show examples. The pair swimdorsal-ward at a same speed of 1.5 mm s–1 while tiltingtheir body axis dorsal-ward at the same angle of 20°, sothat only pairs of two side-by-side or one-above-the-othercopepods are considered in this category and the flowfields are computed accordingly. In calculating the powerand volumetric flux for the swimming cases, equations (1)and (2) can still be used; however the velocity v is


Fig. 3. Meshes for the two ventral-to-ventral copepods separated by a distance of (a) 1.06 mm (shortest) and (b) 4.95 mm (longest); the inter-mediate separation distances between the two ventral-to-ventral copepods are 1.41 mm, 1.78 mm, 2.18 mm and 3.09 mm. Meshes for the twodorsal-to-dorsal copepods separated by a distance of (c) 0.83 mm (shortest) and (d) 4.10 mm (longest); the intermediate separation distances betweenthe two dorsal-to-dorsal copepods are 1.23 mm, 1.86 mm, 2.33 mm, 2.84 mm and 3.43 mm. Meshes for the two side-by-side copepods separatedby a distance of (e) 3.43 mm (shortest) and (f) 4.88 mm (longest); the intermediate separation distance between the two side-by-side copepods is4.10 mm. Meshes for the two one-above-the-other copepods separated by a distance of (g) 1.16 mm (shortest) and (h) 4.18 mm (longest); the inter-mediate separation distances between the two one-above-the-other copepods are 1.63 mm, 2.14 mm and 3.40 mm.

Table I: Normalized power/volumetric flux for a copepod in a stationary two-copepod pair

Separation Normalized Normalized

distance (mm) power volumetric flux

A solitary copepod infinity 1.000 1.000

Two ventral-to-ventral copepods 1.06 0.988 0.996

1.41 0.993 0.997

1.78 0.998 1.000

2.18 1.000 1.001

3.09 1.000 1.001

4.95 1.000 1.000

Two dorsal-to-dorsal copepods 0.83 1.002 1.003

1.23 1.002 1.002

1.86 1.001 1.001

2.33 1.001 1.001

2.84 1.001 1.001

3.43 1.001 1.001

4.10 1.001 1.001

Two side-by-side copepods 3.43 1.000 1.001

4.10 1.000 1.000

4.88 1.000 1.000

The upper one of the two 1.16 1.006 1.015

one-above-the-other copepods 1.63 1.005 1.012

2.14 1.004 1.011

3.40 1.002 1.009

4.18 1.001 1.008

The lower one of the two 1.16 1.014 1.020


2.14 1.007 1.010

3.40 1.004 1.005

4.18 1.003 1.004

Power/volumetric flux is normalized by that for the solitary copepod.

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measured in a frame of reference fixed on the copepods’body. The results are listed in Table II, in which the powerand volumetric flux are normalized by those for a swim-ming solitary copepod. From Table II, almost no energysaving relative to the swimming solitary copepod isobserved for the swimming paired copepods. Meanwhile,the volumetric flux through the capture area of a copepodin every swimming two-copepod pair is almost equal tothe volumetric flux for the swimming solitary copepod,despite the fact that different separation distances betweenthe two copepods and different relative body positions andorientations have been considered.

Geometry of the flow field around twocopepods in close proximity

In order to display the geometry of the simulated flowfields, particle tracking is used to construct stream tubesthrough each copepod’s capture area [for methodologiessee Jiang et al. (1999)]. For each copepod in each case, 40points on a small ellipse (as shown in Figure 1c) are chosenas the initial conditions for the particle tracking. The smallellipse is ventral to the copepod and circumscribes thecopepod’s capture area. For example, for the baseline caseof a solitary model-copepod, the ellipse is centred at (0.22mm, 0.0, 0.16 mm) with a semi-major axis of 0.3 mm inthe y-direction and semi-minor axis of 0.14 mm in the x-direction. For the purpose of comparison, the stream-tube through the capture area of a solitary model-copepod is calculated for both stationary and swimmingcases. The stream tube through the capture area of thestationary solitary copepod is plotted in Figure 4, whichclearly shows that the water flow is directed from theventral-anterior direction to the copepod’s capture area.Figure 5 shows the stream tube through the capture area

of the solitary copepod swimming at a speed of 1.5 mms–1 while tilting its body axis dorsal-ward at an angle of20°. The stream tube for the swimming solitary copepodstarts from the place dorsally far away from the copepod;after passing just above the antennules, the stream tubeturns to go through the capture area.

