HANDBOOK OF SCALING METHODS IN AQUATIC ECOLOGY MEASUREMENT, ANALYSIS, SIMULATION - PART 3 doc

149
10
Signaling during Mating in the Pelagic Copepod,
Temora longicornis
Jeannette Yen, Anne C. Prusak, Michael Caun, Michael Doall, Jason Brown,
and J. Rudi Strickler
CONTENTS
10.1 Introduction 149
10.2 Methods 150
10.2.1 Trail Visualization 150
10.2.2 Copepod Pheromones 151
10.3 Results and Discussion 151
10.3.1 Scent Preferences 151
10.3.2 Tracking Behavior 154
10.3.3 Quantitative Analyses of Trail Structure and Odorant Levels 155
10.4 Conclusion 157
Acknowledgments 158
References 158
10.1 Introduction
In 1973, Katona depicted the mate-Þnding response of copepods as occurring when the male copepod
detects the edge of the diffusing cloud of pheromone emanating from the female at a distance of 4 mm.
Here, the process of diffusion is needed to
transport the signal molecules to the sensors of the male
copepod to alert him of the presence and location of a female copepod. However, using the equation
for the characteristic diffusion time (Dusenbery, 1992;
t = r
2
/4D, where the diffusivity coefÞcient D = 10
–5
cm
2

/s for small chemical molecules), it would take approximately 45 min for the pheromone to diffuse
this distance. It is unlikely the female copepod would remain in the same three-dimensional (3D) position
in the ocean for that length of time. Instead, in 1998, we (Doall et al., 1998; Weissburg et al., 1998; Yen
et al., 1998) reported that male copepods detect discrete odor trails left in the wake of the swimming
diffusion of the odor trail to molecular processes. Diffusion does not transport the pheromone to the
male and, instead, acts to
restrict odor dispersion. The scent persists as a coherent trail, with little dilution
of signal strength, and hence remains detectable for a period that gives enough time for the male to
encounter it. In the case of the copepod
Temora longicornis, this aquatic microcrustacean could Þnd
trails that were less than 10.3 s old. Trails were followed for distances as long as 13.8 cm (~100 body
lengths), greatly extending the encounter volume of the copepod (Gerritsen and Strickler, 1977). Past
studies showed that copepods could detect signals only within a few body lengths (see Haury and
Yamazaki, 1995, for a review). Our Þndings show the perceptive distance can be 10 to 100 times greater.
© 2004 by CRC Press LLC
female copepod (Figure 10.1). Within this low Reynolds number regime, viscosity limits the rate of
150 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
Documenting copepods following 3D aquatic trails provoked the question of the species speciÞcity
of chemical trails. A capability to discriminate the scent of conspeciÞcs enables mate preference,
important in maintaining species integrity (Palumbi, 1994; Orr and Smith, 1998; Higgie et al., 2000).
For such widely dispersing species as aquatic marine organisms, an understanding of mate recognition,
and the mechanisms for mate Þnding and mate choice, may be key steps in the cascade that enforces
reproductive isolation in pelagic environments. To evaluate odor preference in copepods, we developed
a bioassay, based on the behavioral response of trail following. The bioassay relies on trail visualization
(a Schlieren optical path: Strickler and Hwang, 1998; Strickler, 1998) so that we can
see the trail and
see which trail the copepod follows. With this technique, we saw male copepods follow scent trails
containing the odor of their conspeciÞc female copepods. We also saw how the male disturbed the odor
trail, making it less likely that another competing male would Þnd and follow the trail.
10.2 Methods

10.2.1 Trail Visualization
To visualize the trail, we mixed high-molecular-weight dextran with Þltered seawater to change the
refractive index of the trail and used Schlieren optics to see the trail (Strickler and Hwang, 1998; Strickler,
1998). For this optical path, an infrared laser light, focused through a pinhole spatial Þlter, was expanded
and diverted with a large (8-in.) spherical mirror to pass through and illuminate an experimental vessel
Þlled with Þltered seawater. The collimated light passed through the vessel, was collected on another
large (8-in.) spherical mirror, and focused onto a small pointlike matched Þlter. The matched Þlter
prevented the light from reaching the image plane and the CCD image looked black. When a phase
object was introduced into the experimental vessel, it diffracted the light rays so the rays passed by the
matched Þlter and reached the image plane. Phase objects, like the translucent copepod or the trail, were
visualized as bright silhouettes against a black background of the image captured by the infrared-sensitive
CCD video camera.
A 1-mm-wide trail was created by dispensing ßuid from a Þne pipette tip (Eppendorf) to ßow down
into the 4-l observation vessel Þlled with Þltered seawater (28 ppt). The pipette was gravity fed, via thin
FIGURE 10.1 (Color Þgure follows p. 332.) (A) Mate-tracking by the copepod, T. longicornis (1.2 mm prosome length).
The male copepod (thin trail) Þnds the trail of the female copepod (thick trail) when the trail is 5.47 s old and the female
is 3.42 cm distant from him (nearly 30 body lengths). Upon encounter, the male spins to relocate the trail, then accelerates
to catch up with the female. The male copepod follows the path of the female precisely in 3D space. When within 1 to
2 mm of the female, the male pauses quietly, then pounces swiftly to capture his mate for the transfer of gametes packaged
in a ßasklike spermatophore. During copulation, the mating pairs spin and remain together for a few seconds or more.
(B) Mate-tracking by the copepod, T. longicornis: backtracking. The male copepod (thin trail) Þnds the trail of the female
copepod (thick trail) because of a strong cross-plume odor gradient. The female is 1.30 cm distant from him. Upon encounter,
the trail is 2.3 s old and the male follows the trail in the wrong direction, away from the female, because of a weak along-
plume gradient. When initially following the trail, the male smoothly and closely follows the trajectory of the female. After
1.27 s when he is 24.4 mm from his mate and the trail is 6.7 s old, the male turns around and backtracks. When backtracking,
he follows the female path erratically, casting back and forth over the trail. After reaching his intersection point, the male
copepod resumes smooth close-following of the undisturbed female path. (From Doall, M. et al., Philos. Trans. R. Soc.
Lond., 353, 681, 1998. With permission.)
Speed
(mm/s)

10-20
20-30
30-40
40+
5-10
a'
t=0 s
b
t=6.00 s
c
t=9.67 s
b'
a
t=2.067 s
a'
b
t=5.47 s
c
t=7.00 s
b'
a
t=0 s
Z (cm)
X (cm)
X (cm)
Y (cm)
Y (cm)
0-5
B
A

© 2004 by CRC Press LLC
Signaling during Mating in the Pelagic Copepod, Temora longicornis 151
tubing, from a beaker Þlled with a mixture of seawater and dextran (MW of 500,000 at 0.5 g/100 ml).
The difference in the refractive index of the dextran–seawater mixture sinking down through the seawater
could be detected, using the Schlieren optical path. When this mixture passed through the plain seawater,
a Þne trail could be observed as a bright vertical line on the video image (Figure 10.2).
10.2.2 Copepod Pheromones
To test the attractiveness of waterborne odorants, different scents were obtained by putting different
copepod types (stage, sex, species) in the dextran-labeled water and using this conditioned water to
create the doubly labeled trails. For these scents, the same number of copepods (20) would be added to
the same volume of dextran-labeled seawater (20 ml) in the beaker, to introduce comparable odorant
levels to the conditioned water. Copepods that had spent the day in this dextran–seawater mixture had
no detrimental effects as they lived for days after being transferred back to plain seawater with phyto-
planktonic food (
Rhodomonas sp.). The rate of inßow of the doubly labeled water, less than 3 mm/s and
more often 1 mm/s, was adjusted by varying the height of the beaker of conditioned water that was
gravity-fed into the tank. This small beaker was Þlled from a stock of unscented dextran-labeled seawater.
The observation vessel held 20 to 50 male copepods to test their interest in the scented trails. Female
copepods do not mate-track.
10.3 Results and Discussion
10.3.1 Scent Preferences
When offered the choice of trails with dextran-only vs. trails with dextran and female copepod scent,
the male copepods followed the trail with the scent of their conspeciÞc 80% of the time and the unscented
FIGURE 10.2 (Color Þgure follows p. 332.) (A) Visualized trail and upwardly directed trail following. A 1-mm-wide
trail was created by dripping ßuid from a Þne pipette tip (Eppendorf) to ßow into the 4-l observation vessel Þlled with
Þltered seawater (28 ppt). The pipette was gravity-fed, via thin tubing, from a beaker Þlled with a mixture of seawater and
dextran (MW of 500,000 at 0.5 g/100 ml). The difference in the refractive index of the dextran–seawater mixture sinking
down through the seawater could be detected, using a Schlieren optical path. The trail on the left is the undisturbed trail.
The disturbed chevron-dotted trails on the right show how the trail structure changes at various time intervals after the male
copepod T. longicornis follows the trail. Copepod indicated at the upper left on last trail, colored orange (B) Following a

