Behavioral
Ecology
The official journal of the
ISBE
International Society for Behavioral Ecology
Behavioral Ecology (2015), 26(4), 1236–1247. doi:10.1093/beheco/arv073
Original Article
Ideal gas model adequately describes
movement and school formation in a pelagic
freshwater fish
Derrick T. de Kerckhove,a Scott Milne,b Brian J. Shuter,a and Peter A. Abramsa
of Ecology and Evolutionary Biology, University of Toronto, 25 Willcocks St., Toronto,
Ontario M5S 3B2, Canada and bMilne Technologies, PO Box 237, Keene, Ontario K0L 2G0, Canada
aDepartment
Received 23 July 2014; revised 8 May 2015; accepted 11 May 2015; Advance Access publication 3 June 2015.
Movement is a fundamental aspect of the population and community ecology of many organisms, yet, until recently, it has been difficult to measure in the wild. Consequently, simple assumptions are often used to represent movement; a key assumption found in
many classic theoretical ecological models (e.g., predator–prey interactions) is that organisms move like ideal gas particles. Here,
we test whether this assumption adequately describes the movement of the Cisco (Coregonus artedi) and its schools using fisheries
acoustic surveys and mathematical models. We find that several of the individual components of an ideal gas model (IGM) have some
inconsistencies with Cisco behavior, yet overall patterns of school formation are close to IGM expectations. For both individual fish and
schools: 1) the spatial distributions were random or slightly clumped; 2) the swimming speed distributions were unimodal but significantly different from normal; 3) horizontal movement was more frequent than depth changes; and 4) movement trajectories across the
acoustic beam sometimes deviated from straight lines. However, including the average individual and school swimming speeds and
known nighttime densities in an IGM generated values that were similar to the observed values for: 1) the time required for schools to
form in the morning and 2) school encounter rates.
Key words: acoustics, animal movement, Cisco, fisheries, ideal gas model, schooling behavior, spatial ecology.
INTRODUCTION
Movement is a fundamental characteristic of life for many organisms and is frequently one of the basic parameters needed to
understand the ecological dynamics of populations, communities
and ecosystems (Gurarie and Ovaskainen 2013). Until recently, the
principle challenge in explicitly incorporating animal movement
patterns into ecological models was the high effort required to collect movement data. Consequently, ecological models often adopted
simpler representations of animal movement, for example the ideal
gas model (IGM), random walks, and simple diffusion (Maxwell
1860; Dicke and Burroughs 1988; Gurarie and Ovaskainen 2013).
More recently, a wide range of technological solutions for collecting movement data have become available (e.g., telemetry, fisheries
acoustics, global position systems), yet it is still not clear how specific movement patterns influence general ecological relationships
(Jonsen et al. 2003). Further, the fundamental ecological models
based on simpler representations of animal movements are still
widely used in academic research and wildlife management (e.g.,
Address correspondence to D.T. de Kerckhove. E-mail:fish.research@
gmail.com.
© The Author 2015. Published by Oxford University Press on behalf of
the International Society for Behavioral Ecology. All rights reserved. For
permissions, please e-mail: [email protected]
most predator–prey models). It therefore seemed useful to determine whether a particular study organism could be characterized
by simpler movement models, for which the analytical behavior is
understood and is known to be consistent with widely used models of species interactions. In this study, we compare the observed
movement of Cisco (i.e., Lake Herring, Coregonus artedi Lesuer) individuals and schools with the expectations of the IGM.
The IGM was derived from the physics of gas particles within a
large and confined space (Maxwell 1860) and subsequently applied
to characterize animal movement in many ecological models and
methods (Hutchinson and Waser 2007; Gurarie and Ovaskainen
2013). Its primary assumptions are: 1) movement between individuals is independent, 2) speed over each dimension of movement (e.g.,
x, y, and z coordinates) follows a Maxwell–Boltzman distribution,
3) movement is equally likely in any direction, and 4) movement
trajectories follow straight lines between particle encounters. The
basic model can be modified to accommodate a few common deviations from these assumptions and has successfully been applied to
a wide range of animals (Gerritsen and Strickler 1977; Evans 1989;
Hutchinson and Waser 2007). The great strength of the IGM is
that encounter rates between individuals can be predicted based
on the density of the organisms within the environment if the
de Kerckhove et al. • IGM describes movement of fish and fish schools
basic movement parameters are known. This relationship can be
used as a model to estimate population sizes during field surveys or
to predict ecological interactions within populations or communities (Hutchinson and Waser 2007). However, an important caveat
is that the behavioral, ecological, and spatial contexts of animal
movements in nature often change with diel, seasonal or stochastic environmental variation (Hughes 1995; Okubo and Levin 2001;
Mazur and Beauchamp 2006).
A large body of research characterizing individual fish movement suggests that strong deviations from simple movement models
are likely to exist and often reflect optimizing fitness under changing environmental conditions (de Kerckhove et al. 2006) or exploring new territories (Radinger and Wolter 2013). However, at larger
spatial scales the dispersal of populations and distributions of species ranges still mimic simple diffusive processes (Muneepeerakul
et al. 2008). The movement of fish schools has not been studied
to the same degree as that of individual fish and those studies that
have been carried out typically involve fine scale measurements of
small schools in laboratory settings (Hoare et al. 2004) or coarse
measurements of school distributions in natural environments
(Nøttestad et al. 1996). Such observations are difficult to incorporate into dynamic models of ecological interactions. Consequently,
ecological models often treat schools as large individuals or mobile
homogenous patches, and prescribe to them the known movement properties of individual fish (Misund 1993; Cosner et al.
1999; Geritz and Gyllenberg 2013). In some cases, properties of
the school can emerge from the combined behavior of individuals following simple movement rules (Parrish and Edelstein-Keshet
1999) or from many fish following 1 or 2 leaders (Viscido et al.
2005). However, such simplistic conceptions often fail when additional ecological questions are posed regarding the interactivity of
school members in school formation, predator evasion, or resource
consumption (Abrahams and Colgan 1985, Swartzman 1991,
Nøttestad and Axelsen 1999).
In this study, we use fisheries acoustic surveys to test whether
individual Cisco (i.e., singletons) and Cisco school movements
adhere to the 4 main assumptions of the IGM by characterizing
their swimming speeds, spatial distributions, headings (i.e., direction of movement), and tortuosity (i.e., deviation from movement
in a straight line). We then create an IGM of Cisco singleton and
schools movement based on data from acoustic surveys, and test its
predictions against observed patterns of Cisco schooling behavior
in our study system including 1) the timing of school formation in
the morning and 2) the expected encounter rates between stationary surveys and schools based on known school density in the late
afternoon. Following Hutchinson and Waser (2007), if the observed
patterns deviate enough from IGM predictions we can reject it as
an adequate descriptor of the animal’s movement.