Two stationary copepods

When two copepods are ventral-to-ventral and in closeproximity (Figure 6a), the stream tube for each copepoddiffers from that for the stationary solitary copepod (bycomparing Figure 6a with Figure 4a). Instead of from theanterior-ventral direction, the water flow is directed fromthe dorsal-anterior direction, passing by the place justabove the antennules, to the capture area. On the otherhand, for two dorsal-to-dorsal copepods (Figure 6b),because the hydrodynamic interaction between the twocopepods is weak, the stream tube for each copepod isquite similar to that for the stationary solitary copepod (bycomparing Figure 6b with Figure 4a). In addition, for twoside-by-side copepods (Figure 6c), the stream tube for eachcopepod becomes asymmetric about the median plane ofeach copepod, different from the symmetry as shown inFigure 4b. Finally, the stream tubes for two one-above-the-other copepods are plotted in Figure 7. For a separationdistance of 1.16 mm between the two copepods (Figure7a), the stream tube for the lower copepod can last less than2 seconds before connecting with the stream tube for theupper copepod; while for a separation distance of 4.18 mm(Figure 7b), the stream tube for the lower copepod can lastmore than 10 seconds before connecting with the streamtube for the upper copepod. In both cases, the deviation ofthe stream tube for the lower copepod from that for the sta-tionary solitary copepod is prominent.


Table II: Normalized power/volumetric flux for a copepod in a swimming two-copepod pair

Separation Normalized Normalized

distance (mm) power volumetric flux

A solitary copepod infinity 1.000 1.000

Two side-by-side copepods 3.43 1.005 1.006

4.10 1.005 1.006

4.88 1.005 1.006

The upper one of the two 1.16 1.006 1.013


4.18 1.002 1.008

The lower one of the two 1.16 1.012 1.015


4.18 1.005 1.006

Power/volumetric flux is normalized by that for the solitary copepod.

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Two swimming copepods

When two copepods are swimming side by side and inclose proximity, instead of following the swimming direc-tion, the two stream tubes respectively associated with thetwo copepods attract each other toward the symmetryplane between the two copepods (Figure 8). In addition,

the strong hydrodynamic interaction between the twocopepods makes the flow along the inner-side (i.e. the partnear the other copepod) of each copepod’s stream tubemuch faster than that along the outer-side. Figure 9 showsthe most interesting cases: for the two one-above-the-other copepods with a separation distance of 1.16 mm,


t=0.0s

t=-1.0s

t=-3.0s

t=-7.0s

t=-12.0s

0

-2

2

4

z (m

m)

0 2-2 4x (mm)

t=0.0s

t=-1.0st=-3.0s

t=-7.0st=-12.0s

0 2-2-4 4y (mm)

0

-2

2

4

z (m

m)

(a) (b)

Fig. 4. Stream-tube through the capture area of the stationary solitary copepod. (a) Lateral view; (b) Ventral view.

(a) (b)

t=0.0s

t=-1.0s

t=-3

.0s

t=-7

.0s

t=-1

2.0s

0

-2

2

4

z (m

m)

-4 -2-6 0x (mm)

Vswimming=1.5 mm/s

t=0.0st=-1.0st=

-3.0

s

t=-7

.0s

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2.0s

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4

y (m

m)

-4 -2-6 0x (mm)

Vswimming=1.5 mm/s

Fig. 5. Stream-tube through the capture area of the solitary copepod swimming in the negative x-direction at a speed of 1.5 mm s–1 while tiltingits body axis dorsal-ward at an angle of 20°. The frame of reference is fixed on the copepod’s body. (a) Lateral view; (b) Anterior view.