curvy trail: When following the scent in the trail, T. longicornis also can stay on the track of curved trails, making the same
turns as taken by the trail. The undisturbed curved trail on left was followed for 1.13 s by the male copepod, colored orange
in panel on right. Note how trail structure changes in wake of swimming male.
© 2004 by CRC Press LLC
152 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
trail less than 10% of the time (Table 10.1A). The dextran in the trail did not preclude the male’s interest
in the trail with the female odors. To determine when, during the reproductive maturation period of the
female, the scent was produced at sufÞcient concentrations, we offered the choice of dextran trails with
the scent of conspeciÞc females that had oocytes within their oviducts vs. females without oocytes.
Chow-Fraser and Maly (1988) found for the freshwater copepod, Diaptomus, more males attempted to
mate with gravid than with nongravid females. Here, we found no preference for
T. longicornis based
on scent-tracking (Table 10.1A). Barthélémy et al. (1998) note that a congener,
T. stylifera, requires a
separate fertilization event prior to each clutch of eggs, which may explain why both gravid and nongravid
females were attractive to their male mates. Other copepods are able to store sperm for days to weeks;
these copepods, which do not need to Þnd mates for every clutch, may be more attractive as virgins or
nongravid females (Kelly et al., 1998).
When given the choice to follow trails scented with the odor of conspeciÞc females (combined gravid
and nongravid), conspeciÞc males, copepods of another species (
Acartia hudsonica), or dextran-only
trails, varying degrees of preference were discovered. Preference, evaluated as the likelihood to follow
an encountered trail, could be ranked from most preferred to least interesting: conspeciÞc females,
conspeciÞc males, the other species, the unscented trail (Figure 10.3, Table 10.1B). As an additional test
of preference, we also measured the tracking speeds of the male along the trail (Table 10.1B). Males
would track conspeciÞc female-scented trails at 15 mm/s, which was faster than their tracking speed
along trails scented with conspeciÞc male odor. For the rare event of tracking the dextran-only unscented
trail, the tracking speed (5 mm/s) was just above the ßow speeds of the trail (1 to 3 mm/s). For the trails
with higher ßow speeds, there was a slightly greater probability that the male would show disinterest
(nick, cross, or escape from the trail; Table 10.1). Flow speeds now are kept below 1 mm/s to reduce

the likelihood that the hydromechanical ßow disturbance would elicit an escape (Fields and Yen, 1997).
The relative tracking speed on preferred trails (copepod swimming speed plus trail ßow speed) was
slightly slower than natural trail-following speeds (Doall et al., 1998).
FIGURE 10.3 (Color Þgure follows p. 332.) (A) Scent preferences of T. longicornis. Male copepods of T. longicornis
were offered seven choices of scent trails (from left to right: Female, Male, Acartia, Dextran, Male, Female, Acartia). The
male copepod showed an 80% preference to follow trails scented with the odor of conspeciÞc females (choice 6). Trails
were followed to the source. The source of the odor was a stock of dextran-mixed Þltered seawater within which 20 females
were swimming. The conditioned water ßowed through Þne pipettes down through the observation vessel, Þlled with Þltered
seawater and 20 to 50 male copepods. Schlieren optics visualized the ßow patterns. (B) Downwardly directed trail-following.
The trail on the left is the undisturbed trail. The disturbed trail on the right was the same trail that was followed down to
the source by a male copepod, appearing in the lower right, colored orange. Trails were created by allowing
scented seawater to ßow out of Þne pipettes at the bottom of the observation vessel. The vessel was Þlled with
a mixture of dextran (20 to 40 g/4 l of 500,000 MW dextran) and seawater to create the difference in refractive
index necessary to visualize the trail using Schlieren optics. The plain seawater would ßoat up to create the
trail. Flow speed of 4.43 mm/s can be calculated by the upward movement of the bright bubble.
© 2004 by CRC Press LLC
Signaling during Mating in the Pelagic Copepod, Temora longicornis 153
Even though dextran-only trails were rarely followed, the copepod does not avoid dextran because
scented dextran trails were followed. In most cases, the male copepod would encounter the scented trail
and follow the trail up to the source. On occasion, the male would follow the trail down and away from
the source and frequently would turn around. To determine if gravity inßuenced tracking direction of
the male copepod, we designed the reverse situation. Here, seawater with and without the scent of other
copepods was used to form trails in a tank Þlled with dextran-labeled water (20 to 40 g dextran/4 l).
The trails began now at the bottom of the tank, as the dextranless water would ßoat up to the surface
of the tank Þlled with dextran-labeled water. When the male encountered the trail, he would follow it
91.7% (11 trails followed/12 female trails encountered) of the time
to the source, at the bottom of the
From these analyses of trail following up or down to the source and the speed at which the trails were
followed, we conclude that the behavior of mate-tracking by
T. longicornis is chemically mediated and

tracking direction is not determined by gravity. We found that most trails were followed to the source,
indicating that the copepods are either able to detect the odor gradient or can detect the directional ßow
in the sheared slow ßow of the manufactured trails. We also conclude that these male copepods can
discern differences in scents between sexes and between species.
However, these male
T. longicornis copepods do not follow female trails exclusively, questioning the
speciÞcity of the pheromone. Male copepods would not only follow female trails, but also male trails.
Here, we would like to describe an event showing the adaptive value of following one’s own track. When
studying mate-tracking, we would start with male-only swarms of copepods (Doall et al., 1998). In these
swarms, there was the infrequent event when a male would follow the trail of another male but upon
contacting the male, he would release him immediately. More interestingly, another event in swarms of
mixed sex showed a male copepod that followed the trail of the female, but somehow got derailed and
wandered off the trail. After a couple seconds, he turned around, retraced his “steps,” found the female
trail, and followed it to her to form the spinning mating pair. Hence, an ability to self-track allowed this
copepod to Þnd his mate successfully.
The speciÞcity of the diffusible pheromone is further questioned by our observations of male
T. longicornis following the trails scented with the odor of another species, Acartia tonsa. This behavior
TABLE 10.1
Scent Choice by Male Copepods of the Species Temora longicornis
Source of Scent Preference = # follows/# encounter [%] % Escapes Tracking Speed Flow Speed
A. Response to Trails with the Scent of Females of Different Reproductive State and of Males
Gravid female Tl 43/54 [80%] 7.41% 13.9 2.77
Nongravid female Tl 21/30 [70%] 6.67% 19.2 3.07
Male Tl 4/20 [20%] 10.0% 17.3 2.54
Dextran-only 0/4 [0%] 23.8% — 2.95
B. Response to Trails with Scent from Different Sexes and Species of Copepod
Female Tl 31/39 [80%] 0 15.8 1.29
Male Tl 10/19 [52%] 0 10.1 1.28
Acartia tonsa 1/8 [12%] 0 13.6 1.0
Dextran-only 1/11 [9%] 0 5.7 1.32

Note: Random encounters by freely swimming male copepods of the species T. longicornis [Tl] with scent trails would
evoke a response to the trail. Vertical scent trails were created from dextran-labeled seawater ßowing down through
Þne pipette tips into the observation vessel Þlled with Þltered seawater (28 ppt). Male copepods, given a choice of
trails scented with the odors of different types of copepods, would follow the trail traveling at an accelerated swimming
speed or show disinterest (escape from, nick, or just cross the trail). Preference for a trail type is presented here as
(# trails followed/# trails of that type encountered). Disinterest is presented as (# escapes/# encounters). Swimming
speeds while trail following were compared to ßow speeds of the trail (mm/s). Experiments in A represent responses
to the scent of conspeciÞcs. Experiments in B include responses to scents from a copepod of another genus.
© 2004 by CRC Press LLC
tank (Figure 10.3).
154 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
suggests another possible scenario: as Temora is an omnivorous copepod, could it be responding to this
scent as a possible means to track its prey? Prey tracking recently has been observed for catÞsh detecting
the hydrodynamic wakes of goldÞsh prey (Pohlmann et al., 2001).
Remote detection via the diffusible pheromone is one means of mate recognition. Other possible
moments in mate recognition involve contact pheromones (Snell and Carmona, 1994) or a key-and-lock
Þt of the complex coupling plate (Fleminger, 1975; Blades, 1977). However, there are copepod species
that do not have coupling devices on their spermatophore that mirror the genital structure of the
conspeciÞc female and only have simple means to cement the base to the female genital area. It also is
not known how many copepods secrete contact pheromones, as this has been conÞrmed for only one
species of benthic harpacticoid copepod (Ting et al., 2000), which exhibits precopulatory mate-guarding
over extended periods of time. It is likely that decisions at every step in the mating process contribute
to Þnal species recognition.
10.3.2 Tracking Behavior
When the male copepod of T. longicornis detects a trail, he follows it in either direction, going to the
scent source
æ the female copepod æ or away from her. In one example where the male correctly
tracked to the female, the position of the male copepod was 1.02
± 0.34 SD (n = 45) mm from the
central axis of the 3D trajectory taken by the female seconds earlier, indicating spatially accurate tracking

way, away from the source of the scent: the female copepod. After 1.27 s, he reoriented and returned
backtracking showed that when the male Þrst follows the trail, his positions precisely overlap the positions
in the 3D trajectory taken by the female seconds earlier. The distance from the central axis of the female
trail is close to 1 mm. When he turns around, we Þnd that the 3D trajectory of the male copepod no
longer matches the path of the female. Instead, the male backtracks the disturbed trail very erratically,
FIGURE 10.4 (Color Þgure follows p. 332.) Changes in scent trail shape caused by tracking behavior of male
T. longicornis. (A) Trail structure changes from a smooth undisturbed vertical trail (left) to a disturbed trail (same portion
of the trail shown on right) after the passage of the male copepod of T. longicornis. (B) Close-up of the disturbed trail. The
red dotted line deÞnes the hypothesized helical trail left by spiraling movements of the male while tracking the trail. The
spiral is longer, thinner, and has a greater surface area than the smooth trail. The action of molecular diffusion can dilute
smooth and helical trail; radius of smooth and helical trail, slope of spiral structure of helix. For each 180∞ segment of the
helix, the radius and slope were repetitively calculated to reconstruct the helix so it cuts through the brightest spots of the
image. Scale: 6.8 pixels/mm.
10
20
30
40
50
60
70
80
90
0
20
40
60
Time = 0 sec
Time = 1.33 sec
AB
© 2004 by CRC Press LLC