METHODS
Field surveys
All acoustic surveys were conducted on the South Arm of Lake
Opeongo in Algonquin Provincial Park in Ontario, Canada
(45°42′N, 78°22′W). See Supplementary Figure SM1, for a map of
the South Arm and the location of surveys. The lake is 58 km2 with
a maximum depth of 49.4 m and contains only 2 pelagic schooling fish, Cisco and Yellow Perch (Perca flavescens Mitchill), although
young of the year Lake Whitefish (Coregonus clupeaformis Mitchill) are
occasionally found within Cisco schools. Night, dawn and daytime
mobile surveys were conducted on 16, 17, 21, and 22 August 2009
1237
to measure the spatial distributions and densities of fish and schools,
and on 27 and 28 July 2010 to measure the length of time Cisco
required to form schools at dawn. Twenty-four hour fixed-platform
acoustic surveys were conducted between 6 and 30 October 2010
to measure the swimming characteristics of individual Cisco and
Cisco schools, as well as daytime school detection rates. Netting
programs were conducted in September 2009 and throughout July–
October 2010, including more than 60 two-hour sets of suspended
pelagic nets between the surface and 20 m using single (50 m ×
2 m with either 7, 19, 25, or 38 mm stretched mesh sizes in 2009
and 2010) and multimesh (25 m × 6 m with 7, 13, 19, 25, 32, and
38 mm stretched mesh sizes in 2010 only) gill nets. This sampling
program provided a comprehensive assessment of the species composition and size distribution of fish found in the parts of the water
sampled by our acoustic surveys. The gill net samples showed that
the sizes of Cisco (mean = 90 mm) were much larger than those of
Yellow Perch (maximum = 60 mm), and that Yellow Perch schools
were typically found only within the top few meters of the water
column. This allowed us to focus mobile acoustic surveys on Cisco
by restricting our analysis to depths from 5 to 15 m, and to fish sizes
(using target strength—size relationships in Frouzova et al. 2005)
20 mm greater than the maximum Yellow Perch size (i.e., 80 mm),
and less than the maximum Cisco size (i.e., 160 mm).
The methods associated with the calculation of the swimming
speeds and density of fish schools for both mobile and fixed-platform acoustic surveys are described elsewhere (see de Kerckhove,
Milne, Shuter, manuscript in review). We will briefly outline those
methods, but refer the reader to the associated publication for additional detail, and on the acoustic surveys in general. The parameters
and settings used for the mobile and fixed-platform acoustic surveys
are listed in Table 1. All transducers were from the Simrad EK60
series (Kongsberg Maritime, Kongsberg, Norway) and mounted on
aluminum poles. All surveys adhered to the Great Lakes Standard
Operating Protocols (Parker-Stetter et al. 2009). All raw acoustic
data were processed using Echoview® (Myriax Software Pty. Ltd
version 5.2.70) following recommendations in the software manual and standard data-analysis procedures (Parker-Stetter et al.
2009). All data analysis parameters are listed in Table 1, including
any settings that were not default for the Echoview’s® 2D School
Detection Module, Echoview’s® Fish Track Detection Module and
Echocalc GPS® (Myriax Software Pty Ltd version 4.8).
Cisco singleton or school densities were estimated following
standard operating procedures for the Great Lakes (ParkerStetter et al. 2009) including a 100-m long elementary distance
sampling unit (EDSU) justified as an appropriate distance to balance the effects of spatial auto-correlation at local scales and
to preserve variance at global scales within Lake Opeongo.
Singleton densities were calculated through echo-integration
with the schools’ backscatter masked from the analysis. School
densities were calculated as the number of school detections
divided by volume of water surveyed. Here, the width of the
ideal acoustic beam (3 dB) is increased on either side of the
beam axis by the radius of the average school size to include
the area outside of the ideal beam, which is surveyed indirectly
when only the edge of a school is ensonified. Last, the singleton and schools densities were added together to estimate the
density of total Cisco “entities” (i.e., groups of Cisco for which
the minimum group size is 1 individual). This process allows
for separate density estimates of singleton and schools, as well
as spatial analysis of the distributions of singleton and schools
across the pelagic zone.
Behavioral Ecology
1238
Table 1
Acoustic survey and data-analysis parameters and settings used for fixed-platform and mobile surveys
Acoustic survey parameters
Transducers
Transducer name
Frequency (kHz)
Beam type
Pulse duration (ms)
Sample interval (ms)
Transmit power (W)
Transducer gain (dB)
Sa correction (dB)
Major beam angle (degrees)
Minor beam angle (degrees)
Major offset angle (degrees)
Minor offset angle (degrees)
Sv noise estimate (dB)
T1
120
Split
0.128
0.023
300
24.78
−0.18
8.16
4.13
0.03
−0.04
−124.3
Acoustic data analysis parameters
Sv analysis threshold (dB)
TS analysis threshold (dB)
Elementary distance sampling unit (m)
Vertical depth bin (m)
2D school detection module
Minimum total school length (m)
Minimum total school height (m)
Minimum candidate school length (m)
Minimum candidate school height (m)
Maximum verticle linking (m)
Maximum horizontal linking (m)
Distance mode
EchoCalc GPS module
dummy variable speed (m/s)
Fish track detection module
Minimum number of pings
Minimum number of single targets
Maximum gap between single targets (pings)
Fixed-platform
−60
−54
100
10
Mobile
−60
−54
100
10
1
0.5
0.15
0.1
0.15
1.5
EchoCalc
1
0.5
0.15
0.1
0.15
1.5
Global position system (GPS)
0.4
N/A
1
1
5
1
1
5
IGM assumption test #1: swimming speed
estimates
The fixed-platform survey data were used to estimate singleton
and school swimming speeds. The acoustic measurement of individual fish swimming behavior is well documented (e.g., Arrhenius
et al. 2000; Mulligan and Chen 2000) and is accomplished using
Echoview’s® Fish Track Detection Module; however, school swimming behavior is not well documented for split-beam transducers.
Generally, school swimming speed and direction were estimated
by measuring the timing, distance, and angle of school movement between 2 acoustic beams set up in parallel to each other
(de Kerckhove, Milne, Shuter, manuscript in review). Swimming
speed frequency distributions for singletons and schools were tested
for normality using a Shapiro–Wilk test in R programming code
(v2.13.1 R Foundation for Statistical Computing 2011).
IGM assumption test #2: independence of
particles
Independence of movement for each singleton and for each school
is difficult to test directly because potentially subtle environmental cues cannot be all controlled or observed (e.g., scent trails,
Hemmings 1966; or fish aggregating devices, Soria et al. 2009).
Instead the degree of “randomness” in their spatial distributions
should give an indirect indication of the independence of singleton and school movement, and was tested here by comparing the
observed nearest neighbor (NN) distances among singleton or
T2
120
Split
0.128
0.023
300
22.56
0.03
8.37
4.19
0.04
−0.04
−124.8
Vertical
120
Split
0.128
0.023
300
24.68
−0.17
6.43
6.39
−0.07
0
−125.6
Vertical
70
Split
0.128
0.023
300
25.73
−1.05
6.45
6.45
−0.03
0
−141.9
schools with NN distances from randomizing their locations within
the surveys (Crawley 2007). All transects covered identical depths,
and so could be aggregated into 1 long continuous transect by
arranging the EDSUs by survey time. The average NN distance was
calculated for singletons and schools within that continuous transect (i.e., observed NN). Then, 1000 additional continuous transects
were generated for which the location of singletons and schools
had been randomized. For each of these randomized transects, an
average NN distance was calculated for singletons and schools (i.e.,
random NNs). The observed NN was compared to the distribution
of random NNs using a z-test, and a NN index (i.e., observed NN/
mean random NN) was calculated such that NN indices that are
>1, =1, and <1 represent even, random, and clumped distributions, respectively (Crawley 2007). We further examined the evenness of singleton densities across the pelagic zone using a Kriging
algorithm (Spatial Analyst toolbox in ArcGIS v10, Environmental
Systems Research Initiative, Redlands, CA) with the EDSU densities as the primary unit. Nighttime densities were assumed to represent the spatial distribution of Cisco across the pelagic zone at
dawn, and were verified with localized dawn surveys.