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the stream tube for the lower copepod can last less than1.5 seconds before connecting with the stream tube for theupper copepod (Figure 9a). While for the two one-above-the-other copepods with a separation distance of 4.18mm, both stream tubes are well separated and not con-nected with each other (Figure 9b). However, in both casesthe stream tube for the lower copepod is significantlydifferent from that for the swimming solitary copepod.

Hydrodynamic force

The hydrodynamic force on each copepod in a two-copepod pair is calculated from the three-dimensionalflow field. The flow pressure and shear stress are

integrated over the body–fluid interface to form the fric-tion and pressure drag. The sum of the friction andpressure drag is the total drag, i.e. the hydrodynamic force.For the purpose of comparison, the hydrodynamic forceon the solitary copepod (stationary or swimming) is alsocalculated.

Table III lists the calculated hydrodynamic forces onthe body surface of a copepod in a stationary two-copepod pair as well as the hydrodynamic forces on a stationary solitary copepod. For the copepods in aventral-to-ventral, dorsal-to-dorsal, or one-above-the-other copepod pair, the y-direction drag force, i.e. the horizontally side-to-side drag force, is negligible


Fig. 6. (a) Lateral view of the two stream tubes respectively throughthe respective capture areas of the two copepods that are ventral-to-ventral, stationary and separated by a distance of 1.06 mm. (b) Lateralview of the two stream tubes respectively through the respective captureareas of the two copepods that are dorsal-to-dorsal, stationary and sep-arated by a distance of 0.83 mm. (c) Ventral view of the two streamtubes respectively through the respective capture areas of the two cope-pods that are side-by-side, stationary and separated by a distance of 3.43mm.

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compared to the forces in the other two directions, asexpected from symmetry. Table III clearly shows thatthe x- and z-direction forces for the paired copepods are

significantly different from those for the stationary soli-tary copepod. When the separation distance betweenthe paired copepods decreases, the difference in theforce magnitude increases substantially. For the cope-pods in a side-by-side copepod pair, the y-directionforces are not negligible due to the asymmetry in theflow field and the two copepods attract each other.Table IV tabulates the hydrodynamic forces calculatedfor the swimming cases. It is shown that the forces on thebody surface of the copepods in a swimming copepodpair are significantly different from those on the swim-ming solitary copepod, increasing substantially as theseparation distance between the paired copepodsdecreases.

Velocity magnitude and deformation rate

To quantify the signals that can be potentially sensed bythe mechanoreceptors on copepods’ antennules, velocitymagnitudes and deformation rates are calculated along aline just above the antennules. The distance between theline and the antennules is about 300 µm (Figure 10).Within this distance we believe the signals can be sensedby the mechanoreceptors, since the antennules areequipped with setae of lengths varying between 200 and500 µm and some setae could be as much as 1 mm longfor certain species (Yen et al., 1992). The deformation rateis calculated as


Fig. 7. Lateral view of the two stream tubes respectively through the respective capture areas of the two copepods that are one-above-the-other,stationary and separated by a distance of (a) 1.16 mm and (b) 4.18 mm.

Fig. 8. Anterior view of the two stream tubes respectively through therespective capture areas of the two copepods that are side-by-side, swim-ming in the negative x-direction at the same speed of 1.5 mm s–1 whiletilting their body axis dorsal-ward at the same angle of 20° and sepa-rated by a distance of 3.43 mm. The frame of reference is fixed on oneof the two copepods.

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Fig. 9. Lateral view of the two stream tubes respectively through the respective capture areas of the two copepods that are one-above-the-other,swimming in the negative x-direction at a same speed of 1.5 mm s–1 while tilting their body axis dorsal-ward at a same angle of 20° and separatedby a distance of (a) 1.16 mm and (b) 4.18 mm. The frame of reference is fixed on one of the two copepods.