the pheromone more quickly. Quantitative measurements of trail geometry (for Table 10.3 calculations) include: length of
along the trail back to the female to capture her. Further analyses (Table 10.2) of this behavior of
(Figure 10.1A). In another example (Figure 10.1B), the male found and followed the trail in the wrong
Signaling during Mating in the Pelagic Copepod, Temora longicornis 155
casting back and forth over the original location of the female trail at distances twice as far from the
central axis of the path. When backtracking, he casts at an average course angle of more than 70° in
either the
x–z or y–z direction, which is greater than the average course angle of 46° taken when following
the trail initially. A tightening of his course angles to 15° begins after passing his initial intersection
with the trail. Here, he again precisely follows the trail, remaining at distances of close to 1 mm from
the axis of the path. Weissburg et al. (1998) found that this pattern of counterturns enabled the copepod
to stay near or within the central axis of the odor Þeld.
The male copepod’s casting behavior suggested that the trail was disturbed by his own swimming activity,
making the trail more difÞcult to follow. Casting behavior also has been noted when the female copepod
swims slowly or hovers (Figure 3b in Doall et al., 1998), producing a trail so thick, the male wanders back
and forth between the edges. When the female copepod hops and creates a diffuse cloud, male casting
behavior also is observed (Figure 5a in Doall et al., 1998) and the male can lose the trail, suggesting that
the female hop diluted her scent to levels below the threshold sensitivity of his chemical receptors.
These observations led us to hypothesize that the trail structure has changed in the wake of the tracking
by the male copepod. To test this hypothesis, we relied on our method of small-scale ßow visualization
to document changes in the structure of the trail, evoking the behavioral response of trail following by
placing the scent of the female in the trail we created. By visualizing the trail, we saw how the tracking
structure in the wake of the tracking male copepod. The undisturbed trail is a smooth vertical line and
we presume from the two-dimensional (2D) image that the trail is a 3D cylinder. When the male copepod
intersects the trail, he spins and turns to relocate the trail. He then accelerates to three times higher than
his normal swimming speed as he follows the trail. Detailed analysis of the structure of the disturbed
trail reveals a herringbone 2D pattern. We can imagine that the male copepod swims around the trail,
deforming it into a 3D helix. Tsuda and Miller (1998) saw a larger male copepod of the species
Calanus
marshallae

that would tilt back and forth as he followed the scent trail of his female. The tilting appears
to allow the copepod to insert one antennule into the odor trail and one antennule out of it, thus assessing
location by bilateral comparison. As these were 2D observations of the large copepod, we make the
assumption that this tilting occurs in 3D, suggesting that the copepod spiraled around the trail. Spiraling
while swimming fast has been observed during the escape response of the large
Euchaeta norvegica
(Yen et al., 2002). Spiraling by the mate-searching copepod Oithona davisae has been suggested to help
the male locate a pheromone source more accurately and to promote diffusion of the pheromone to
prevent other males from pursuing the source (Uchima and Murano, 1988).
10.3.3 Quantitative Analyses of Trail Structure and Odorant Levels
To deÞne the characteristics of the disturbed trail, we assume that the brightness of the Schlieren image
represents the location of the chemical trail, where areas with similar brightness represent similar
concentrations (Gries et al., 1999). Here, we assume the structure of the deformed trail to resemble a
TABLE 10.2
Kinematics of Backtracking in a Mate-Seeking Copepod, Temora longicornis
Mate-Tracking of
T. longicornis
AWAY from Female
(n = 39)
Back TO Female on
Disturbed Trail
(n = 34)
TO Female on
Undisturbed Trail
(n = 23)
Normal Swimming
(n = 40)
Velocity (mm/s) 33.9 ± 9.7 50.4 ± 10.6 33.5 ± 12.4 14.4 ± 2.78
Course angle (x–z) 46.4 ± 41.5 73.3 ± 50.7 15.3 ± 15.6
Course angle (y–z) 42.3 ± 37.0 71.3 ± 43.1 26.2 ± 18.5

Track distance (mm) 1.16 ± 0.49 2.09 ± 1.14 1.15 ± 0.66
Note: The velocity, course angle (in both x–y and x–z directions), and track distance (closest distance between 3D trajectory
of the male copepod and central axis of female trajectory) are given for the event when the male copepod crossed
the trail of the female copepod, but went the wrong way: away from the source of the scent, the female. While
tracking and retracing his steps, course angles and track distance nearly double in value, indicating the erratic casting
behavior exhibited by the male as he returns along the disturbed trail.
© 2004 by CRC Press LLC
male disturbed the scented trail. Close analyses of the trail (Figure 10.4) show dramatic changes in trail
156 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
3D helix, where the relocation of odorant due to copepod movements producing an outward curve of
the helix equals the amount relocated on an inward curve of the 3D structure. Using geometric and
image processing techniques, we estimated the length, volume, and surface area of the scent trail before
and after the copepod swam through it (Table 10.3). The undisturbed trail was assumed to be a cylinder.
A section of the image of this cylinder was isolated; length and diameter measurements were taken and
used to calculate volume and surface area. The disturbed trail was assumed to have a helical character.
A section of the image of the disturbed trail corresponding to the section of the image of the undisturbed
trail was isolated. This image section was thought of as a 2D projection of the helical trail onto the CCD
imager: the bright spots of the image were assumed to indicate points where the helix crosses the plane
of the image projection. Taking these bright spots as data points, an “adaptive helix” was Þt to the image
in 180°
segments as follows. Two successive bright spots were located and their positions used to
determine parameters for a helical line segment passing from the Þrst to the second bright spot
(Figure 10.4). Next the third bright spot was located and its position was used in conjunction with the
second bright spot’s position to generate the parameters for another helical line segment to link the
second and third bright spots. Proceeding in a similar fashion produced a 3D helical line joining the
bright spots of the scent trail. A consequence of the construction of this helical line was that each 180°
segment was itself naturally divided into many smaller subsegments and this characteristic was exploited
to calculate length, volume, and surface area. The length was found by summing the lengths of each
individual subsegment. For the remaining measurements, each subsegment was taken to be the axis of
a short cylinder, the diameter of which was found by measuring several such “diameters” in the image

and averaging. The volume and surface area of the whole helical scent trail then were found by summing
the volumes and surface areas of the individual cylinders thus deÞned.
TABLE 10.3
Geometric Changes in Trail Structure Dilutes Chemical Signal
Scent Trails Smooth Trail (A) Disturbed Trail (A) Natural Trail (B) Disturbed Trail (B)
Initial diameter
a
0.14 cm ± 5.6%
SE (n = 82)
0.083 cm ± 17.3%
SE (n = 15)
0.05 cm 59%
Initial length 1.2 cm 2.9 cm 0.10 cm 240%
b
Initial V 1.09 ¥ 10
–2
cm
3
1.6 ¥ 10
–2
cm
3
2.0 ¥ 10
–4
cm
3
1.6 ¥ 10
–4
cm
3

Initial SA 5.4 ¥ 10
–1
cm
2
7.5 ¥ 10
–1
cm
2
1.6 ¥ 10
–2
cm
2
2.2 ¥ 10
–2
cm
2
Initial C 100% 100% 100% 100%
Final radius 0.09 cm 0.06 cm 0.05 cm 0.04 cm
Final length 1.25 cm 2.91 cm 0.14 cm 0.277 cm
Final V 0.0318 cm
3
0.0363 cm
3
8.9 ¥ 10
–4
cm
3
10.4 ¥ 10
–4
cm

3
Final SA 0.707 cm
2
1.15 cm
2
4.0 ¥ 10
–2
cm
2
6.1 ¥ 10
–2
cm
2
Final C 59% 43% 22% 15.2%
Volume (V) = pr
2
h; surface area (SA) = 2prh.
Possible initial odorant concentration (C) = 10
–5
M (Poulet and Ouellet, 1982).
Length increment = characteristic diffusion length = SQRT[4Dt]; t = time.
D = diffusion coefÞcient = 10
–5
cm
2
/s for small chemical molecules (Jackson, 1980).
Note: The initial volume (V), surface area (SA), and pheromone concentration (C) of the smooth and
disturbed trail mimic (A) and natural trail (B) were compared to the Þnal volume (V), surface area,
and concentration after 10 s of molecular diffusion (radius increases by 0.2 cm). The geometric
measurements of the undisturbed natural trail are from Table 2 in Yen et al. (1998).