IGM assumption test #3: direction and linearity
of movement
The direction of movement was estimated from the first and last
observed spatial positions in the fixed-platform survey data and
was characterized more than 360° across the x (West to East)
and y (Surface to Benthos) coordinates. It was also separately
de Kerckhove et al. • IGM describes movement of fish and fish schools
characterized more than 180° across the z coordinate (North
to South) which is roughly onshore and offshore movement.
Movement was separated into these 2 vectors because 1) onshore
movement of strictly pelagic fish species is not expected to occur
very often; 2) the geometry and orientation of the stationary survey
beams were biased toward recording a higher frequency of alongshore movements. The linearity of Cisco movement trajectories
was characterized by an index of tortuosity (i.e., the path length of
a fish’s trajectory divided by the linear distance separating their first
and last positions). This index was calculated for single fish tracks
made up of at least 8 separate positions. It could not be accurately
calculated for schools because the number of separate positions was
typically small (<4).
The ideal gas model
The IGM equations used in this study estimate the encounter
rate (ε) more than 1 min of 1 member of a focal population (i.e.,
singletons or schools) with any member of a target population, of
which the targets can also be from the same population as the focal
individual (e.g., the encounter rates among singletons). We distinguish between singletons and schools with a numerical subscript
(n) representing the number of individuals within the group (i.e.,
1 = singleton; 2 = schools with 2 individuals; … n = schools with
n individuals; and 2+ = all schools of all sizes), or more generally
between focal and target individuals with the letter subscripts f and
t, respectively. See Supplementary Table SM1 for a full list of symbols for the IGM parameters. The rates of encounters are calculated using 3 main parameters: density, visual acuity, and relative
speed. Relative speed is the change in distance over time between
the focal and target individuals based on both of their swimming
speeds and bearings of movement. Visual acuity and relative speed
determine the volume of water surveyed by the focal individual (S),
and the density (ρ) determines the number of individuals expected
to be encountered within that volume as,
ε = ρS(1)
This relatively simple relationship, and the aforementioned 4
assumptions of the IGM model, describes a Poisson distribution
of encounters. Therefore, the negative exponential function of
the encounter rate determines the probability of not encountering
another individual (Hutchinson and Waser 2007).
P ( ε = 0) = e − ε (2)
The IGM equation parameters are dependent on time because
the visual environment and densities of different entity types
change over time. Densities of singletons and schools of all sizes
change over time as school formation proceeds. Visual acuity
changes with ambient light levels and as larger, more easily visible, schools are encountered more frequently later in the day.
However, relative speeds do not change over time; although swimming speeds differ between singletons and schools, the speeds
of each category do not change during daylight hours in Lake
Opeongo (de Kerckhove, Milne, Shuter, manuscript in review),
nor are they greatly influenced by school size (Soria et al. 2007).
The IGM equations incorporating changing ambient light levels
are thus used to estimate the change in singleton and school densities over a user-defined period of time. We used 1 min as our
fundamental unit of time, so encounter parameters and school
densities were updated each minute assuming no schools were
present at dawn nautical twilight.
1239
The starting density of Cisco was estimated from the mobile
nighttime field studies on Lake Opeongo, using data collected over
a depth range 7–11 m. This depth range was chosen because dawn
surveys suggested that schools formed primarily in this region.
Density was measured in the model as the total number of entities
(δ) found in volume of the lake (V) between 7 and 11 m.
The volume of water surveyed by an entity was estimated using
a formula presented in Hutchinson and Waser (2007, Equation 29
and Figure 9) which is the integral of all possible angles of encounter among entities moving at different bearings, and their associated survey volume. This formula is appropriate because it uses
the observed probability of each bearing of movement expected
to occur for Cisco and Cisco schools within Lake Opeongo, rather
than assume that all entities move in random directions. Hutchinson
and Waser’s (2007) formula is based on calculating the X, Y, and Z
vectors of the survey volume for each interaction. We present the
vector equations using an illustrative example (Figure 1a) of an
interaction between a focal singleton encountering a target school
with an elliptic cylindrical survey volume with length X, major
axis Y, and minor axis Z (in some cases Y and Z could be the same
length). The basic form of the equation for survey volume is,
π
S = ∫π X Y Z P (θ) d θ(3)
0
where π X Y Z is the volume of the survey area at a particular
angle of encounter for the focal entity and P(θ) is the probability of
a particular angle of encounter between the focal and target entities. Head-on encounters occur at an angle of π radians, encounters from above at 0.5π radians and encounters from behind at 0
radians. Note that it is also possible to integrate over the range of
swimming speeds observed in Lake Opeongo if their frequency distributions deviate greatly from normal. In our preliminary model
runs this approach did not lead to any quantitatively different
results than using a constant mean speed for singletons and schools
because the frequency distributions were bell-shaped. For simplicity,
we only present the model with constant swimming speeds.
The angle of encounter (θ) and the swimming speeds (v) of the
focal (f) and target (t) entities over time (T) determine the length of
the cylinder (see X, a thick black arrow in Figure 1a), which is calculated by trigonometry,
X = T * v 2f + vt2 − 2v f vt cos θ(4)
The total length of axes of the elliptic cylinder (Y and Z) are determined by the size of the focal and target entities (specifically the
axes that cross perpendicular to the direction of relative movement,
and at right angles to each other), and the maximum distance
the focal entity is able to detect a target. The size of the entities
is important for 2 reasons: 1) the focal entity needs to only detect
the edge of a school for an encounter to occur, and 2) even in the
absence of light (i.e., at minimal detection distances) the physical
displacement through water by moving entities could lead to an
encounter through a collision. Y and Z are the sum of 1) the axes
of focal (yf and zf) and target entities (yt and zt) that are aligned perpendicular to the direction of relative movement (i.e., vector X),
within their respective dimensions, and 2) twice the detection distance (yd and zd; see Figure 1b–d). For singletons, the length of these
axes are calculated using the body length (l1) and thickness (w1) of
a Cisco, respectively, following Hutchinson and Waser (2007, their
equation 29; see solid green lines in our Figure 1b and c),
Behavioral Ecology
1240
(a)
(b)
Y
0.5 yt
yd
yf
Z
yd
X
0.5 yt
(d)
(c)
Z
0.5 z
Y
zd
zf
zd
0.5 zt
Figure 1
Encounters between a singleton and a school demonstrating the X, Y, and Z vectors of the dimensions of the elliptical cylinder representing the search
volume of the focal individual. (a) the relative distance travelled of the focal individual is the length of the cylinder (thick black arrow) which is calculated
using the lengths of the actual distance travelled by the focal and target entities (thin black arrows) and the angle of their encounter (θ); (b) the components
of the Y vector including the relative length of the focal individual (yf), the detection distance (yd), and the half-axis of the school (yt), (c), the components of
the Z vector including the thickness of the focal individual (zf), the detection distance (zd), and the horizontal radius of the school (zt); (d) the elliptical base of
the cylinder made up of the Y and Z vectors.
y1 =
l1v1 sinθ
(5)
X
z1 = w1(6)
Note that y1 is the aspect of Cisco’s body length that displaces water
as it moves relative to its target and so will always be smaller or
equal to the total body length (see dotted green lines in Figure 1b),
whereas because Cisco move with their head oriented forward, their
body thickness will always contribute equally to the Z axis along
the path of relative movement (see dotted green lines in Figure 1c).