Table III: Drag forces on the body surface of a copepod in a stationary two-copepod pair

Separation x-dir. force y-dir. force z-dir. force

distance (mm) (10–9 N) (10–9 N) (10–9 N)

A solitary copepod infinity –0.025 0.009 –36.40

Two ventral-to-ventral copepods 1.06 3.272 0.003 –39.02

2.18 1.137 0.006 –37.27

4.95 0.176 0.009 –36.43

Two dorsal-to-dorsal copepods 0.83 0.463 0.013 –37.17

2.33 0.172 0.014 –36.66

4.10 0.073 0.014 –36.44

Two side-by-side copepods 3.43 –0.018 0.152 –36.57

4.10 –0.019 0.101 –36.49

4.88 –0.020 0.067 –36.42

The upper one of the two 1.16 0.204 0.010 –37.12

one-above-the-other copepods 2.14 0.073 0.009 –37.00

4.18 0.001 0.009 –36.70

The lower one of the two 1.16 –0.250 0.008 –38.50

one-above-the-other copepods 2.14 –0.143 0.010 –37.70

4.18 –0.077 0.010 –36.92

The positive x-direction: for a solitary copepod or for copepods in pairs of two ventral-to-ventral, side-by-side, or one-above-the-other copepods, is thedirection from dorsal side to ventral side; for copepods in pair of two dorsal-to-dorsal copepods, it is the direction from ventral side to dorsal side.The positive y-direction: for the pairs of two side-by-side copepods, is the direction side to side from one copepod to the other.The positive z-direction: is the direction from posterior to anterior.

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S S S

xu

yv

zw

yu

xv

zu

xw

zv

yw

21

21

21

*ij ij

2 2 2 2

2 2

22

22

22

22

22

22

22

22

22

/ =

+ + + +

+ + + +

c c c c

c c

m m m m

m m

(3)

where

Sxu

x

u

21

ijj

i

i

j

22

2

2= +

J

L

KK

N

P

OO (4)

stands for the components of the strain rate tensor S,and the summation convention applies. The indicial

notations are adopted with the xyz axis referred to as xi,i =1, 2, 3, and the uvw velocity components referred toas ui, i =1, 2, 3. The (u, v, w) velocity components havebeen calculated through the numerical simulations. Thedeformation rate defined in this form is invariant underany transformation of the coordinate system. Thedeformation rate S* is related to the viscous dissipationrate �:

S� ��

2 * 2= (5)

where � is the density of the fluid and µ is the dynamic viscosity of the fluid.


Table IV: Drag forces on the body surface of a copepod in a swimming two-copepod pair

Separation x-dir. force y-dir. force z-dir. force

distance (mm) (10–9 N) (10–9 N) (10–9 N)

A solitary copepod infinity 8.47 0.005 –36.84

Two side-by-side copepods 3.43 10.0 0.169 –38.26

4.10 8.66 0.091 –38.02

4.88 7.58 –0.003 –37.82

The upper one of the two 1.16 4.59 0.008 –37.65


4.18 4.77 0.009 –37.23

The lower one of the two 1.16 2.44 0.011 –38.02


4.18 3.52 0.011 –37.12

The positive x-direction: is the direction from dorsal side to ventral side.The positive y-direction: for the pairs of two side-by-side copepods, is the direction side to side from one copepod to the other.The positive z-direction: is the direction from posterior to anterior.

Fig. 10. Illustration of the line 0.3 mm right above the antennules. Along this line velocity magnitudes and deformation rates are calculated.

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The velocity magnitude is calculated according toU u v wu 2 2 2/ = + + .

For two ventral-to-ventral, dorsal-to-dorsal, side-by-side, or one-above-the-other stationary copepods, thevelocity magnitudes along the line just above the anten-nules are plotted versus the y-coordinates of the line inFigure 11. Also, for the swimming cases of two side-by-side or one-above-the-other copepods, the velocity mag-nitudes along the line are plotted in Figure 12. Since 20 µm s–1 in the velocity difference between the tip andbase of the setae is enough to elicit a neural response fromthe antennules of copepods (Yen et al., 1992), based on thedistribution of the velocity magnitudes along the line justabove the antennules of the solitary copepod (stationaryor swimming), a range of velocity magnitudes along thesame line can be defined as