a
Using MATLAB image analytical techniques, the edges were determined as that corresponding to a
threshold luminance value of 40%. As the entire trail was only 10 pixels wide, different thresholds will
yield different values.
b
the central axis of the trajectory of the female copepod. Similarly, after tracking, we found here that the
trail expanded 2.37 times wider.
© 2004 by CRC Press LLC
As noted in Table 10.1, when backtracking natural trails, the male casts at nearly twice the distance from
r = radius of cylinder; h = length of cylinder (see Figure 10.4 for illustration).
Signaling during Mating in the Pelagic Copepod, Temora longicornis 157
disturbed trail was 60% thinner, 2.4 times longer, with a surface area 40% greater than the original trail.
Although the helix extends over the same linear length occupied by the trail, it expands to a larger helix
width and therefore may change the probability of encounter. Once encountered, the next male would
have to manage to stay on the track of a thinner trail and also follow a much longer trail. Copepods are
female copepod so if they found the helix, they could spend extra time following every helical loop or
take longer casts suggesting exploration of a more diffuse odorant cloud.
Comparing the volume of the smooth trail to that of the helical trail, we found the volume of the trail
decreased by only 17%. The similarity in trail volume suggests that, at these Reynolds numbers, turbulent
eddy diffusion is limited by viscous forces. The Reynolds number for the male copepod, while swimming
along natural trails, can reach up to values of 60 (maximum tracking velocity = 50 mm/s in contrast to
the males’ normal swimming velocity of 10 mm/s), quite different from the initial Reynolds number for
the trail of 7 (swimming speed of female copepod = 6 mm/s). Here, the Reynolds number of the
manufactured trail is close to 1 and that in the wake of the male is between 15 and 20. At these Reynolds
numbers, diffusion acts by molecular processes only to slowly disperse the odor. The spiraling copepod
reshapes the odor trail but there appears to be little dilution of trail contents.
To determine if the odorant levels would differ after diffusion from a smooth vs. helical trail, we
calculated how much larger the trail would become after 10 s of molecular diffusion (radius increases
by 0.2 cm after 10 s). It is possible that adjacent loops of the helix would diffuse and fuse to reform the
smooth trail. To estimate the time needed for this change to occur, we measured the average separation

distance between loops of the helix as 1.92 mm. The loops would need to diffuse 0.96 mm to meet the
next loop. This would take more than 3 min. Because trails older than 10 s are rarely followed, it is not
necessary to consider the coalescing of odors between loops of the helix. Instead, we considered the
change in concentration if the helix were a straight cylinder to compare to the change in concentration
of the undisturbed smooth trail. We found that, after 10 s, the Þnal odorant concentration was 59% of the
original level (= 100%) in the smooth trail and 43% of the initial concentration in the disturbed trail, with
a 60% increase in surface area over the original trail. For natural mating trails, the oldest trail followed
by the male T. longicornis was 10.3 s old (Doall et al., 1998). This suggests that trails of this age stimulate
the copepod sensors just at their threshold. Trails any older are undetectable. Similarly, the backtracking
copepod turned around when the trail age was 6.7 s old, close to the time limit beyond which diffusion
reduced the concentration levels below the detection threshold. Considering the geometry of a natural
trail (from Table 2 in Yen et al., 1998), we calculated that the odorant would decline to 22% of the original
odorant levels excreted into the natural trail after 10 s. We conclude that 22% is the threshold for detection
by copepod chemoreceptors. If we now disturb the natural trail in a geometrically similar fashion as was
determined for the post-tracking manufactured scent trail, the odorant level in post-tracking helical natural
trail would decline to 15.2% of the original levels after 10 s (Table 10.3B). This is less than the 22%
threshold level. Therefore, in addition to becoming a longer and more convoluted trail, the odor became
less distinct where odorant concentrations may drop below the threshold sensitivity of the male’s receptors.
These structural changes, along with the changes in odorant levels in the disturbed trail plus possible
chemical degradation of pheromones over time, have the potential to confound the tracking ability of
another chemoreceptive plankter. Hence, by disturbing the trail due to his scent-tracking behavior, the
male may lower the possibility that another competing mate or threatening predator will Þnd his female.
10.4 Conclusion
To summarize, the study of mate-tracking, using this new behavioral bioassay along with the observation
of 3D copepod trajectories in a 4-l container, indicates that T. longicornis is able to detect and follow
scent trails. As he follows these pheromonal trails to his mate, the male copepod disturbs the signal and
effectively lowers the possibility that another mate or chemoreceptive predator will Þnd his female.
However, for T. longicornis, mate recognition by remote detection of a diffusible pheromone is not
certain. Temora longicornis appears capable of following different trail types. To understand this seeming
© 2004 by CRC Press LLC

We compared the structure of the trail before and after tracking (Table 10.3A) and found that the
able to follow curvy natural (Doall et al., 1998) and manufactured (Figure 10.2B) scent trails of the
158 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
lack of speciÞcity, we consider the biology of sparse populations. Plankton, although common, form
sparse populations because they are widespread at low densities; they seldom encounter each other and
so are effectively rare (Gerritsen, 1980). In sparse populations, the probability that mates encounter each
other is reduced, resulting in reduced birth rate and reduced population growth rates. If the population
density is low enough, mating encounters are so rare that few individuals reproduce to sustain the
population, leading to possible extinction. Because extinction is such a strong selective force, adaptations
will be favored that increase the probability of mating encounters. Here it is adaptive to follow any trail
because the probability of encountering anything is low for these small (1 to 10 mm), slowly swimming
(1 mm/s), widely dispersed (1/m
3
), aquatic microcrustaceans with limited ranges for their sensory Þeld
(1 to 100 body lengths). As copepods are considered some of the most numerous multicellular organisms
on earth (Humes, 1994), this study of their mating strategies has elucidated their reliance on unusually
precise mechanisms for survival and dominance.
Acknowledgments
This work was supported by National Science Foundation Grants OCE-9314934 and OCE-9402910 to
J.Y. and J.R.S., with continued support to J.Y. from IBN-9723960. All thank the editors: Laurent Seuront
and Peter Strutton.
References
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families of Centropagoidae (Copepoda: Calanoida). Philos. Trans. R. Soc. Lond. B, 353: 721–736.
Blades, P.I. 1977. Mating behavior of Centropages typicus (Copepoda: Calanoida). Mar. Biol. (Berlin), 40: 47–64.
Chow-Fraser, P, and E.J. Maly. 1988. Aspects of mating, reproduction, and co-occurrence in three freshwater
calanoid copepods. Freshw. Biol., 19: 95–108.
Doall, M.H., S.P. Colin, J.R. Strickler, and J. Yen. 1998. Locating a mate in 3D: the case of Temora longicornis.
Philos. Trans. R. Soc. Lond., 353: 681–689.
Dusenbery, D.B. 1992. Sensory Ecology. New York: W.H. Freeman.

Fields, D.M. and J. Yen. 1997. The escape behavior of marine copepods in response to a quantiÞable ßuid
mechanical disturbance. J. Plankton Res., 19: 1289–1304.
Fleminger, A. 1975. Taxonomy, distribution, and polymorphism in the Labidocera jollae group with remarks
on evolution within the group (Copepoda: Calanoida). Proc. U.S. Natl. Mus., 120: 1–61.
Gerritsen, J. 1980. Sex and parthenogenesis in sparse populations. Am. Nat., 115: 718–742.
Gerritsen, J. and J.R. Strickler. 1977. Encounter probabilities and community structure in zooplankton: a
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Gries, T., K. Johnk, D. Fields, and J.R. Strickler. 1999. Size and structure of “footprints” produced by Daphnia:
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Haury, L.R. and H. Yamazaki. 1995. The dichotomy of scales in the perception and aggregation behavior of
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Higgie, M., S. Chenoweth, and M.W. Blows. 2000. Natural selection and the reinforcement of mate recognition.
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Jackson, G.A. 1980. Phytoplankton growth and zooplankton grazing in oligotrophic oceans. Nature, 284:
439–441.
Katona, S.K. 1973. Evidence for sex pheromones in planktonic copepods. Limnol. Oceanogr., 18: 574–583.
Kelly, L.S., T.W. Snell, and D.J. Lonsdale. 1998. Chemical communication during mating of the harpacticoid
Tigriopus japonicus. Philos. Trans. R. Soc. Lond., 353: 737–744.
Orr, M.R. and T.B. Smith. 1998. Ecology and speciation. TREE, 13: 502–506.
Palumbi, S.R. 1994. Genetic divergence, reproductive isolation and marine speciation. Annu. Rev. Ecol. Syst.,
24: 547–572.
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Signaling during Mating in the Pelagic Copepod, Temora longicornis 159
Pohlmann, K., F.W. Grasso, and T. Breithaupt. 2001. Tracking wakes: the nocturnal predatory strategy of
piscivorous catÞsh. PNAS, 98: 7371–7374.
Poulet, S.A. and G. Ouellet. 1982. The role of amino acids in the chemosensory swarming and feeding of
marine copepods. J. Plankton Res., 4: 341–361.
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copepod mating pheromones. Am. Soc. Limnol. Oceanogr. Abstract, February 2001.