Schools are modelled here as oblate spheroids with equal horizontal
diameters (i.e., twice the school radius; r), and a height (h), that is,
three-fifth diameter (as found in previous studies on Lake Opeongo,
see Milne et al. 2005). For the Y dimension, yn is dependent on the
angle of encounter between the focal entity and the school, and
thus can be solved by the trigonometric definitions of the elliptical
school’s axes (see solid blue lines in Figure 1b),
yn = 2r sin θ + h cos θ (7)
Note that with this equation at perpendicular angles of encounter,
the axes are simply the diameter of the school, whereas head-on
encounters would involve only the school height. For the Z dimension, we assume that the schools will maintain their disk-like shape,
and so their contribution to the Z vector will simply be the school
horizontal diameter (see blue lines in Figure 1c),
z f = 2r (8)
Although our illustrative example uses a singleton as the focal individual, if 2 schools were encountering each other, the 2 school axes
equations above would be required for both the focal and target
individuals.
The visual acuity of Cisco was estimated from a range of studies including allometric relationships in fish, laboratory work on
Cisco and experimental work on closely related species (i.e., the
Salmonidae family). The basic model we employed combined the
visual acuity of the fish with the ambient light intensity of the surrounding environment.
McGill and Mittelbach (2006) developed an allometric relationship (r2 = 0.29) describing the maximum detection distance (a in m)
for a fish of mass m (in kg) observing a fish target of length l (in m):
(
)
a = 13.8 m 0.107 * l (9)
The detection distance refers to the radius of the hemisphere in
front of a fish’s eyes that bounds the limits of its visual range, and
the target length refers to the greatest length of the target that will
present itself to the eye of the fish. The simplest assumption for
individual fish is that l1 equals the typical body length for a singleton, and for schools that l2+ equals the school diameter (Table 2).
Although Cisco could encounter each other head on and be
exposed to a much smaller target, their sinuous and intermittent
swimming patterns (Arrhenius et al. 2000; Cech and Kubecka
2002), combined with a broad and hemispheric visual range (Link
1998), likely allow them to observe the full length of the target in
the great majority of interactions. School volumes were calculated
by multiplying the total number of fish in the school (n) by an individual fish volume, which included a buffer region of half a body
length all around 1 fish. This meets the convention that fish in
schools position themselves with respect to their neighbors by about
1 body length (Partridge et al. 1980; Misund 1993; Viscido et al.
2005), regardless of the exact architecture of the Cisco schools
which is not known. Thus, a school radius (r in m) for n fish is
defined under the dimensions of the oblate spheroid,
r = 3 0.0038 *
n
(10)
π
School formation begins shortly after the onset of dawn nautical twilight (i.e., when ambient light at the surface rises above 2
de Kerckhove et al. • IGM describes movement of fish and fish schools
Table 2
IGM parameters used in this study
IGM parameters
Values
Study zone volume (m3)
Length of Cisco (m)
Mass of Cisco (kg)
Volume of 1 Cisco with buffer (m3)
Mean speed of Cisco (m/s)
Mean speed of Cisco school (m/s)
100 000 000
0.1
0.02
0.00226
0.24
0.3
lux), and therefore school formation will begin at a suboptimum
light intensity for Cisco eye physiology and then surpass a saturation threshold as the sun rises (Aksnes and Giske 1993; Holbrook
et al. 2013). To model the sun rise and fall, ambient light levels
were collected from the surface (Φ) using a Lux Light Meter (PMA,
Solar Light Co.) every 10 min starting at 4:00 AM for 13, 14, 19,
and 24 August 2009. The pattern of sunrise was the same across
the 4 days but cloud cover affected the maximum light intensity at
solar noon (i.e., peak sun height in the sky). A Generalized Additive
Model with Gaussian errors (P < 0.001) was used to model the light
intensity at different times of the day (T in seconds). Each day of
ambient light monitoring, a depth profile was taken from 1 to 12 m
to calculate a mean light extinction coefficient of 0.73/m (Crawley
2007). From the output of the generalized additive model and the
light extinction coefficient, we could estimate the light intensity at
any time of the day at a 9-m depth (ψ), which is the mid-point of
the depth range of our study:
ψ = Φe −0.73* 9(11)
Cisco reaction distances do not change beyond 40 lux (Link 1998)
and so 40 lux is assumed to be a reasonable saturation threshold.
To address this phenomenon, the detection distance was modelled
as a Michaelis–Menten function (d) dependent on light levels at 9 m
depth (ψ), the maximum detection distance (a) and a half-saturation
constant (α). The half-saturation constant was given a value of 5
to ensure visual acuity would increase from 0 to approximately the
maximum allometric acuity between 1 and 40 lux (as in Holbrook
et al. 2013):
d=
a*ψ
(12)
α+ψ
Importantly, we do not include the school size of the focal individual in calculating our detection distance. Although every additional
school member contributes an additional set of eyes to the group,
we do not have enough information about Cisco school structure
to determine how many individuals have unobstructed views or
whether individuals on the periphery can influence the direction
of travel for the entire school (see Viscido et al. 2005, Soria et al.
2007). And so we assume that the leading fish at the tip of the
sphere directs the group, and thus that schools have the same visual
acuity as an individual. However, even if we assumed that all the
Cisco along the perimeter of the school could detect targets and
influence school swimming directions, we calculated that this would
confer an increase of 10% of the visual field for very large schools
(i.e., 200 individuals) but much less for small schools. Further, running the model with this assumption did not greatly influence the
rate of school formation.
As depicted in Figure 1a, the X, Y, and Z vectors describe the volume surveyed by a focal individual for 1 angle of encounter. These
1241
calculations must be completed for every possible angle, and then
integrated with the probability of these angles of encounter occurring in Lake Opeongo. The frequency of bearings of movement for
Cisco and all Cisco schools observed in Lake Opeongo were used
to estimate the probabilities of a range of bearings around vertical (0) to horizontal (π/2) movement. We assumed movement to be
symmetric between ascending and descending depths, and across
all cardinal directions. This simplification allows for the probability of all potential angles of encounter to be calculated by adding
the angles and multiplying the probabilities of the focal individual’s
movement with the target’s (e.g., 2 entities belonging to a population whose members move randomly at a bearing of 0 or π/2
would have an angle of encounter of 0, π/2, and π with probabilities of 0.25, 0.5, and 0.25, respectively). From our field measurements of Cisco and Cisco school movements, we fitted normal
probability distributions to the frequencies of bearings of movement around a set mean of π/2, resulting in standard deviations
(SDs) of 0.249 for Cisco and 0.082 for schools. These probability
distributions describe mainly horizontal movements, and thus most
of the encounters in the model are head-on. With a mean of π, the
SD for all angles of encounter for Cisco–Cisco, Cisco–School, and
School–School interactions were modelled as 0.52, 0.365, and 0.15,
respectively. As these probability distributions only describe half of
the interactions (0 to π and not π to 2π), we multiply it by 2 within
the integral.