U U Um s m s� �20 20solitary paired solitary1 1$ $# #- +- - (6)

for a copepod in a two-copepod pair. In this range thecopepod could not detect the presence of the nearbycopepod. Furthermore, using this range (i.e. the rangegraphically between the paired dotted lines in Figures 11and 12), one can determine the separation distance between

two copepods within which the detection could not happen.Figure 11a indicates that a stationary copepod could detectthe presence of another stationary copepod at a separationdistance of 3.09 mm (more than 2.3 body lengths) when thetwo copepods are ventral-to-ventral. When two stationarycopepods are dorsal-to-dorsal (Figure 11b), the detectionwould barely happen when the two copepods are separatedby the short distance of 0.83 mm. When two stationarycopepods are side-by-side (Figure 11c), the velocity magni-tudes at the side closest to the other copepod are just highenough to enable the detection. Figure 11d shows the plotsfor two stationary one-above-the-other copepods. It isshown that the lower copepod could potentially detect thepresence of the upper copepod in a separation distance of4.18 mm (more than 3.2 body lengths), while the uppercopepod does not notice the lower copepod. For two side-by-side swimming copepods, Figure 12a shows that onecopepod could potentially detect another copepod in aseparation distance of 4.88 mm (more than 3.7 bodylengths). For two one-above-the-other swimming copepods,it is shown that the copepods could potentially detect eachother in an even larger separation distance (Figure 12b).

The deformation rates along the line just above the anten-nules are also plotted versus the y-coordinates of the line inFigure 13 for two ventral-to-ventral or one-above-the-other


1.06 mm

2.18 mm

3.09 mm

4.95 mm

The paired dotted lines represent thevelocity magnitude of the solitary copepod

+/- 20 µm/s. Any velocity magnitude outside thesedotted lines can be detected, based on a detection threshold

of 20 µm/s velocity difference.

0.83 mm

4.10 mm2.33 mm


+/- 20 µm/s. Any velocity magnitude outsidethese dotted lines can be detected, based on a

detection threshold of 20 µm/s velocity difference.

3.43 mm4.10 mm

4.88 mm




L1.1

6 m

m

L2.14 mm

L4.18 mm




a b

c d

Fig. 11. Velocity magnitudes along the line right above the antennules. (a) For the two ventral-to-ventral stationary copepods. (b) For the twodorsal-to-dorsal stationary copepods. (c) For the two side-by-side stationary copepods. (d) For stationary copepods, one-above-the-other. The labelsin mm indicate the separation distance between the two copepods. Note that in (d) the letter ‘L’ indicates ‘the lower copepod’ and that the un-labelled lines between the two dotted lines are for ‘the upper copepod’.

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stationary copepods and in Figure 14 for two side-by-side orone-above-the-other swimming copepods. Clearly, theresults show the deformation-rate differences between thepaired copepods and the solitary copepod for both station-ary and swimming situations. The differences in defor-mation rates actually reflect the hydrodynamic signals thatenable the potential detection between copepods in close

proximity. However, the threshold deformation rateobtained from some previous work ranges from 0.5 to 5 s–1

[see (Kiørboe et al., 1999)] and therefore may not help us tofind the detection distance between two copepods of com-parable size. Even using these threshold deformation rates,we may not determine the detection distances but thereaction distances.


L1.16 mm

L2.14 mm

L4.18 mm

U1.16 mm U4.18 mm

The paired dotted lines represent the velocity magnitude of the solitarycopepod +/- 20 µm/s. Any velocity magnitude outside these dotted lines

can be detected, based on a detection threshold of 20 µm/s velocity difference.

3.43 mm

4.10 mm

4.88 mm




a b

Fig. 12. Velocity magnitudes along the line right above the antennules. (a) For the two side-by-side copepods swimming in the same fashion as inFigure 8. (b) For the two one-above-the-other copepods swimming in the same fashion as in Figure 9. The labels in mm indicate the separation dis-tance between the two copepods. Note that in (b) the letter ‘L’ indicates ‘the lower copepod’ and that the letter ‘U’ indicates ‘the upper copepod’.

