## SUMMARY

Migrating fish traversing velocity barriers are often forced to swim at
speeds greater than their maximum sustained speed (*U*_{ms}).
Failure to select an appropriate swim speed under these conditions can prevent
fish from successfully negotiating otherwise passable barriers. I propose a
new model of a distance-maximizing strategy for fishes traversing velocity
barriers, derived from the relationships between swim speed and fatigue time
in both prolonged and sprint modes. The model predicts that fish will maximize
traversed distance by swimming at a constant groundspeed against a range of
flow velocities, and this groundspeed is equal to the negative inverse of the
slope of the swim speed–fatigue time relationship for each mode. At a
predictable flow velocity, they should switch from the optimal groundspeed for
prolonged mode to that for sprint mode. Data from six migratory fish species
(anadromous clupeids: American shad *Alosa sapidissima,* alewife *A.
pseudoharengus* and blueback herring *A. aestivalis*; amphidromous:
striped bass *Morone saxatilis*; and potomodromous species: walleye
(previously known as *Stizostedion vitrium*) and white sucker
*Catostomus commersonii*) were used to explore the ability of fish to
approximate the predicted distance-maximizing behaviors, as well as the
consequences of deviating from the optima. Fish volitionally sprinted up an
open-channel flume against fixed flow velocities of 1.5–4.5 m
s^{-1}, providing data on swim speeds and fatigue times, as well as
their groundspeeds. Only anadromous clupeids selected the appropriate
distance-maximizing groundspeed at both prolonged and sprint modes. The other
three species maintained groundspeeds appropriate to the prolonged mode, even
when they should have switched to the sprint optima. Because of this, these
species failed to maximize distance of ascent. The observed behavioral
variability has important implications both for distributional limits and
fishway design.

## Introduction

Diadromous and other migratory riverine species often encounter zones of
high-velocity flow that impede their migrations. Where these flows exceed
maximum sustained swim speed (*U*_{ms}), successful passage may
still be possible, provided that fish select an appropriate swim speed. The
focus of this paper is to identify optimal swim speeds for traversing such
velocity barriers, to test whether such strategies are employed, and to
explore the consequences of failure to optimize, when it occurs.

Because of the fitness consequences of swimming performance, locomotor
behavior is a good candidate for optimization. There is disagreement, however,
on what constitutes optimal behavior. Most work in this area has applied
hydraulic equations to generate predictions of optimizing behaviors (Weihs,
1973,
1977;
Webb, 1993;
Videler, 1993). These authors
used equations based on the combined energetic costs of basal metabolic rate
and drag on swimming fish, predicted from hydraulic principles, to create
cost-of-transport models that yield predictions of optimal swim speed; Weihs
(1974) and Videler and Weihs
(1982) further described how
burst-and-coast swimming can afford energetic advantages. Trump and Leggett
(1980) used a different
approach, calculating the metabolic cost of transport directly from empirical
equations derived from respirometry data. Both models generated similar
predictions, namely that fish could maximize energetic efficiency by swimming
at speeds corresponding to about one body length per second (*BL*
s^{-1}). Data supporting these predictions are sparse, due to the
difficulties of monitoring swimming fish in their native habitat, and a review
of the literature by Bernatchez and Dodson
(1987) found that such
optimizing behavior is rare, characterizing only certain populations with long
migrations.

Both the hydrodynamic and metabolic cost-of-transport models require the assumption of ready availability of energy. While this may be reasonable for aquatic animals swimming at sustained speeds, it does not hold at faster speeds. At these speeds, contributions of anaerobic metabolism to power production create limits to endurance, with energy supplied increasingly from stores contained within muscle fibers, and insufficient time and circulation to remove metabolic waste products (Brett, 1964).