In this study, we tallied the density of every school sizes between
2 and 199 individuals. Any schools larger were tallied as generically
“large” schools with a volume corresponding to 200 individuals.
Limiting school volumes to 200 individuals provides an upper limit
to the detection distance of schools by Cisco. Under the allometric
relationship there is no size-independent detection distance limit,
which is unrealistic because even large objects will not be visible at
large distances due to the extinction coefficient of water. The maximum detection distance for a school of 200 individuals is assumed
to be 2.4 m, which is similar to that found for Lake Trout foraging
on Cisco schools in Lake Opeongo (Dunlop et al. 2010) who are
expected to have roughly similar visual acuity.
Using the equations described above, the encounter rate more
than 1 min can be calculated for every focal entity (i.e., singletons
and 199 size classes of schools) as long as the time of day and the
abundance of each group of entities is known (see Supplementary
Data for full equations). The encounter rate is calculated for every
focal entity with every other entity of equal or larger size to ensure
that encounters are not counted twice. Starting at dusk, the encounter rates are used to adjust the abundances of each entity following
a minute of movement, for up to 2 h.
IGM test #1: school formation
The emergent properties of school formation were tested against
expectations from our IGM by comparing observed changes in
overall densities of Cisco entities (i.e., single fish or schools) and
schools alone, subsequent to dawn nautical twilight, with IGM
simulated density changes. This was achieved for the Cisco entities
using ordinary least squares linear regression between the average
maximum and minimum plateau of densities over the first 2 h, and
for the schools the model predictions were compared with a Local
Polynomial Regression Fitting algorithm of survey data (i.e., loess
function in R with an α of 0.75; Crawley 2007). The visual acuity of the Cisco is likely the weakest parameter in the IGM model
because of the lack of targeted field experiments in Lake Opeongo
Behavioral Ecology
1242
respectively, and T is the length of the survey in seconds. In an
acoustic survey, the IGM detection term has been replaced with a
more general detection area parameter estimated directly from the
beam geometry and the diameter of the average school. The average school length identified in the 2010 mobile surveys was 3.6 m;
given the ideal beam geometry between 5 and 10 m, this translates
to an area of 47 m2. For the fixed-platform acoustic beam, the relative velocity is determined by the average school swimming speed,
and the detection area is larger because the acoustic beam is set
at an angle of 10° (i.e., A is 140 m2). The variance in daily densities was reported over the 11 day fixed-platform surveys. To determine the variance in the density estimate for the mobile survey, the
EDSUs were randomly sampled with replacement up to the number required for the survey (N = 230), for 11 different iterations.
measuring actual detection distances (Aksnes and Giske 1993). We
conducted sensitivity analyses of the visual acuity value by varying
the allometric model (McGill and Mittelbach 2006) by 50%, 75%,
90%, 110%, 125%, and 150% from the model’s parameter, and
comparing these results with the observed changes in overall Cisco
entities.
IGM test #2: encounter rates
Test #1 is mainly qualitative because there were only 2 dawn surveys with which to compare the model predictions. However, we
were able to quantitatively test the movement patterns of fish
schools under the model while excluding the Cisco’s visual acuity
parameter by following a methodological concept from Turesson
and Brönmark (2007). This uses the model to generate 2 estimates
of school density: 1 based on the school encounter rates observed
in the mobile acoustic surveys and the other based on the school
encounter rates observed in the stationary acoustic surveys. In
this context, the beam geometries are well known, and, within
the IGM equations, they represent the visual acuity parameter.
The remaining parameters are simply the survey vessel and school
swimming speeds. As the vessel and fixed platform survey information is known, and both act as ideal gas particles (i.e., moving
in straight lines at a constant speed, or not at all), the estimated
densities depend on the movement patterns of the schools only.
If the schools also act approximately like ideal gas particles, then
both surveys should generate similar density estimates; a large and
consistent discrepancy between these estimates would suggest that
school movement patterns consistently deviate from that assumed
by the IGM.
A 4 h mobile fisheries acoustic survey was conducted on 23rd
September 2010, followed by a set of fixed-platform surveys conducted over the same time of day between 14 and 24 October
2010. School densities (ρ2+) were estimated from the mobile and
fixed-platform survey data using the following simplified IGM
equation:
ρ2+ =
RESULTS
IGM assumption test #1: swimming speed
estimates
The Cisco singleton and school swimming speed distributions were
unimodal (Figure 2) but significantly different from normal according to the Shapiro–Wilk test. However, the mean swimming speeds
were similar at 0.24 m/s for singletons (n = 102), and 0.3 m/s for
schools (n = 160), although singletons had a higher modal speed
(about 0.2 vs. 0.1 m/s) and a much lower proportion of observations in the lowest velocity class. Further, the amount of variation
around the mean was lower for singletons then for schools. Faster
swimming speeds in schools were often associated with an observed
predation event (i.e., interaction with a large single fish leading to
predator-evasion behaviors). There did not appear to be any difference in speed depending on the time of day for schools, or between
night and twilight for singletons.
IGM assumption test #2: independence of
particles
The nearest neighbor (NN) analysis indicated that singletons and schools are generally distributed in random (NN
index = 1) and slightly clumped patterns (NN index < 1),
respectively. Singleton NN indices ranged from 0.87, 0.9, and
0.92 over 3 surveys. Although these NN values are significantly
different from the value of 1.0 expected given a random distribution, they are very close to 1.0 and the power of the test was
ε
(13)
v AT
where ε is the number of encounters between the acoustic beam
and Cisco schools over the survey period, v is the vessel speed for
mobile surveys and school swimming speeds for fixed-platform
surveys (m/s), A is the detection area (m2) of the acoustic beam,
60
(b)
0
0
10
10
20
Count
30
20
Count
40
30
50
(a)
0.1
0.2
0.3
0.4
Swimming Speed (m/s)
0.5
0.0
0.2
0.4
0.6
Swimming Speed (m/s)
0.8
Figure 2
Swimming speeds of (a) Cisco and (b) Cisco Schools measured with fixed-platform acoustics in Lake Opeongo in September and October 2010.
de Kerckhove et al. • IGM describes movement of fish and fish schools
1243
very high due to the large sample size (i.e., roughly 4200 individuals per test). Examining the results of the kriging analysis
(Figure 3a), we note that the distribution of singleton densities
is relatively even across the main lake basins. School NN values
ranged from 0.69, 0.79, and 1.02 over 3 surveys although only
the lowest estimate was considered significantly different from
expectation under randomized transects. For the schools there
were much smaller sample sizes of about 60 schools per test
(Figure 3b).
the location of the fixed-platform transducer. Schools also generally move alongshore; however, more schools moved East over
our study period and appeared to gradually reduce their depth.