1.06 mm

2.18 mm

3.09 mm4.95 mm

solitary copepod(the dotted line)

L1.1

6 m

m

L2.14 mm

L4.18 mm


a b


L1.16 mm

L2.14 mm

L4.18 mm

3.43

mm

4.10 mm

4.88 mm


a b

Fig. 14. Deformation rates along the line right above the antennules. (a) For the two side-by-side copepods swimming in the same fashion as inFigure 8. (b) For the two one-above-the-other copepods swimming in the same fashion as in Figure 9. The labels in mm indicate the separation dis-tance between the two copepods. Note that in (b) the letter ‘L’ indicates ‘the lower copepod’ and that the unlabelled lines very close to the dottedline are for ‘the upper copepod’.

Fig. 13. Deformation rates along the line right above the antennules. (a) For the two ventral-to-ventral stationary copepods. (b) For the two one-above-the-other stationary copepods. The labels in mm indicate the separation distance between the two copepods. Note that in (b) the letter ‘L’indicates ‘the lower copepod’ and that the unlabelled lines very close to the dotted line are for ‘the upper copepod’.

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D I S C U S S I O N

Energetic considerations

Our results of the power and volumetric flux have con-firmed that in contrast to schooling fishes or flocking birdsno energetic advantages exist for copepods in close prox-imity (e.g. in dense swarms) even for two aligned swim-ming copepods that are very close to each other. Theseresults are understandable. Members in a fish school orbird flock adopt appropriate positions relative to theirneighbours to benefit from the vortices shed by the neigh-bours. Because of their relatively large size and high swim-ming or flying speeds the Reynolds numbers are high, sothat vortex shedding due to flow separation is guaranteedand hence they can save energy by forming an organizedschool or flock. On the other hand, for members incopepod swarms, the Reynolds numbers are low due totheir small size and low swimming speeds. As such, vortexshedding due to flow separation is not expected forswarming copepods and no energetic benefits are avail-able. No other mechanism could enable them to saveenergy by swarming together. Thus, we expect that incopepod swarms there are no preferred positions for eachswarming member relative to its neighbours. This is sup-ported by a study of a laboratory-induced copepodswarm, in which swarming copepods showed seeminglychaotic movement and had no readily apparent orien-tation or predictable trajectory (Yen and Bundock, 1997).However, more observational evidence and detailedanalyses are needed to prove this expectation further.Indeed, in real copepod swarms, two copepods are seldomaligned when swimming. Copepods in swarms are sepa-rated by even larger distances than those considered in thepresent numerical simulations, where our calculationsshow the presence of other swarming copepods will notaffect the swimming energetics of individuals.

Influences of the hydrodynamic interactions

The stream tubes show that the hydrodynamic interactionbetween two copepods changes the geometry of the flowfield around each copepod. The interaction is dependenton the separation distance between the two copepods,their relative body positions and orientations, their relativeswimming velocities and their respective movements ofmouthparts and appendages. The results may have impli-cations on copepods’ swarming behaviours. Whenforming dense swarms, there inevitably exist the hydrody-namic interactions between the swarming copepods. Ournumerical results and some observational facts suggestthat the hydrodynamic interactions play two roles incopepod swarms. First, since the hydrodynamic inter-actions enable the swarming copepods to become awareof the presence of nearby copepods, they may