Snell, T.W. and M.J. Carmona. 1994. Surface glycoproteins in copepods: potential signals for mate recognition.
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Strickler, J.R. and J S. Hwang. 1998. Matched spatial Þlters in long working distance microscopy of phase
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© 2004 by CRC Press LLC
161
11
Experimental Validation of an Individual-Based Model
for Zooplankton Swarming
Neil S. Banas, Dong-Ping Wang, and Jeannette Yen
CONTENTS
11.1 Introduction 161
11.2 Theory 163
11.2.1 Differentiating between Swarming and Diffusion 163
11.2.2 Diffusion in an Aggregative Force Field 164
11.2.3 Further Model Predictions 165

11.2.4 Swarming in Two and Three Dimensions 166
11.2.5 The Acceleration Field 166
11.3 Experiment 167
11.4 Analysis 168
11.4.1 Constructing a Statistical Ensemble 168
11.4.2 A Procedure for Testing Model Consistency 169
11.5 Results 170
11.5.1 Velocity Distributions 170
11.5.2 Velocity Autocorrelations and Fit Parameters 171
11.5.3 Acceleration Fields 173
11.6 Discussion 174
11.6.1 Model Consistency 174
11.6.2 Model Interpretation 175
11.6.2.1 Damping 175
11.6.2.2 Excitation 177
11.6.2.3 Concentrative Force 177
11.6.2.4 Physical–Behavioral Balances 177
11.7 Conclusion 178
Acknowledgments 178
References 178
11.1 Introduction
The ecology of marine planktonic assemblages depends, in essential, intricate ways, on the behavior of
individual zooplankters. Swarming behavior is among the most crucial, and also least charted, of the
territories that span population and organismal biology in this way. On large scales in the ocean, and
possibly in some small-scale environments like frontal zones, aggregation into patches is probably a
physics-driven, passive process. At the same time, active swarming behavior — that is, a type of motion
that resists dispersion without orienting or distributing animals in an organized way — is well known,
© 2004 by CRC Press LLC
162 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
and controls small-scale zooplankton distribution in the ocean to an unknown extent. Passive and active

aggregations are essential for setting encounter rates between predators and prey, between grazers and
patchily distributed food sources (Lasker, 1975; Davis et al., 1991) and between conspeciÞcs in search
of mates (Brandl and Fernando, 1971; Hebert et al., 1980; Gendron, 1992).
Swarming differs from schooling (which is known but rare among the zooplankton; Hamner et al.,
1983) in that it is stochastic, unarrayed, and largely uncoordinated between individuals. At the same
time swarming differs from truly random motion — that is, Brownian motion or diffusion — in that a
swarm does not spread out over time and disperse as does a cloud of molecules or drifting particles.
Behaviors that maintain swarms against diffusion may be either social and density dependent or nonsocial
and density independent; we consider each category brießy in turn.
A variety of models for social swarm maintenance have been proposed, centered on mechanisms in
which, for example, animals seek a target density (Grünbaum, 1994) or correlate their motion with their
neighbors’ motion (Yamazaki, 1993). The possibilities for such mechanisms in a given species are
strongly constrained by sensory ability. Perhaps the most fundamental constraint is that the majority of
zooplankton lack image-forming eyes (Eloffson, 1966) and, accordingly, do not appear to orient to their
conspeciÞcs visually. Long-distance interactions along scent trails have been observed in copepod swarms
(Katona, 1973; Weissburg et al., 1998), and similar interactions are possible along trails in the shear or
pressure Þelds (Fields and Yen, 1997), as has been observed in schools of Antarctic krill (Hamner et al.,
1983). Nevertheless, most intraspecies communication in zooplankton appears to be local and intermit-
tent, as opposed, say, to the long-range and constant visual coordination that gives Þsh schools their
character. Leising and Yen (1997), for example, found Þve copepod species to be insensitive to the
proximity of their conspeciÞcs except at nearest-neighbor distances of a few body lengths. Swarms of
this density are rarely found in nature (Alldredge et al., 1984). It is important to note that even intermittent
social interactions may play a large role in swarm dynamics. Leising and Yen argue that the number
density of the laboratory swarms they observe is controlled by avoidance reactions to chance close-range
encounters in the swarm center.
A number of nonsocial aggregative behaviors have also been observed
in situ. Phototaxis maintains
swarms of the cyclopoid copepod
Dioithona oculata in shafts of light between mangrove roots (Ambler
et al., 1991), and similar responses to light gradients are known in several other species (Hamner and

Carlton, 1979; Hebert, 1980; Ueda et al., 1983). Attraction to food odors, and increased turning in food
patches, which aids foraging and increases forager density, have also been observed in a number of
species (Williamson, 1981; Poulet and Ouellet, 1982; Tiselius, 1992; Bundy et al., 1993). Some zoo-
plankton in fact may respond directly to water temperature and salinity (Wishner et al., 1988; Gallager
et al., 1996).
This enormous range of social and individual behaviors does not consolidate readily into a simple,
general account of zooplankton swarming. Still, a unifying thread runs through them: whatever the
driving behavioral mechanism, the tendency that counters dispersion in a swarm is not just a collective,
statistical property, but rather must be observable in each swarmer’s individual motion as a hidden
regularity. This is an experimentally powerful notion, for it suggests that we may be able to apprehend
the dynamics of a large, observationally unwieldly aggregation by studying the behavior of a few typical
individuals. Indeed, in a dynamic, rather than statistical (one could also say ethological, rather than
ecological) approach to animal swarming, the larger aggregation may often be close to irrelevant. Writes
Okubo (1986):
There is an interesting observation by Bassler that a swarm could be reduced to a single individual
of mosquito, Culex pipiens which yet continued to show the characteristic behavior of swarming
dance [Clements, 1963]. Also Goldsmith et al. [1980]… noted that a single and a few midges
did show movements characteristic of swarming by a large number of animals.
Provocatively, many zooplankton may behave like mosquitoes and midges in this respect, and form
swarms in which interaction between individuals plays at most a secondary role. Yen and Bundock
(1997), for example, found no social interaction within phototactic laboratory swarms of the harpactacoid
copepod
Coullana canadensis.
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Experimental Validation of an Individual-Based Model for Zooplankton Swarming 163
When interaction does occur, as noted above, it is generally chemical or rheotactic, rather than visual,
and thus occurs slowly, along spatially torturous paths. These paths are very difÞcult to observe or map,
and in a large aggregation, especially outside the laboratory, would not easily be differentiated from a
diffuse, continuous sensory cue. Indeed, if such interactions are fundamental to the dynamics of a swarm
(as opposed to simply being facilitated by the swarm, as mating encounters might be), then their

importance is not particular but cumulative, statistical, parameterizable. While the details of social
communication are crucially relevant to swarming biology on one level, for analyzing balances between
dispersion and counterdispersion — for understanding the kinematics of an individual swarmer — it
seems more apt to average over many encounters, and to model social effects as a net dispersive or
concentrative tendency in each individual.
In this modeling approach, then, whether a swarm is socially or nonsocially driven, we regard it
as an interaction not between animals, but between each animal and its local stimulus Þeld. This
approach has the advantage of generality. While one can parameterize a social, density-dependent
response — a series of avoidance reactions, for example, or motion through a network of pheromonal
trails — as an individual response to a steady cue, it is hard to imagine modeling, say, phototaxis by
reversing the analogy.
Attempts to produce a general quantitative description of zooplankton swarming have been frustrated
by a lack of marine observations. Okubo and Anderson (1984) present a simple and general individual-
based model of steady-state swarming (see also Okubo, 1980, 1986), but write that for purposes of
model veriÞcation, “no such data really exist in the marine Þeld.” They proceed, with reservations, by
examining data from “aeroplankton,” midges swarming in a forest clearing. Since the time of their
publication, optical and acoustic technologies for observing and recording
in situ zooplankton distribution
and behavior have become available (Alldredge et al., 1984; Schultze et al., 1992; Smith et al., 1992;
Davis et al., 1992). These and laboratory methods have occasionally been applied to the problem of
aggregation dynamics. McGehee and Jaffe (1996), for example, examine the relationship between path
curvature and swimming speed in a zooplankton patch observed through acoustic imaging; Buskey et al.
(1995) used three-dimensional imaging in the laboratory to analyze the phototactic formation of a swarm
in
Dioithona oculata.
Still, no dynamic account has been given of the swarming motion of individual animals followed for
sustained periods. In the present study we use optical methods developed by Strickler (1998) to evaluate
Okubo and Anderson’s model — the “aggregating random walk” (Yamazaki, 1993) — in the case of
two species of zooplankton, the calanoid copepod
Temora longicornis and the cladoceran Daphnia

magna
, swarming phototactically in the laboratory.
11.2 Theory
In this section we derive quantitative predictions of the aggregating random-walk model, which we
can directly compare with data, beginning from kinematic Þrst principles, following Okubo and
Anderson (1984).
11.2.1 Differentiating between Swarming and Diffusion
Assume for simplicity that the swarming motion is one dimensional, in the x direction — we generalize
to two and three dimensions later — and assume that the swarm has reached a steady state and is
isotropic. If all the swarmers are responding to the attractive stimulus in the same way, then their paths
will be centered on the same point: call this the origin. Then the spatial variance of an individual’s
path also measures the size of the swarm.
A standard result in the theory of diffusion (Okubo, 1980) is that under these assumptions, for large
t,
increases as
(11.1)
x
2
x
2
xDt
2
2Æ
© 2004 by CRC Press LLC
164 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
where D is the diffusivity. D can be written as
(11.2)
where
u ∫ dx/dt. R is the Lagrangian velocity autocorrelation coefÞcient, a function of time lag t:
(11.3)