Alongshore movement for both singletons and schools can be
represented by normal distributions where the tails represent
angles of deviation from the mean heading (data not shown).
Singletons generally exhibited lower levels of tortuous movement
(Figure 5) indicating for the most part that swimming occurred in
straight lines.
IGM assumption test #3: direction of movement
IGM test #1: school formation
Singletons and schools moved most often along the horizontal
plane with little change in depth (Figure 4). Further, on- or offshore movements were rare. These patterns suggest that singletons generally move along shore in a particular depth range at
The IGM model predicted a rate of school formation that corresponded well with changes in Cisco entity (i.e., fish and schools)
densities over time (Figure 6a). The functional form of the model
was a sigmoidal decreasing curve, however, the model predicts that
(a)
(b)
Density (m–3)
0.001-0.005
0.005-0.01
0.001-0.05
0.05-0.1
Figure 3
Spatial distributions of (a) individual Cisco densities at night and (b) Cisco schools in the day across the South Arm of Lake Opeongo in July 2009. Note that
1) the map only depicts the part of the lake that is deeper than 5 m, 2) the individual Cisco densities are from 1 survey, and 3) the Cisco school distributions
are from 2 surveys (dots of different color represent schools from separate surveys) and have been offset slightly for improved clarity.
(a)
(b)
(c)
–90º
0º
–90º
90º –90º
90º
90º
0º
0º
0º
270º
270º
270º
90º
90º
90º
180º
180º
180º
Figure 4
Swimming directions of (a) individual and (b) Cisco schools in Lake Opeongo in October 2010, in the 3-dimensional space formed by the coordinates x (East
to West), y (Shallow to Deep), and z (North to South ~ Offshore to Onshore). The upper panels represent the frequency of fish/school movement in the z–x
coordinate plane where the degrees characterizes movement that is towards shore (−90°), alongshore (0°) and offshore (90°). The lower panels represent the
frequency of fish/schools in the x–y coordinate plane where the degrees characterizes movement that is toward the surface (0°), alongshore in the Western
direction (90°), toward the lake bottom (180°), and alongshore in the Eastern direction (270°). (c) The headings are taken relative to the acoustic beam
depicted here for visual reference emanating from the transducer.
Behavioral Ecology
1244
40
Count of Cisco Tracks
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
>10
Tortuosity
Figure 5
Frequency distribution of the tortuosity index for dawn movements of individual Cisco (straight-line movements are valued at 1.0, and the value of the index
increases as the movement becomes more convoluted).
(a)
(b)
0.012
Density of Cisco School (m–3)
Density of Cisco Entities (m–3)
0.05
0.04
0.03
0.02
0.01
0
0
20
40
60
80
Time (minutes)
100
120
0.01
0.008
0.006
0.004
0.002
0
0
20
40
60
80
Time (minutes)
100
120
Figure 6
The change in the density of (a) Cisco entities (i.e., single fish or schools of any size) or (b) only Cisco schools following the onset of nautical twilight on Lake
Opeongo for 2 mobile surveys (open and filled circles) and the IGM (solid line). Additional lines are (a) the sensitivity analysis of the model where visual
acuity is reduced (dotted line) or increased (long dashes) by 50%, and (b) the local polynomial regression fits for the closed (dotted line) and open (dashed line)
data points.
the decline in entities is close to linear (see Figure 6a; between 40
and 60 min). For this period, the decelerating decrease in the number of singletons is combined with a decelerating increase in the
number of schools, making the predicted relationship nearly linear.
The fitted slope of a linear approximation to the model over this
time frame was steeper than those for the fitted linear relationships
for the 2 survey days (−0.00145 vs. −0.00079 and −0.00055). In
our sensitivity analysis, we found that a 50% decrease in visual acuity result in a slope much closer to the survey data at −0.0008, indicating that the parameterization of the IGM may be missing an
important factor.
A bell-shaped relationship between time and school density is
expected from the model. The 2 sets of survey data were generally
too variable to clearly show the initial acceleration in the number
of schools over the early part of the morning (Figure 6b), but the
later deceleration is apparent as the school density becomes low
at around 80 min. The IGM was able to characterize the changes
in school density and fell within the range of the observed acoustic data but was generally lower and narrower than the smoothed
curves on both survey days.
IGM test #2: encounter rates
The IGM model applied to school encounter rates for 1 mobile
acoustic survey and 11 days of fixed-platform acoustic surveys predicted equivalent school densities between surveys (Figure 7). The
mean school density predicted from the mobile survey was within 1
SD of the mean school density observed over the 11 days of fixedplatform surveys.
DISCUSSION
The movements of individual Cisco and Cisco schools appear
to violate some of the core properties of ideal gases, yet an IGM
parameterized based on observations of Cisco behavior in Lake
Opeongo was reasonably good at describing both the fairly rapid
decrease in Cisco singleton density, and the increase and subsequent decrease in school density, from 40 to 80 min following nautical twilight. However, the fit was not ideal, suggesting that there
may have been an important factor to school formation that was
not properly addressed by the IGM. Although it is possible that the
de Kerckhove et al. • IGM describes movement of fish and fish schools
0.00006
Density (m-3)
0.00005
0.00004
0.00003
0.00002
0.00001
0
Mobile
Fixed-Platform
Figure 7
Two estimates of Cisco school density calculated using IGM equations
from school encounter data from a mobile (filled), and fixed-platform
(open) acoustic surveys. The error bars represent ± 1 SD: error bars were
calculated for the single mobile survey by randomly sampling EDSUs with
replacement to generate 11 different randomized survey results; error bars
for the fixed-platform data represent the between-day variation over the 11
survey days.
violations of the core properties of ideal gas movement may be the
missing factor, we found that decreasing the visual acuity parameter
by 50% conferred a better fit in our model output. Further, controlling the visual acuity parameter by substituting the known beam
geometry of mobile and fixed-platform acoustic surveys allowed
us to successfully predict Cisco school densities in the pelagic zone.
Both these findings would suggest instead that the missing factor
is related to our parameterization of visual acuity, and that on the
whole, Cisco and Cisco schools indeed move much like ideal gas
particles. To explore this conclusion, we examine each of the violations as well as the visual acuity parameterization below.
First, the frequency of singleton and school swimming speeds did
not follow a Maxwell–Boltzman distribution, yet we modelled them
as a constant speed (i.e., the mean speed found for Cisco and Cisco
schools in Lake Opeongo). However, as mentioned earlier, our
model formulation allowed us to compare using their actual distributions with constant mean speeds (Hutchinson and Waser 2007)
and we found very little difference in model output. This suggests
that IGM expectations are fairly robust against slight deviations
from Maxwell–Boltzman distribution of speeds, and because the
observed distribution of Cisco and Cisco schools speeds were both
approximately bell shaped, they were likely not complex enough to
significantly violate this particular property of ideal gases.
