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SHORT COMMUNICATION
Lagrangian description of zooplankton
swimming trajectories
MARCO UTTIERI*, MARIA GRAZIA MAZZOCCHI, AI NIHONGI1, MAURIZIO RIBERA D’ALCALÀ, J. RUDI STRICKLER1
AND ENRICO ZAMBIANCHI2
1
LABORATORY OF BIOLOGICAL OCEANOGRAPHY, STAZIONE ZOOLOGICA ‘ANTON DOHRN’, VILLA COMUNALE, 80121 NAPOLI, ITALY, GREAT LAKES WATER
INSTITUTE, UNIVERSITY OF WISCONSIN MILWAUKEE, WI
53204, USA AND 2INSTITUTE OF METEOROLOGY AND OCEANOGRAPHY, UNIVERSITY OF NAPLES
‘PARTHENOPE’, VIA DE GASPERI, 80121 NAPOLI, ITALY
*CORRESPONDING AUTHOR:
[email protected]
Three-dimensional swimming, in 20 trajectories of Daphnia pulex, was characterized in terms of
its kinematic properties. Results show that the random component was stronger than the deterministic component at time scales >1–2 s. Such random movements may have evolved to outwit
potential predators and prey.
Zooplankton swimming behaviour corresponds to a
trade-off between searching for prey or mates and avoiding predators. Motion can be considered a basic adaptive trait and therefore a relevant source of information
on the biological and behavioural responses of animals
to variable environmental conditions. Moreover, it is a
key aspect for understanding aspects of autoecology
(Strickler, 1977), including niche separation between species (Strickler, 1985).
Zooplankton trajectories have been quantitatively
analysed using parameters such as the net-to-gross
displacement ratio (Buskey, 1984) or the realized encounter volume (Bundy et al., 1993); their fractal (Coughlin
et al., 1992) or multifractal (Schmitt and Seuront, 2001)
characteristics have also been considered. These parameters describe the morphological complexity of swimming tracks, without considering other aspects of the
motion such as the kinematic properties. Additional tools
are necessary to parameterize the paths exhaustively.
In the present paper, we decompose the swimming
motion of 20 individuals of the freshwater cladoceran
Daphnia pulex into two superimposed fractions: a relatively larger scale one and a smaller scale component
of irregularly fluctuating character. For the latter, we
propose a characterization following a Lagrangian
approach. In Lagrangian analysis, each particle is followed in space and time (Pickard and Emery, 1982;
Emery and Thomson, 1998) and its trajectory is reconstructed and characterized.
The trajectories were characterized in terms of their
kinematic properties by means of: (i) the velocity autocovariance (Kundu, 1990); (ii) the spectral analysis (Press
et al., 1989; Kundu, 1990; Emery and Thomson, 1998);
and (iii) the reconstruction of the kinetic energies associated with the motion (Patterson, 1985). The Lagrangian approach allows us to classify and compare
trajectories effectively, thus improving our knowledge
about the complex behaviour of zooplanktonic organisms. This approach has been tested on three-dimensional trajectories described by D. pulex exposed to two
different light conditions.
Forty-four D. pulex adults were sorted from a batch
culture (started from some females kindly provided by
Dr S. I. Dodson, University of Wisconsin, Madison, WI,
USA) and transferred into a vessel (30 30 15 cm)
filled with 11 L of artesian water (temperature 20 1 C), without food to avoid any prey influence. The
vessel was covered to prevent any turbulent mixing.
Animal trajectories were recorded for 2 h, with
filming equipment and techniques as described in detail
by Strickler (Strickler, 1998). Swimming behaviour of
D. pulex was analysed under two different light conditions: a red laser (He–Ne laser, wavelength l = 632 nm)
as ‘background light’, and a blue laser (Ar laser, l = 514.5
doi: 10.1093/plankt/fbg116, available online at www.plankt.oupjournals.org
Journal of Plankton Research Vol. 26 No. 1, Ó Oxford University Press 2004; all rights reserved
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Fig. 1. Four examples of swimming trajectories of D. pulex. It is evident that, in the presence of the blue laser light (BL condition), the trajectories
are more convoluted than in the RL condition. The trajectories are scaled in a reference frame that represents the volume of observation.
nm), which is reported to attract D. pulex, Cyclops scrutifer
and Chydorus sphaericus (Strickler, 1998) (Figure 1).
Through the entire filming, the area was always illuminated by the red laser. During the first 60 min, the room
lights were on (RL condition), then during the following
45 min the blue laser was added to the red one and the
room lights were off (BL condition). For the next 5 min, the
blue laser was off and then it was switched on again; room
lights were still off. Prior to filming, the animals were
allowed to acclimatize to the light condition for 10 min.
Here we report the analysis of the 20 longest clips
recorded, each one lasting nearly 90 s (10 clips in RL
condition, 10 clips in BL). Shorter clips, besides not
being comparable with the 90 s ones, would not yield
as reliable statistics. For each clip, the three-dimensional
coordinates of the displacement were obtained using
an Amiga-running tracking software (Strickler, 1998).