Eventually any individual’s velocity decorrelates from its earlier values, since no animal, swarming,
diffusing, or otherwise, moves in the same pattern forever, and so
R(t) Æ 0 for large t. The shape of
R(t) as it tends to this asymptotic limit, however, is variable. In the case of simple diffusion, R(t) decays
exponentially. Its integral is a positive number, so that
D, by Equation 11.2, is a positive number, and
, by Equation 11.1, increases linearly. In the case of swarming, in which trajectories do not spread
over time, however, remains constant;
D must then be zero; and R(t) oscillates about the axis in such
a way that its area converges likewise to zero (Figure 11.1).
Thus the shape of
R(t) — speciÞcally, the presence or absence of an axis crossing — is the key to
distinguishing kinematically between swarming animals and diffusing ones.
R(t) can be calculated
directly from position and velocity data by Equation 11.3.
11.2.2 Diffusion in an Aggregative Force Field
We can recast the problem in dynamic Newtonian terms, again following Okubo and Anderson, as a
balance between diffusion and a deterministic concentrative force. In this model the concentrative force,
an inward acceleration that grows with distance from an attractor at the swarm center, keeps the swarmers
from dispersing, while diffusive motion keeps the swarmers from collapsing onto the central attractor.
The resulting equation of motion can be written:
(11.4)
The Þrst two terms on the right-hand side constitute a standard “random-ßight” model of diffusion:
k is
a frictional coefÞcient in a Stokes model of drag, and
A(t) is a random acceleration, some sort of white
noise, with a delta-function autocorrelation and Þnite power . Note that we can interpret this
random excitation and frictional damping either as true external, turbulent forces acting upon a passive
particle, or simply as accelerations that express behavior patterns.
The third term on the right-hand side, the concentrative force, can be interpreted either as a harmonic

(linear) restoring force, such as the force that gravity exerts on a pendulum, or as a local approximation
to a more complicated, anharmonic restoring force. Okubo and Anderson suggest anharmonicity as a
mechanism for maintaining a uniform, rather than Gaussian, density distribution through the swarm
center. Our experiment neither conÞrms nor refutes this idea, and so in the interest of empiricism we
have retained only the harmonic concentrative term. We view our dynamic model (Equation 11.4) as a
FIGURE 11.1 Schematic representation of the distinction in velocity autocorrelations between swarming and diffusion.
The shaded area in each case is equal to the diffusion coefÞcient D.
Diffusion Swarming
Time lag
Time lag
Autocorrelation
Autocorrelation
Du Rd∫
()
•
Ú
2
0
tt
R
u
ututtt
()
∫
()
+
()
1
2
x

2
x
2
du dt A ku x=- -w
2
BA
u
∫
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Experimental Validation of an Individual-Based Model for Zooplankton Swarming 165
Þrst-order approximation to a behavior that may well involve a number of unmodeled, higher-order
effects, such as advection of momentum, in both the diffusive and concentrative motions. Higher-order
models may be necessary to parameterize the effects of foraging, mate-Þnding, escapes, and other
behaviors simultaneous with swarming.
As shown by Okubo (1986), Equation 11.4 does indeed yield a velocity autocorrelation of the form
(11.5)
where
(11.6)
The integral of
R(t) is identically zero. Note that as the attractive forcing weakens — that is, as w Æ 0
— Equations 11.4 and 11.5 both approach the results for simple diffusion:
(11.7)
(11.8)
In practice, we necessarily calculate the autocorrelation
R(
t
) on discrete time series, either from
experiment or from numerical simulation, and such a discrete autocorrelation is not fully equivalent to
Equation 11.5, derived from continuous dynamics. Yamazaki and Okubo (1995) show that the discrete
autocorrelation does not integrate to zero, i.e., that discretization introduces an apparent (but artiÞcial)

net diffusion rate. This mathematical inconsistency is potentially a limit on the precision with which we
can determine
k and w from observations.
11.2.3 Further Model Predictions
In this model, a swarm is characterized primarily by position variance (a measure of the size of the
swarm), velocity variance (a measure of the kinetic energy of its members) and by
k, w, and B
(measures of the strength of the damping, attractive, and excitational forces). We can derive from
Equation 11.4 some useful and experimentally veriÞable relationships between these parameters.
Multiplying Equation 11.4 by
u and taking the time average, we obtain
(11.9)
is the power of the random forcing, or
B. Since we assume the swarm is in a steady state, both
and are zero. Thus, Equation 11.9 becomes
(11.10)
Note that this relationship is independent of
w and therefore true of purely diffusive motion as well.
Thus the attractive force has no bearing on the kinetic energy of a swarm. Nor does it inßuence the
speed distribution of the swarmers: an individual subject to Equation 11.3 follows the Maxwellian speed
distribution for free particles (like gas molecules):
(11.11)
where
p(u) is the fraction of a statistical ensemble found between velocities u and u + du (Okubo and
Anderson, 1984).
Re t
k
t
k
tw

w
w
t
()
=-
Ê
Ë
Á
ˆ
¯
˜
- 2
1
1
1
2
cos sin
ww
1
222
4=-k
du dt A ku=-
Re
k
t
t
()
=
-
x

2
u
2
u
du
dt
ku xu Au=- - +
22
w
A
u
ududt
()
xdxdt
()
u
B
k
2
=
pu
u
u
u
()
=-
Ê
Ë
Á
ˆ

¯
˜
1
2
2
2
2
p
exp
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required by kinematic considerations, as discussed above and illustrated in Figure 11.1:
166 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
The strength of the attractive force does, however, have a very direct bearing on the steady-state swarm
size . For large
t, approaches the value
(11.12)
(Uhlenbeck and Ornstein, 1930). Thus swarm size is inversely proportional to the strength of the attractive
forcing. When
w is large — that is, when the attractive tendency increases rapidly with distance from
the swarm center — the swarm is concentrated tightly. As
w approaches zero — the case of pure diffusion
— the swarm’s limiting size increases toward inÞnity.
11.2.4 Swarming in Two and Three Dimensions
These results can easily be generalized to more than one dimension. Assume, for example, that the same
forces that act in the
x direction act in the y. Because our model involves no coupling of motion along
different axes, the results so far derived still hold, and and (where
v ∫ dy/dt) are related to k and
w by expressions analogous to Equation 11.10 and 11.12. We can then write expressions for the two-
dimensional parameters of interest:

(11.13)
(11.14)
The generalization to three dimensions is analogous.
Velocity distributions for two- and three-dimensional swarms are simply the two- and three-dimensional
Maxwellian distributions. In two dimensions,
(11.15)
Note that animals swarming in a three-dimensional ocean are not necessarily engaged in three-
dimensional swarming. The number of dimensions in which the swarming model applies depends on
the symmetries of the attractor. If the attractor is planar (for example, a front or thin layer containing
high food concentrations: Yoder et al., 1994; Hanson and Donaghay, 1998), then the swarming forces
do indeed act in only one dimension, the vertical, and motion in the two horizontal dimensions, in which
the attractor is isotropic, is likely to be diffusive. If the attractor is one dimensional (for example, the
light shaft in our experiment, or a light shaft through mangrove roots: Ambler et al., 1991), then we
might expect swarming motion in two dimensions and diffusive motion in the third. Only if the attractor
is a point or spherically symmetrical (for example, a diffusing chemical signal in an isotropic environ-
ment) does the swarming model apply to three dimensions.
11.2.5 The Acceleration Field
Finally, returning to the one-dimensional case, we can derive predictions about the mean and variance
of the spatially varying acceleration Þeld.
Taking the mean of Equation 11.4, holding
x constant, we obtain
(11.16)
where
a ∫ du/dt. Thus, the mean acceleration Þeld consists of the center-directed concentrative force alone.
x
2
x
2
x
u

2
2
2
=
w
y
2
v
2
Vuv
B
k
222
2
∫ +=
rxy
V
222
2
2
∫ +=
w
pV
V
V
V
V
()
=-
Ê

Ë
Á
ˆ
¯
˜
2
2
2
2
exp
ax x
()
=-w
2
© 2004 by CRC Press LLC
Experimental Validation of an Individual-Based Model for Zooplankton Swarming 167
Fluctuations about this mean consist of the random-ßight accelerations A(t) – ku. Okubo (1986) derives
an expression for the expected variance of these ßuctuations, . The quantity we actually calculate
from data, however, is not acceleration but a Þnite-difference approximation to acceleration:
(11.17)
where Dt is the sampling period, and the theoretical variance of Du/Dt is not the same as that of the
acceleration itself. (Note that the distinction does not affect the predicted mean acceleration Þeld, or the
predictions concerning the velocity Þeld given above.) Without loss of generality, set t
0
= 0. Then the
Þnite-difference acceleration variance can be written
(11.18)
since the velocity variance is assumed constant in time. The second term can be evaluated using the
deÞnition of the autocorrelation (Equation 11.3) and the diffusive result (Equation 11.8), so that
(11.19)

predicts the variance about the mean acceleration for a time series sampled with a time step Dt.
Just as the acceleration Þeld has a Þnite variance about the mean, so does the autocorrelation curve
R(t). Only the mean autocorrelation was derived above. Because of the complexity of the problem,
however, in the data analysis below we estimate the theoretical variance of the autocorrelation numeri-
cally, through direct integration of the equation of motion (Equation 11.4), rather than deriving an
expression for it analytically.
11.3 Experiment
Data were collected for homogeneous groups of two species, the freshwater cladoceran Daphnia magna
(Hussussian et al., 1993) and the calanoid copepod Temora longicornus. In each experiment the animals
were placed in a small (10 ¥ 10 ¥ 15 cm) Plexiglas tank, a light down the center axis of the tank turned
on, and the trajectories of the animals swarming to the light recorded on videotape and then digitized.
The geometry of the tank-and-light system is shown in Figure 11.2. Description of the optical methods
employed can be found in Strickler and Hwang (1998) and Strickler (1998); digitization methods are
FIGURE 11.2 Geometry of the tank and light system.
a
2
D
D
D
D
u
t
ut t ut
t
=
+
()
-
()
00