Second, the spatial distributions of singletons and schools were
either random or only slightly clumped, so for the most part the
random distribution assumption was met in our system. However,
we have observed over the course of these surveys that under
high winds the distributions of schools can become dramatically
clumped, and so under certain conditions this assumption could be
violated and lead to large differences in the observed rates of school
formation (as found in Soria et al. 2009).
Third, school movement directions were clearly biased to eastern
alongshore movement at the location of the fixed-platform transducer, yet we modelled the bearing of movement as being equally
likely across all cardinal directions. Notwithstanding our finding, it
is less likely that any bias is found in the middle of the lake basin
1245
or else 1) school distributions would be clumped along the Eastern
shoreline, and 2) schools would generally be caught on the western side of gill nets (a bias not observed in Lake Opeongo gill net
catches). Further, if this bias was indeed significant the Cisco school
encounter rates at the fixed-platform station would have led to an
underestimate the true school density in the pelagic zone, which we
did not find. Thus, we can be fairly confident that Cisco schools
generally moved equally among the cardinal directions and that
this violation did not bias the IGM results.
Last, singletons were found to move in fairly straight lines within
the acoustic beams, and coordinated movements of individuals in
groups suggest that straight line movement for schools will also be
common (Viscido et al. 2005; Hemelrijk and Hildenbrandt 2012).
Regardless, this assumption is expected to have very little influence
on the IGM results because tortuous paths are primarily a concern if multiple encounters with the same individual are expected
to occur (Waser and Hutchinson 2007). In our model, repeated
encounters would not occur because the first encounter always
results in a union of the 2 Cisco entities.
The results from converting school encounter rates to densities
between mobile and fixed-platform acoustic surveys also suggest
that the movement of Cisco schools generally adhered to IGM
expectations. There was large variation in observed encounter rates
with Cisco schools over the 10 days of the fixed-platform surveys—
the number of encounters over a 4-h period ranged from 6 to 39
schools (mean = 20, SD = 10). However, even under this large variation the mean density predicted from the fixed-platform survey
was close to that of the mobile survey (i.e., 0.000034 vs. 0.000046
schools/m3). Further, while not included in the results, a second
mobile transect survey (this 1 shorter and directly across the main
basin), was carried out concurrent with the fixed-platform data
collection on 6 October 2010 between 5:30 and 6:30 PM during
windy weather. This brief test run resulted in the same though generally higher predicted densities in both survey types (i.e., 0.0001
schools/m3 for both fixed-platform and mobile surveys). Thus, natural variation in school distributions caused by weather differences
may explain some of the variation in the mean encounter rates
over our study period, however, when detection rates (analogous
to visual acuity) are known directly (i.e., through the known beam
geometries of the fixed-platform and mobile transducers), resultant
abundance estimates are consistent with Cisco movements following IGM assumptions.
The movement parameters on the whole did not deviate strongly
from the ideal gas assumptions, yet our IGM of school formation
rates did not result in a perfect fit among modelled and observed
data. Through the discussion above and our sensitivity analysis,
it would appear that the Cisco’s visual acuity may not have been
well parameterized, leading to an overestimate of encounter rates
between Cisco entities. General improvements to the visual acuity
model could include obtaining Cisco-specific estimates of detection
distance (as in Mazur and Beauchamp 2006) rather than relying
on an allometric model developed for another species (McGill and
Mittelbach 2006), or using a physiological approach to limit the
maximum distance the Cisco can see through water (as in Aksnes
and Utne 1997) rather than set maximum school sizes based on
field data (Dunlop et al. 2010). However, obtaining an exact fit
with the field data is going to be challenging as there are likely
many other behaviors and physiological aspects that have not been
addressed under this framework. Instead, the ability of IGM model
to provide a reasonable fit is encouraging for conducting future
work on Cisco behavior and ecology.
Behavioral Ecology
1246
Our results suggest some interesting hypotheses regarding how
the movement patterns of singletons and schools interact and are
affected by environmental conditions. The low variation in swimming speeds for singletons is consistent with the low variability in
body size we observed within age classes. In contrast, the higher
variability found for schools suggests that swimming speeds are a
behavioral choice based on external conditions (Hoare et al. 2004).
The fastest school swimming speeds were 1.7× faster than the
maximum speeds observed for singletons and followed predation
attempts, suggesting that burst swimming may be reserved for predation evasion maneuvers (Nøttestad and Axelsen 1999). Further,
under exploratory analyses no relationships were found between
school sizes or school packing density (measured through volume
backscattering) and swimming speed. This suggests that schools
may not be sensitive to size effects on the efficiency of movement,
and can remain organized as they get larger (Partridge and Pitcher
1979; Viscido et al. 2005). Instead, slower swimming speeds may
be a behavioral choice to improve foraging efficiency (Milne et al.
2005) as well as reduce encounter rates with predators (Hutchinson
and Waser 2007). Finally, a low tortuosity of movement at dawn is
compelling because it suggests that singletons are swimming with a
purpose (Dicke and Burroughs 1988). As a comparison, we characterized the evening movement of 108 Cisco just as schools were
breaking up and found significantly higher levels of tortuosity (t-test
between samples with unequal variance, t208 = 1.97, P < 0.05, data
not shown).
From this study, we conclude that Cisco singleton and school
movements appear to generally approximate the assumptions and
the expectations of ideal gas movement (Maxwell 1860). Cisco are
the primary prey of large piscivores and so it seems appropriate to
characterize these trophic relationships using predator–prey models
based on mass action encounter rates with Cisco entities. Analytical
models involving predation on fish schools are rare (Cosner et al.
1999; Ioannou et al. 2011; Geritz and Gyllenberg 2013) and often
have trouble incorporating realistic modes of prey aggregation as
population sizes changes. The IGM provides a direct link between
changes in prey density and school formation, and with a further
extension could also provide an estimate of predator–prey encounter rates. Further, as Cisco schools are predators themselves on very
small and slow, but relatively dense swarms of plankton prey, their
movement may be driven by strategies that optimize those encounter rates (Hutchinson and Waser 2007; Gurarie and Ovaskainen
2013). If so, we might expect Cisco movement patterns to be more
closely related to the movement and distributions of their prey than
their predators.
SUPPLEMENTARY MATERIAL
Supplementary material can be found at http://www.beheco.
oxfordjournals.org/
We wish to thank Kongsberg Maritime, Milne Technologies Inc. and the
Ontario Ministry of Natural Resources (OMNR) Harkness Fisheries
Laboratory for the loan of acoustic equipment. We thank M. Ridgway and
T. Middel for their many contributions to this research, and we also thank
L. Rudstam, M.-J. Fortin, N. Lester, and H. Cyr for helpful comments on
the methodology. We thank the Natural Science and Engineering Research
Council of Canada for financial support of PAA, BJS, and DTK. We also
thank the OMNR for research funding to BJS. Last we thank 2 anonymous
reviewers and Dr S. Nakagawa for their constructive critique and suggestions on improving the manuscript.
Handling editor: Shinichi Nakagawa
REFERENCES
Abrahams MV, Colgan PW. 1985. Risk of predation, hydrodynamic efficiency and their influence on school structure. Environ Biol Fishes.
13:195–202.