contribute to maintaining the integrity of the swarms.Second, the hydrodynamic interactions could interferewith each swarming copepod’s independent feeding, sincetwo copepods in close proximity share the food resources.[This differs from the interference competition betweendifferent species, in which chemicals play an essential role(Folt and Goldman, 1981).] One of the functions of theflow field around a copepod is to bring food particles tothe copepod’s capture area. When two copepods are inclose proximity, in a limited space the hydrodynamic inter-action between the two copepods will interfere with eachcopepod’s independent feeding. The results show that insome situations, such as for two ventral-to-ventral (Figure6a) or two one-above-the-other (Figures 7 and 9) cope-pods, strong interference occurs due to the strong hydro-dynamic interactions. The patterns of the stream tubes fortwo one-above-the-other copepods (Figures 7 and 9)support the conclusion by O’Brien (O’Brien, 1989) thatindividuals in an aggregation avoid positions directlyabove or below their neighbours, since the lowercopepod’s feeding is strongly interfered with and thecopepod cannot obtain food since it actually sampleswater from the upper copepod which has potentiallyremoved the best food. Gut-pigment analysis indicatedthat the feeding rates of Dioithona oculata copepods wereoften reduced in swarms during the day compared towhen the copepods were dispersed at night (Buskey, 1998).The diel difference in gut contents could be a direct resultof the hydrodynamic interference interactions betweenswarming copepods or just a diel feeding rhythm (E. J.Buskey, personal communication). Further experimentalstudies are needed to clarify this point.

Copepods in a swarm have to be separated by suitabledistances. Large nearest neighbour distances (NNDs)between individuals have been observed for copepodswarms. For example, the mean NND between individualcopepods in a monospecific swarm of Oithona oculata was14 body lengths (Hamner and Carleton, 1979). A meanNND of 9.5 body lengths was estimated for an Acartia

plumosa swarm (Ueda et al., 1983). Ambler et al. reported amean density of 9.1 ml–1 of a Dioithona oculata swarm,which resulted in a mean NND of seven body lengths(Ambler et al., 1991). These observations may indicate thatthe hydrodynamic interactions between the swarmingcopepods are weak when separated by these distances, sothat each copepod’s feeding may not be affected by thepresence of others. A good strategy for swarming whilekeeping suitable separation from others in the swarm is tofrequently change the swimming direction. The NNDshould be greater than the distance at which the hydrody-namic interactions become weak; if the separation dis-tance between two copepods is less than this distance, theinteraction between the two copepods lets them know the


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presence of each other. As an analogy, an individual in acopepod swarm might have a ‘territory’ around itself [Yenand Strickler (1996) also mentioned a ‘familiar territory’of the feeding current], just as many terrestrial animalsdo. Here, the hydrodynamic interaction is the confine-ment.

There is the potential for influence of the hydrody-namic interaction on each copepod’s orientation. TablesIII and IV show quantitatively that for copepods sepa-rated by a distance of four body lengths, the changes inthe drag forces (relative to the drag forces on the solitarycopepod) are still notable. [However, the swimming speedis relatively low (1.5 mm s–1) in the present work. Manyspecies can swim at much higher speeds and significantchanges in the drag forces are at even greater separationdistances.] The results indicate that the interactionbetween copepods affects the equilibrium of the forces bychanging the hydrodynamic forces. A solitary copepodmaintains the equilibrium (or near equilibrium) of theforces by using the complex arrangement and movementof its appendages and by adopting certain body orien-tations. When two copepods are in close proximity orapproach each other, the strong interaction between themdistorts the geometry of the flow field (as shown in above)and hence the distributions of the shear stress and flowpressure over the body–fluid interface differ from whenthe copepods are alone. As such, the drag forces exertedby water on each copepod differ from those on the solitarycopepod, and the forces on each copepod will be modi-fied. In dealing with this, each copepod may adopt a newbody orientation (or possibly rearrange its appendages’movement). This is similar to the phenomenon that theorientation of a prey is the result of the interaction of theprey with the predator’s feeding current (Fields and Yen,1997). But for the present situations involving two cope-pods of comparable size, the influence on the body orien-tation is mutually applied. However, if the hydrodynamicinteraction is too strong, at least one copepod will jumpaway. Such reactions are often observed in the laboratory.