All clips were digitized with the sampling frequency
( fs) at 6 Hz. Since the maximum appendage beat rate
of D. pulex is 7 Hz (Porter et al., 1982), this value of
fs allowed us to reconstruct the trajectories without
redundant information on the movement of the appendages. The data so obtained were neither smoothed nor
fitted.
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The swimming trajectories performed by D. pulex were
quite complex (Figure 1). The scales involved in the
motion spanned between some centimetres (i.e. the
length spanned by the trajectory as a whole) and 1–2 mm
(i.e. the typical scale of the organism displacement). These
two scales were clearly separated. The larger one was
characterized by repeated, even though non-periodic,
abrupt changes or even complete reversals of direction,
showing strong differences between RL and BL conditions (Figure 2).
We evaluated the autocovariance of the fluctuating
part of the three components of the velocity, in order
to determine the degree of randomness of the swimming
tracks, where we define a random, or stochastic, variable
as an irregularly fluctuating one (Gardiner, 1990;
Gnedenko, 1997).
Velocities were computed using a central difference
method. The autocorrelation of the i-th component of
the velocity at an arbitrary time t with itself at a later time
t + t can be expressed as (Bendat and Piersol, 1966):
Rii ðtÞ ¼ ui ðtÞui ðt þ tÞ
ð1Þ
Dividing the equation by u2i (mean square velocity), we
obtain the autocovariance (Kundu, 1990):
r ii ðtÞ Rii ðtÞ
u2i
in terms of their peak frequency composition (Emery and
Thomson, 1998). The power spectral density function
represents the way the energy associated with the process
is distributed over the spectrum of frequencies (Kundu,
1990; Legendre and Legendre, 1998).
In both RL and BL, the autocovariance of the three
velocity components dropped sharply to zero (deltashaped pattern) indicating very short correlation of velocities (Figure 3). Accordingly, the values of the integral
time scale T were very small (Table I).
The spectral analysis showed that the motion could be
considered as the superposition of a large number of
frequencies. The PSD plots displayed a flat spectrum without any pronounced peak (Figure 4), showing that there
was no preferential frequency in the motion, either during
RL or during BL. Moreover, the PSD plots were characteristic of white noise (no defined slope) confirming the
absence of any apparent memory in the velocity.
The distinction between different kinetic energies
associated with the motion allows the characterization
of the strictly kinematic properties of the organisms’
motion itself. Based on the kinematic or Reynolds
decomposition, we can assume that the velocity of an
organism is the sum of a larger-scale mean field and of a
smaller-scale one, fluctuating around zero (Pond and
Pickard, 1983; Mann and Lazier, 1996):
u ¼ u þ u0
ð2Þ
which is as a normalized measure of the correlation of the
velocity with itself at subsequent times. For short memory
velocities, rii(t) is significantly different from zero only at
short time lags. At distant times randomness implies little
self-correlation (Bendat and Piersol, 1966), therefore, for
t > 0, rii (t) will rapidly decrease and asymptotically
approach zero as t increases. The problem of the presence of small-scale coherent motions detectable only in
the off-diagonal elements of the autocovariance matrix
[rij (t) with i 6¼ j] was recently discussed in the Lagrangian
oceanographic context (Reynolds, 2002). None of our
trajectories presented any such occurrence.
The parameter T, the integral time scale, given by
(Kundu, 1990):
1
ð
T ¼ r ii ðtÞdt
ð3Þ
0
measures the ‘memory’ of the process, being an estimate
of how long the velocity at a certain time influences the
later motion.
Further information can be drawn from the Fourier
transform of the velocity autocorrelation [power spectral
density (PSD), in the following], which describes the data
ð4Þ
where the underlines represent vector quantities. The
total kinetic energy (TKE) per unit mass is:
TKE ¼
u2
2
ð5Þ
whereas the kinetic energies associated with the mean
(MKE) and stochastic (eddy) (EKE) fields are, respectively:
u2
ð6Þ
MKE ¼
2
u0 2
EKE ¼
ð7Þ
2
It is worth noticing that the above energies are kinematic
properties associated with the organisms’ motion, and do
not reflect any metabolic process.
In particular, we evaluated the EKE/TKE ratio over
the three orthogonal planes (XY, XZ and YZ) by combining the three velocity components (Patterson, 1985);
for example, the three kinetic energies associated with
motion on the XY plane are given, respectively, by:
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TKE ¼
u2 þ v2
2
ð8Þ
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(b)
Fig. 2. Histograms of the frequency of the reversals of the three velocity components in (a) RL and (b) BL conditions, evaluated after low-pass
filtering the velocity data by means of a 25-element running average.
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Fig. 3. Autocorrelation of one velocity component of a freely
swimming D. pulex.
Fig. 4. Power spectral density plot calculated on one velocity component of a D. pulex. Frequencies are in Hz, whereas the power spectral
density is in mm2 s2/s.