D
DD
DD
D
D
u
tt
ut t ut ut t ut
t
uututt
2
22
0000
2
2
00
1
1
22
=+
()
-
()
[]
+
()
-
()
[]
=+

()
+
()
()
D
DD
D
u
t
u
t
e
kt
2
2
2
2
2
1=-
()
-
ax
()
Light
shaft
z
x
y
© 2004 by CRC Press LLC
168 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation

described in Doall et al. (1998). The light source in the Daphnia experiment was a 3-mW argon laser
(wavelength 488 and 514.5 nm), which produced a collimated beam 4 mm wide throughout the water
column. The Temora experiment used a 2-mm-diameter Þber-optic light guide, projecting light from a
halogen lamp at the top of the tank. This produced a narrow cone-shaped beam with more attenuation
through the water column, although the beam reached clearly to the bottom of the tank.
All three axes of motion were captured for the Temora experiment, using a pair of orthogonal
projections. One camera recorded motion in the x–z plane, and a second, motion in the y–z plane, so
that by matching z motions one could recover full three-dimensional trajectories. The Daphnia experi-
ment was recorded with an earlier generation of the optical system and thus contains only a single x–z
projection. Because of the radial symmetry of the light source, and the fact that our model predicts no
coupling of motion between orthogonal axes, the lack of a third dimension in the Daphnia data does
not hinder the statistical analysis.
Phototaxis in marine and freshwater macrozooplankton is well known, primarily in the context of
light-gradient-driven diel migration (Russell, 1934; Stearns, 1975; Bollens et al., 1994). The behavioral
signiÞcance of the strong positive phototaxis that zooplankters often show in the laboratory, however,
is poorly understood. Some animals placed in the tank showed no attraction to the light, and the response
of other animals was intermittent. Some of the Temora simply remained at the bottom of the tank, while
others collected at the surface, where the Þber-optic illumination was intensiÞed. Flux between these
subpopulations and the swarm in the mid-water column was continual. Trajectories for analysis were
taken from the animals that remained in the swarm for longer than one sample period.
These swarms tended to be small, generally consisting of fewer than ten animals, so that mean nearest-
neighbor distances were many body lengths. This low density is consistent with our treatment of
swarming behavior not as a density-dependent, social interaction, but rather as an individual response
to a sensory cue. The simultaneity or nonsimultaneity of the swarmers’ motions does not enter into the
analysis, because each animal is in effect swarming alone.
11.4 Analysis
11.4.1 Constructing a Statistical Ensemble
Trajectories were sampled at 1-s intervals. This time step was chosen both to resolve the fundamental
timescales of motion (the damping time 1/k and the attractive force period 2p/w, which were estimated
by an initial calculation of the velocity autocorrelation) and simultaneously to Þlter out higher-frequency

behaviors such as avoidance reactions and trail following (Weissburg et al., 1998). Long-time step
sampling is an unreÞned method for low-pass-Þltering a trajectory record; but a test performed on Temora
data sampled at 0.2 s suggested that the choice of this method over tapered low-pass Þltering of a higher-
resolution data set had a negligible effect on the analysis. Longer-time step sampling makes the digiti-
zation of trajectories (which was done by hand here rather than by computer program to ensure accuracy)
far less labor intensive.
Recorded trajectories were divided into uniform 20-s segments for the analysis. This length of time
was chosen to balance two considerations. The segments had to be long enough to resolve the inherent
dynamic timescales mentioned above, and at the same time as short as possible — i.e., as close to the
autocorrelation timescale as possible — to ensure uniformity of behavior within each record. They thus
become independent samples in the statistical sense. Within each 20-s segment, velocities and acceler-
ations were calculated using the Þnite-difference equations
(11.20)
(11.21)
ut
xt xt t
t
()
()

()
~
D
D
at
ut t ut
t
xt t xt xt t
t
()

=
+
()
-
()
=
+
()
-
()
+-
()
D
D
DD
D
2
2
© 2004 by CRC Press LLC
Experimental Validation of an Individual-Based Model for Zooplankton Swarming 169
Even among the animals that remained suspended in the water column, not all were engaged in
swarming behavior. A few simply drifted through the swarm without, in a kinematic sense, being part
of it, like waiters carrying trays across a lively ballroom. The shape of the velocity autocorrelation, as
goal is not to test a hypothesis about the phototactic behavior of Daphnia and Temora, but rather to
describe the swarming that a light gradient can evoke in these animals, we winnowed the trajectory set
to remove those animals who were not participating. (Because this winnowing eliminated only a small
number of candidate trajectories, as described below, it is fair to assume that the swarming dynamics
thus described account for the bulk of the animals’ response to this stimulus.) We eliminated from both
the Daphnia and Temora data sets trajectory samples whose velocity autocorrelation did not cross the
time axis, as well as samples whose mean position lay more than two standard deviations from the

z axis, the swarm center. These criteria eliminated 4 of 24 Temora trajectories and 2 of 60 Daphnia
trajectories. Some diffusers were also eliminated by eye before digitization of trajectories began. Lateral
The Þnal step in creating a uniform statistical ensemble of trajectories is to verify that the process
captured is kinematically stationary. Figure 11.3B and Figure 11.4C show the means and standard
deviations of position and velocity for both data sets. Note that there are no strong outliers, and that the
11.4.2 A Procedure for Testing Model Consistency
We proceed as follows for both the Daphnia and Temora experiments.
1. Verify that the velocity distribution is Maxwellian. This conÞrms (in the absence of an independent
measure of the excitation variance B) the kinetic energy balance of Equation 11.10.
2. Calculate the velocity autocorrelation R(t), and verify that its form Þts the kinematic requirement
Fit R(t) to Þnd w and k. This is done by minimizing, by inspection, the square-error function:
(11.22)
With these parameters, directly integrate Equation 11.4 to create a simulated ensemble of
velocity autocorrelations, and thus a numerical prediction of the autocorrelation variance
(just as Equation 11.5 gives an analytical prediction for the autocorrelation mean).
3. With w from step 2, conÞrm the relation between swarm size and kinetic energy predicted by
Equation 11.12.
4. Compare the mean and variance of the spatial acceleration Þeld with those predicted by
Equations 11.16 and 11.23.
5. Examine motion in the z (vertical, along-laser) direction, which we have not to this point
discussed. In this dimension the attractor is more-or-less isotropic and we might expect diffu-
sive, not swarming, dynamics. The animals’ vertical motion can be compared with theory under
the assumption that the diffusion-driven parameters , B, and k will be the same in the
z direction as they are in the x and y (cross-laser, swarming) directions. Here we follow an
abbreviated version of the outline above.
Look for a Maxwellian velocity distribution, as in step 1.
Calculate the velocity autocorrelation, compare its ensemble mean with Equation 11.8, and
compare its ensemble variance with numerical results, as in step 2.
Calculate the mean and variance of the acceleration Þeld as in step 4. We now expect the mean
to be zero (by Equation 11.16, with w = 0) and the variance to be as predicted by Equation 11.19.

Jk R R
observed theoretical
wtt
t
,
()
=
()
-
()
()
Â
2
u
2
© 2004 by CRC Press LLC
illustrated in Figure 11.1, quantitatively distinguishes swarmers from drifters, i.e., slow diffusers. As our
projections of all remaining Daphnia and Temora trajectories are shown superimposed in Figure 11.3A
variances of the ensembles are on the same order as the variances of individual trajectories within them.
and Figure 11.4A, B.
illustrated in Figure 11.1.
170 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
11.5 Results
that the calculations for the horizontal dimensions in the Temora data are two dimensional, as both horizontal
components of motion were captured and analyzed, and one dimensional for the Daphnia data. The
parameters k, w, and B, derived from the horizontal analysis, along with rms velocity and position (which
swarm sizes) predicted by Equation 11.12 and the error between these predictions and the values observed.
11.5.1 Velocity Distributions
Figure 11.5A, B and Figure 11.6A, B show observed velocity distributions for the horizontal (swarming)
and vertical (diffusive) dimensions, along with theoretical curves based on Equations 11.11 and 11.15.