Because of this, high-speed swimming is not associated with continuous
behaviors like filter feeding or migration through lentic environments (e.g.
Ware, 1975); instead it is
associated with short-term, fitness-critical behaviors, such as capture of
mobile prey, predator avoidance, and traversing velocity barriers during
migrations. For these behaviors, the trade-off between swim speed and fatigue
time may define performance. This relationship has been well studied at
prolonged speeds (Beamish,
1978; Videler,
1993; Webb, 1994),
and is generally thought to follow a log-linear model:
1
where *T* is fatigue time, *U*_{s} is relative swim
speed (in *BL* s^{-1}), and *a* and *b* are the
intercept and slope coefficients, respectively; *b<*0. Given this
swim speed–fatigue time relationship, I propose the following new models
to predict distance-maximizing behaviors.

The maximum distance a fish can swim (*D*_{s}) can be
described as *U*_{s} × *T*, or, from
Equation 1:
2
This is a nonlinear function, with a clear distance-maximum, dependent on the
values of coefficients *a* and *b*.
Figure 1 is derived from
Bainbridge (1960)'s data on
rainbow trout (table 10.1 in Videler,
1993; data have been transformed from 10-base logarithms and
minutes to natural logarithms and seconds), which predict that distance can be
maximized by fish swimming at 2.8 *BL* s^{-1}. This, however,
is below the range of speeds for which the coefficients were developed; at the
relevant burst speeds (greater than ∼5 *BL* s^{-1}), a
trade-off exists, and distance is maximized by minimizing speed.

This relationship is more complex for diadromous migrants confronted with
velocity barriers, such as might be found at rapids, culverts or other
constrictions in a river. Because distance through a velocity barrier is often
unknown, and because of the sometimes dire fitness consequences of failing to
traverse the barrier, fish traversing such barriers should pursue a
distance-maximizing strategy. Moreover, the distance that needs to be
maximized is not the distance swum, or through-water distance, but rather the
distance over ground, as this is what defines the boundaries of the velocity
barrier. In the presence of flow, the ground distance (*D*_{g})
attained at fatigue time becomes:
3
where *U*_{g} is ground speed, or the difference between swim
speed and the speed of flow (*U*_{f}). Thus,
4
Here, the resulting response surface shows a clear optimum swim speed
(*U*_{opt}), with the maximum distance over ground dependent on
both flow and swim speeds (Fig.
1B). By taking the first derivative of *D*_{g} with
respect to *U*_{s}, and solving for zero slope, it can be shown
that the optimal swim speed within a given mode is defined by:
5
and thus the optimal groundspeed *U*_{gopt} is:
6
In other words, the distance-maximizing strategy for fish swimming against
flow velocities equal to or greater than *U*_{ms} is to swim at
a constant groundspeed, regardless of *U*_{f}, and equal to the
negative inverse of the slope defined in
Equation 1.

This relationship, however, is not constant across modes. Brett
(1964) first observed, and
numerous subsequent studies have confirmed (see tables and figures in
Beamish, 1978;
Videler, 1993), the existence
of two distinct unsustainable modes of steady swimming: prolonged mode, which
can be maintained for durations of 20 s to 200 min, and sprint mode, which
results in fatigue in less than 20 s. Although a biological explanation for
the existence of these two modes has never been definitively established, the
change is generally thought to be the result of a shift from mixed
contributions of aerobic and anaerobic metabolism and muscle groups in
prolonged mode to almost pure anaerobic metabolism and muscle groups in sprint
mode (Brett, 1964;
Webb, 1975). Regardless of the
underlying cause, the slopes of the swim speed–fatigue time
relationships of these two modes vary in ways that are consistent across taxa,
namely *b* is steeper (larger negative magnitude) in prolonged mode
than in sprinting, and the transition between modes is discrete. This means
that the distance-maximizing swim speed, while still a constant groundspeed
within a particular mode, will show a discrete shift to a higher value as fish
shift from prolonged to sprint mode.