Table I: Mean values ( standard deviation) of
the integral time scale T (in seconds) along the
three directions (X, Y, Z ) for 20 swimming
paths in D. pulex in two different light
conditions
The mean flow evaluated in order to derive the
fluctuating part is the result of averaging the velocities
in quasi-cubic subregions (‘bins’) into which the domain
spanned by the organisms was divided. Particular attention was paid to the width of the bins (bs ) used; this issue
is subject to debate within the oceanographic community [e.g. (Poulain and Niiler, 1989)]. Narrow bins allow
the small-scale variability of the process to be accounted
for, but wide ones, containing more data-points, yield
more reliable statistics (Patterson, 1985). Moreover, a
crucial point for the operational usefulness of the kinematic decomposition is the presence of a scale separation
between the mean and the residual portions, i.e. bs has to
be much larger than the typical length scale of the
displacement fluctuations. For these reasons, for each
trajectory the evaluation of the different EKE/TKE
ratios was reiterated over a range of bs (from 2 to
7 cm). Not only did these values respect the scaleseparation, but also they were large enough so that a
minimum of 15 points was present in every bin, thus
making the averaging statistically more consistent.
The characterization by means of the kinetic energies
confirmed the relevant random component in the paths.
The role of the kinetic energy associated with the
fluctuating portion in the total energetic budget was
evident from the values of the EKE/TKE ratios. In
both RL and BL they were %0.94 over the three planes
considered (Table II), i.e. >90% of the total energetic
budget was due to the random component. In addition,
the values of the EKE/TKE ratios were comparable for
each of the three planes, so that the movement could be
assumed to be energetically isotropic independent of the
light condition.
Integral time scale
Mean values SD (s) (n = 10)
RL
T(X)
1.19 0.41
T(Y)
1.07 0.19
T(Z)
1.18 0.32
BL
T(X)
1.34 0.22
T(Y)
1.53 0.32
T(Z)
1.18 0.17
RL, blue laser off; BL, blue laser on.
u2 þ v2
2
ð9Þ
u02 þ v02
2
ð10Þ
MKE ¼
EKE ¼
The EKE/TKE ratio determines the relative importance of the fluctuating component of the velocity in the
overall motion, and we therefore used it as an estimate of
the irregularity of the motion. Values close to 1 would be
typical of random motions, for which most of the total
kinetic energy budget is represented by the fluctuating
part.
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Mean values SD (n = 10)
RL
XY
0.94 0.03
XZ
0.94 0.02
YZ
0.95 0.02
0.94 0.02
XZ
0.94 0.03
YZ
0.95 0.02
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ACKNOWLEDGEMENTS
BL
XY
j
random when clues are absent (as in our experimental
conditions) and that only once they detect a signal (either
mechanical or chemical) does their motion become more
coherent and deterministic, reflecting a change in the
kinematic properties of the trajectories. Additional tests
using experimental conditions resembling those experienced by the animals in nature will be carried out,
integrating all the above analysis, so as better to compare
the different responses.
Table II: Mean values ( standard deviation)
of EKE/TKE ratio (non-dimensional),
measured over three planes (XY, XZ and YZ )
in the swimming trajectories performed by
D. pulex in two different light conditions
EKE/TKE
26
EKE, eddy kinetic energy; TKE, total kinetic energy; RL, blue laser off; BL,
blue laser on.
Based on our results, some conclusions can be drawn.
The autocovariance and the spectral analysis characterized the organism’s motion, allowing us to evaluate the
degree of randomness of the tracks and the role of the
different frequencies involved in the motion. Our results
showed that the swimming velocities of D. pulex were not
self-correlated even at short times and resembled a random process. This suggests that after few time steps the
motion was independent of the previous dynamics and
could be in this sense assumed to be stochastic (in the
sense described above). It has indeed been shown by the
presence of white energy spectra that no dominant frequency occurred in the motion: all the frequencies sum
up and the motion of the animal generates a complex
hydromechanic signal. The kinetic energies associated
with the trajectories confirmed the strong random character of the motion, in agreement with what is suggested
by the velocity autocovariance and by the spectral
analysis.
We suggest that D. pulex adopts such complex swimming behaviour to outwit the sensory perception of
potential prey and predators, which could lack cues by
which to determine the source of the signal. This
hypothesis is in agreement with the ‘fluid-dynamical
camouflage’ proposed by Hwang and Strickler (Hwang
and Strickler, 2001).
The analysis of the kinematics of the swimming tracks
may be needed to understand the behaviour of zooplanktonic organisms. Based on our application, some
general behavioural hypotheses can be formed. In zooplanktonic organisms, different stimuli trigger specific
responses (escape, attack, mate following). Generalizing
our results, we propose that these animals swim at
The authors owe Thomas Kiørboe for useful comments
and suggestions. M.U. is grateful to Howard Leu and
Sarah Lovern for their support and collaboration during
experiments; thanks are also due to Priscilla Licandro for
her suggestions in the statistical analysis. This work was
partly supported by the Italian Space Agency and by
the Italian Ministry of Scientific and Technological
Research.
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Received on May 14, 2003; accepted on September 17, 2003
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