FIGURE 11.3 (A) Superimposition of all 58 Daphnia trajectories included in the analysis, shown in x–z projection. Plot
boundaries indicate the edges of the tank, and the dotted line shows the position of the light shaft. (B) Position (horizontal
axis) and velocity (vertical axis) statistics for each trajectory sample. The crosshairs, each representing a single animal
trajectory, are centered on the mean horizontal position and velocity, and indicate standard deviations by their extent.
A
B
Daphnia trajectories (
x
–
z
projection)
x
(mm)
x
(mm)
Individual mean and std. dev.
z
(mm)
u
(mm/s)
150
100
50
0
5
0
2.5
–2.5
–5
–50

–20 –10 0 10 20
–25 0 25 50
2 2
© 2004 by CRC Press LLC
represent u and x ), are summarized for both species in Table 11.1. Also given are rms positions (i.e.,
Results for the Daphnia experiment are shown in Figure 11.5 and results for Temora in Figure 11.6. Note
Experimental Validation of an Individual-Based Model for Zooplankton Swarming 171
All velocity Þelds are close to stationary . Velocity distributions for both species lie close to
the predicted Maxwellian curves, with r
2
= 0.65 and 0.98 for the horizontal and vertical axes of the
Daphnia motion and r
2
= 0.91 and 0.86 for the horizontal and vertical axes for Temora. The horizontal
Daphnia distribution is more platykurtic than predicted, and the horizontal Temora distribution more
sharply peaked, but these deviations could be either statistical artifacts or true dynamic effects related to
these species’ swimming styles, and are second-order effects in either case. The Þrst-order match with
theory supports our assumption of a Stokes form for drag and a stochastic, spatially symmetrical excitation.
Note also that, with the exception of a possible upward drift along the laser axis in the Temora data
for each animal, in the horizontal and vertical dimensions. In fact, velocity variances are indistinguish-
able for the two horizontal dimensions in the Temora data ( = 16.7 mm
2
/s
2
, = 16.6 mm
2
/s
2
). These
patterns support the spatial symmetries we have assumed and our supposition of dynamic consistency between

the along-laser and cross-laser diffusion processes. Note that an upward drift in the Temora experiment would
be consistent with a weak phototactic response to the attenuation of the light shaft through the water column.
11.5.2 Velocity Autocorrelations and Fit Parameters
Daphnia motion. The horizontal theoretical mean is a Þt to the data shown, while the vertical theoretical
FIGURE 11.4 (Superimposition of the 20 Temora trajectories included in the analysis, shown in (A) x–z and (B) y–z
projection. Plot boundaries indicate the edges of the tank, and the dotted line shows the position of the light shaft. (C) Position
(horizontal axis) and velocity (vertical axis) statistics for each trajectory sample. The crosshairs, each representing a single
animal trajectory, are centered on the mean horizontal position and velocity, and indicate standard deviations by their extent.
A
C
Temora
trajectories
(
x
–
z
projection)
Temora
trajectories
(
y
–
z
projection)
x
(mm)
z
(mm)
V
(mm/s)

150
100
50
0
z
(mm)
150
100
50
0
–50 –25 0 25 50
B
y
(mm)
–50 –25 0 25 50
15
10
5
0
5025
r
(mm)
(
)
uu
2
2
<<
u
2

v
2
© 2004 by CRC Press LLC
(Figure 11.6B), the velocity Þelds are very close to symmetrical, and that velocity variances are similar,
Figure 11.5C, D show observed and theoretical velocity autocorrelations for the two dimensions of
172 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
mean is not a Þt but a prediction based on Equation 11.8 with k from the horizontal analysis. Theoretical
for Temora in the same format, except that here the observed horizontal velocity autocorrelation
(Figure 11.6C) is the average of the x and y autocorrelations, a quantity that is invariant under rotation
of the horizontal axes.
Horizontal autocorrelations for both species are Þt very closely by theoretical curves (Daphnia, r
2
=
0.995; Temora, r
2
= 0.90). This match is a strong validation of the kinematic theory underlying our
analysis. Variance of individuals around the ensemble mean for each species is close to the level that
simulation predicts (Daphnia, r
2
= 0.95; Temora, r
2
= 0.71). Note that in contrast to the underdamped,
highly orbital trajectories of members of the midge swarm analyzed by Okubo and Anderson (1984),
both the Daphnia and Temora swarms appear to be near critical damping, with individuals’ velocities
decorrelating almost entirely in less than one period of the harmonic attractive force.
Values of k derived from the horizontal swarming motion suggest decaying autocorrelation curves for
the vertical motion that correspond well with observations (Figure 11.5D and Figure 11.6D), conÞrming
our supposition that motion along the laser axis is primarily random and diffusive, and that the same
damping force acts on horizontal and vertical swimming. The Daphnia vertical autocorrelation, however,
has a longer tail than predicted, suggesting that the animals tend to drift, actively or passively, along the

FIGURE 11.5 Results for Daphnia. Velocity distributions (A, B), velocity autocorrelations (C, D), and acceleration Þelds
(E, F) for the horizontal (A, C, E) and vertical (B, D, F) axes of motion. In all panels, thick lines indicate means and shaded
areas and thin dotted lines indicate standard deviations. Full explanations of plotted quantities are given in the text.
A
B
C D
E
F
x
z
τ
τ
u
w
© 2004 by CRC Press LLC
standard deviations are calculated from numerical simulation. Figure 11.6C, D show autocorrelations
Experimental Validation of an Individual-Based Model for Zooplankton Swarming 173
laser axis, in combination with their diffusive behavior. The Temora vertical autocorrelation may or may
not indicate the same tendency.
Fitting the horizontal autocorrelations for k and w lets us test the consistency of the theoretical
relationship (Equation 11.12) between swarm size, swimming speed, and the strength of the concentrative
— i.e., the relationship between parameters is consistent with theory — to 4% for Daphnia and 45%
for Temora.
11.5.3 Acceleration Fields
standard deviations from Equations 11.16 and 11.19. The x and y acceleration Þelds for Temora
(Figure 11.6E) are superimposed.
The horizontal Daphnia acceleration Þeld (Figure 11.5E) agrees well with theory in both its mean
slope (which is not signiÞcantly different from the predicted value of zero) and its mean standard
FIGURE 11.6 Results for Temora. Velocity distributions (A, B), velocity autocorrelations (C, D), and acceleration Þelds
(E, F) for the horizontal (A, C, E) and vertical (B, D, F) axes of motion. In all panels, thick lines indicate means and shaded

areas and thin dotted lines indicate standard deviations. Full explanations of plotted quantities are given in the text.
A
B
C
D
EF
ττ
x,y
z
V
w
© 2004 by CRC Press LLC
force. Table 11.1 gives observed and predicted values of the swarm size for both species. They agree
Figure 11.5E, F and Figure 11.6E, F show the observed acceleration Þelds with theoretical means and
174 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation
(error in mean standard deviation = 8.9%), although superimposed on the diffusive Þeld are an upward
mean acceleration near the bottom of the tank and a downward mean near the top. These perturbations
suggest that the animals feel the top and bottom boundaries of the domain, perhaps hydrodynamically,
and turn away from them. The vertical Temora acceleration Þeld suggests the same behavior.
The variances of the Temora acceleration Þelds are similar to those predicted by our random-ßight
model (error in mean standard deviation 2.2% for the x direction, 30% for y, 11% for z), but the horizontal
mean Þeld is much smaller than predicted. In the absence of other results this might be taken to mean
the absence of a concentrative acceleration and thus primarily diffusive behavior, but the horizontal
the poor deÞnition of the mean horizontal Temora acceleration Þeld to noise in the data, i.e., a combination
of sampling error and real, high-speed behaviors not accounted for by our model. There are several
reasons to suspect this type of error. The Temora, unlike the Daphnia, make their turns quickly, generally
in less than the sampling time step of 1 s. Their trajectories wobble and jitter at high (~10 Hz) frequencies,
in ways that may be related to the trail-following that Weissburg et al. (1998) observed in the same
series of Temora experiments. Furthermore, in general, successive derivatives of a data set (calculation
of the acceleration requires two) ampliÞes noise at the high-frequency end of a power spectrum. One

would expect a moderate level of noise in the acceleration Þeld to wash out the small mean Þeld but
have only a secondary effect on the Þeld’s variance, as is observed.
11.6 Discussion
11.6.1 Model Consistency
Our goal here has been to provide an internally consistent model description of a steady-state zooplankton
swarm, by a method that could be applied generally to the problem of assessing physical and biological
controls on zooplankton aggregation. The analysis above veriÞes the suitability of the model presented
here to the artiÞcially stimulated swarms we have described. It also suggests the level of quantitative
agreement we can expect for real organisms, which do not move like idealized particles and inevitably
are engaged in other behaviors at the same time they are swarming: errors up to a factor of two, with
closer correspondences in most measures. The median unexplained variance (1 – r
2
) for all theoretical
Þts for which it was calculable was 11%.
TABLE 11.1
Dynamic and Kinematic Parameters along the Axes of Swarming Motion for the Daphnia and Temora
Experiments
Source Daphnia Temora
Damping coefÞcient k Fit to velocity autocorrelation 0.54 s
–1
0.80 s
–1
Concentration coefÞcient
w
Fit to velocity autocorrelation
0.49 s
–1
0.63 s
–1
Power of excitation B Equations 11.10, 11.13 3.6 mm

2
s
–3
13.3 mm
2
s
–3
Kinetic energy Observed 2.6 mm s
–1
(rms velocity) 5.8 mm s
–1
Swarm radius Observed 5.5 mm
(rms position) 16.6 mm
Predicted swarm radius Equation 11.12 5.3 mm
Equation 11.14 9.1 mm
Error in swarm size (observed–predicted) 4% 45%
u
2
uv
22
+
x
2
xy
22
+
x
p
red
2

xy
pred
22
+
© 2004 by CRC Press LLC
deviation (error = 11%). The vertical Daphnia acceleration Þeld (Figure 11.5F) shows similar agreement
velocity autocorrelation (Figure 11.6C) suggests a swarming balance very strongly. Instead, we attribute