Detection between copepods of comparablesize and pattern recognition

Essentially, the bending of a seta on a copepod is due to achange in the forces acting on the seta. A solitary copepodmaintains a flow field with a certain geometry around itsbody, so that the distributions of the shear stress and flowpressure over the surface of a seta on the copepod aregiven and the hydrodynamic forces on the seta are deter-mined. On the other hand, the seta’s orientation deter-mines the seta’s internal stress, which balances theexternal forces. Reasonably, we can take this orientationas the baseline orientation of the seta. Since there aremany setae along the antennules, each seta has its own

baseline orientation. When two copepods approach eachother, the flow field around each copepod is distorted(Figures 6–9). The distortion results in the changes in thedistributions of the shear stress and flow pressure over thesurface of each seta on each copepod, leading to changesin the hydrodynamic forces on each seta. According to the20 µm s–1 velocity threshold of Yen et al. (Yen et al., 1992),if the disturbance in the fluid velocity at the tip of a setais larger than 20 µm s–1 (since the velocity at the setal baseis always zero relative to the antennules), it is plausible thatthe internal stress associated with the seta’s baseline orien-tation must be modified by displacement and/or deflec-tion of the seta from its baseline orientation.

Our results show that at different positions along theantennules, the disturbances in the fluid velocity (or defor-mation rate) are different (Figures 11–14), so that the setaeat different positions along the antennules will be dis-placed and/or deflected to different orientations (relativeto their baseline orientations). The ensemble of all thesesetal orientations forms a pattern along the antennules.Obviously, this pattern is greatly dependent on the separ-ation distance between the two copepods, their relativebody positions and orientations and their relative swim-ming velocities (which can be seen by comparing eachframe in Figures 11 and 12). By recognizing the setal-orientation pattern along its antennules, each copepodcould potentially know from how far away, from whichdirection and at what speed it is being approached by theother copepod.

S U M M A RY A N D C O N C LU S I O N S

The present numerical simulations have detailed thehydrodynamic interaction between two copepods of com-parable size in close proximity.

The results of the power and volumetric flux indicateno energetic advantages are available for copepods to bein close proximity. The power and volumetric flux calcu-lated for each copepod in a two-copepod pair (stationaryor swimming) are almost the same as those calculated fora solitary copepod (stationary or swimming), insensitive tothe separation distance between the two copepods andtheir relative body positions and orientations. Thenumerical results may also suggest that in copepodswarms there are no particularly advantageous positionsfor each swarming member relative to its neighbours.

When two copepods are in close proximity, the hydro-dynamic interaction between them can greatly distort thegeometry of the flow field around each copepod. It hasbeen shown that the influence of the hydrodynamic inter-action on each copepod’s flow field is dependent on theseparation distance between the two copepods, theirrelative body positions and orientations, and their relative


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swimming velocities. Our results of the hydrodynamicforces suggest the possible influence of the hydrodynamicinteraction on each copepod’s orientation.

Hydrodynamic interactions play two roles in copepodswarms. First, they enable swarming copepods to beaware of the presence of nearby copepods and thereforecan contribute to maintaining the integrity of the swarms.Second, the hydrodynamic interactions may interferewith each copepod’s independent feeding. Strong hydro-dynamic interactions between copepods do not increase,and in most cases decrease, the feeding rates. In order toavoid the strong hydrodynamic interactions and to feedwithout the strong interference, copepods in swarms haveto be separated by large NNDs (usually more than fivebody lengths) and avoid positions of strong interactions,such as the positions directly above or below their neigh-bours. Frequently changing the swimming direction is agood strategy for the swarming copepods to keep suitabledistances away from other swarming copepods whileremaining in the swarms.

The hydrodynamic interaction between two copepodsgenerates the hydrodynamic signals that can be poten-tially detected by the mechanoreceptors (the setae) locatedon each copepod’s antennules. By recognizing the setal-orientation pattern along its antennules in response to thehydrodynamic signals, each copepod could potentiallyknow from how far away, from which direction and atwhat speed it is being approached by the other copepod.Our results show that the detection distance between twocopepods of comparable size is about two to five bodylengths.

AC K N OW L E D G E M E N T S

We are grateful to Professor J. Yen and an anonymousreviewer for very helpful comments on the manuscript.The financial support of the Office of Naval Research(contract number N000149710429) is gratefully acknow-ledged.

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Received on January 28, 2000; accepted on October 24, 2001


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