Aksnes D, Giske J. 1993. A theoretical model of aquatic visual feeding. Ecol
Modell. 67:233–250.
Aksnes D, Utne A. 1997. A revised model of visual range in fish. Sarsia.
82:137–147.
Arrhenius F, Bennedheij BJAM, Rudstam LG, Boisclair D. 2000. Can stationary bottom split-beam hydroacoustics be used to measure fish swimming speed in situ? Fish Res. 45:31–41.
Cech M, Kubecka J. 2002. Sinusoidal cycling swimming pattern of reservoir fishes. J Fish Biol. 61:456–471.
Cosner C, DeAngelis DL, Ault JS, Olson DB. 1999. Effects of spatial
grouping on the functional response of predators. Theor Popul Biol.
56:65–75.
Crawley MJ. 2007. The R book. West Sussex (UK): John Wiley and Sons
Ltd.
Dicke M, Burroughs PA. 1988. Using fractal dimensions for characterizing
tortuosity of animal trails. Physiol Entomol. 13:393–398.
Dunlop ES, Milne SW, Ridgway MS, Condiotty J, Higginbottom I. 2010.
In Situ swimming behavior of Lake Trout observed using integrated multibeam acoustics and biotelemetry. Trans Am Fish Soc. 139:420–432.
Evans GT. 1989. The encounter speed of moving predator and prey. J
Plankton Res. 11:415–417.
Frouzova J, Kubecka J, Balk H, Frouz J. 2005. Target strength of some
European fish species and its dependence on fish body parameters. Fish
Res. 75:86–96.
Geritz SA, Gyllenberg M. 2013. Group defence and the predator’s functional response. J Math Biol. 66:705–717.
Gerritsen J, Strickler JR. 1977. Encounter probabilities and community structure in zooplankton: a mathematical model. J Fish Res Board
Canada. 34:73–82.
Gurarie E, Ovaskainen O. 2013. Towards a general formalization of
encounter rates in ecology. Theor Ecol. 6:189–202.
Hemelrijk CK, Hildenbrandt H. 2012. Schools of fish and flocks of birds:
their shape and internal structure by self-organization. Interface Focus.
2:726–737.
Hemmings C. 1966. Olfaction and vision in fish schooling. J Exp Biol.
45:449–464.
Hoare D, Couzin I, Godin J-G, Krause J. 2004. Context-dependent group
size choice in fish. Anim Behav. 67:155–164.
Holbrook BV, Hrabik TR, Branstrator DK, Mensinger AF. 2013. Foraging
mechanisms of age-0 lake trout (Salvelinus namaycush). J Great Lakes Res.
39:128–137.
Hughes BD. 1995. Random walks and random environments. Oxford:
Clarendon Press.
Hutchinson JMC, Waser PM. 2007. Use, misuse and extensions of “ideal
gas” models of animal encounter. Biol Rev. 82:335–359.
Ioannou CC, Bartumeus F, Krause J, Ruxton GD. 2011. Unified effects of
aggregation reveal larger prey groups take longer to find. Proc Biol Sci.
278:2985–2990.
Jonsen ID, Myers RA, Flemming JM. 2003. Meta-analysis of animal movement using state-space models. Wildl Res. 21:149–161.
de Kerckhove D, McLaughlin RL, Noakes DL. 2006. Ecological mechanisms favouring behavioural diversification in the absence of morphological diversification: a theoretical examination using brook charr (Salvelinus
fontinalis). J Anim Ecol. 75:506–517.
Link J. 1998. Dynamics of lake herring (Coregonus artedi) reactive volume for
different crustacean zooplankton. Hydrobiologica. 368:101–110.
Maxwell JC. 1860. Illustrations of the dynamical theory of gases. Part
1. On the motions and collisions of perfectly elastic spheres. Philos Mag.
19:19–32.
Mazur MM, Beauchamp DA. 2006. Linking piscivory to spatial-temporal
distributions of pelagic prey fishes with a visual foraging model. J Fish
Biol. 69:151–175.
McGill BJ, Mittelbach GG. 2006. An allometric vision and motion model to
predict prey encounter rates. Evol Ecol Res. 8:691–701.
Milne S, Shuter B, Sprules W. 2005. The schooling and foraging ecology of
lake herring (Coregonus artedi) in Lake Opeongo, Ontario, Canada. Can J
Fish Aquat Sci. 62:1210–1218.
Misund O. 1993. Dynamics of moving masses: variability in packing density, shape, and size among herring, sprat, and saithe schools. ICES J Mar
Sci. 50:145–160.
de Kerckhove et al. • IGM describes movement of fish and fish schools
Mulligan TJ, Chen DG. 2000. Comment on “Can stationary bottom splitbeam hydroacoustics be used to measure fish swimming speed in situ?”
by Arrhenius et al. Fish Res. 49:93–96.
Muneepeerakul R, Bertuzzo E, Lynch HJ, Fagan WF, Rinaldo A,
Rodriguez-Iturbe I. 2008. Neutral metacommunity models predict fish
diversity patterns in Mississippi-Missouri basin. Nature. 453:220–222.
Nøttestad L, Aksland M, Beltestad A, Ferno A, Johannessen A, Misund
O. 1996. Schooling dynamics of Norwegian spring spawning herring
(Clupea harengus L.) in a coastal spawning area. Sarsia. 80:277–284.
Nøttestad L, Axelsen BE. 1999. Herring schooling manoeuvres in response
to killer whale attacks. Can J Zool. 77:1540–1546.
Okubo A, Levin S. 2001. Diffusion and ecological problems: modern perspectives. New York: Springer Verlag.
Parker-Stetter SL, Rudstam LG, Sullivan PJ, Warner DM. 2009. Standard
operating procedures for fisheries acoustic surveys in the Great Lakes.
Great Lakes Fishery Commission Special Publication #09-01. Ann
Arbor (MI).
Parrish JK, Edelstein-Keshet L. 1999. Complexity, and pattern, in animal
evolutionary aggregation. Science. 284:99–101.
1247
Partridge BL, Pitcher TJ. 1979. Evidence against a hydrodynamic function
for fish schools. Nature. 279:418–419.
Partridge BL, Pitcher T, Cullen JM, Wilson J. 1980. The three-dimensional
structure of fish schools. Behav Ecol Sociobiol. 6:277–288.
Radinger J, Wolter C. 2013. Patterns and predictors of fish dispersal in rivers. Fish Fish. 15: 456–473.
Soria M, Dagorn L, Potin G, Fréon P. 2009. First field-based experiment
supporting the meeting point hypothesis for schooling in pelagic fish.
Anim Behav. 78:1441–1446.
Soria M, Freon P, Chabanet P. 2007. Schooling properties of an obligate
and a facultative fish species. J Fish Biol. 71:1257–1269.
Swartzman G. 1991. Fish school formation and maintenance: a random
encounter model. Ecol Modell. 56:63–80.
Turesson H, Brönmark C. 2007. Predator-prey encounter rates in freshwater piscivores: effects of prey density and water transparency. Oecologia.
153:281–290.
Viscido SV, Parrish JK, Grünbaum D. 2005. The effect of population size
and number of influential neighbors on the emergent properties of fish
schools. Ecol Modell. 183:347–363.