Because of the parameters describing the swim speed–fatigue time
relationship, maximum distance in prolonged mode (*D*_{maxP})
greatly exceeds that in sprint mode (*D*_{maxS}) at low
*U*_{f}. However, as *U*_{f} increases, this
difference declines, and eventually *D*_{maxS} exceeds
*D*_{maxP} (Fig.
2A). This suggests that there exists a critical speed of flow
*U*_{fcrit}, at which a mode shift should occur
(Fig. 2B). At
*U*_{f} values less than this, fish should swim at the optimum
prolonged speed; at greater values, they should swim at the optimum sprint
speed. This critical flow speed can be calculated as the point where the
predicted distance maxima are equal, i.e. from Equations
1,
4 and
6, where
7
Thus,
8
and fish should select prolonged or sprint modes, respectively, (along with
their optimal groundspeed), depending on whether current velocity is greater
or less than *U*_{fcrit}.

The above leads to the following hypotheses: (1) when confronted with
velocity challenges, fish should swim at a constant groundspeed of–
1/*b BL* s^{-1}; (2) selected groundspeed will vary with
mode, being lower at prolonged than at burst speeds; (3) The velocity of flow
at which the shift to the sprint optimum will occur is described by
Equation 8; and (4) to the extent
that fish fail to approximate *U*_{gopt}, the deviation will
reduce maximum distance of ascent. This paper describes tests of these
hypotheses with data from six species, volitionally swimming up a large scale,
open-channel flume.

## Materials and methods

### The flume

Data for this study come from a series of experiments on sprinting fish
using a large-scale hydraulic flume, which is described in detail elsewhere
(Castro-Santos, 2002;
Haro et al., 2004). The flume
apparatus was built at the fish passage facility of the Conte Anadromous Fish
Research Center in Turners Falls, MA, USA, which was designed for such
studies, and consisted of a bulkhead that retained a headpond, supplied with a
maximum of 10 m^{3} s^{-1} of water through pipes leading from
an adjacent power canal. The flume proper was an open channel, 1 m × 1 m
in cross-section, and 23 m long. At the interface of this channel and the
headpond bulkhead was an adjustable control gate (headgate). This gate,
combined with the headpond level and the level of the water in the flume,
served to regulate the flow through the flume. At the downstream end of the
channel was a large staging area (0.75–1.5 m × 3 m × 5 m,
depth × width × length) that, with its greater cross-sectional
area and length provided fish with a low-velocity zone from which to stage
their attempts at swimming up the flume. A variable-crest weir (tailwater
weir) fitted with 2.4 cm^{2} screening installed at the downstream end
served the dual functions of controlling the depth of the water in the staging
area and preventing the fish from escaping downstream. At the interface of the
staging area and the flume channel was a v-shaped gate (exclusion screen) used
to prevent fish from swimming up the flume before beginning a trial, while the
velocities were brought to their desired levels.

Water levels were monitored and recorded every 60 s in the headpond, at
three locations within the flume, and in the staging area. Settings for the
headgate and tailwater weir were likewise monitored. Instantaneous velocity
estimates were generated for each point of a 10 cm grid over the full
cross-section of the flume using a combination of physical modeling and
hydraulic equations, confirmed with direct measurements in the flume
(Fig. 3;
Castro-Santos, 2002;
Haro et al., 2004). Mean
cross-sectional velocities were then controlled to within ±5% of their
average values within each nominal velocity, corresponding to 1.5, 2.5, 3.5
and 4.5 m s^{-1}, the variability arising from fluctuating water
levels, both in the power canal and within the facility
(Table 1).

### Data collection

To avoid using coercive measures to motivate fish to swim, we relied
instead on the innate rheotactic behaviors of six fish species that migrate
annually through rivers of Northeast USA. Collections and testing were
performed between the months of April and July, 1997–1999, on dates
corresponding with periods of upstream migration for each species. Test fish
were captured from traps at nearby fishways (American shad *Alosa
sapidissima* Wilson 1811, striped bass *Morone saxatilis* Walbaum
1792, and white suckers *Catostomus commersonii* Lacepède 1803),
coastal streams (alewife *A. pseudoharengus* Wilson 1811) from the
Herring River, Bourne MA, and blueback herring *A. aestivalis* Mitchill
1814 from the Charles River, Watertown, MA), or electrofished [blueback
herring, striped bass, walleye *Sander vitreus* Mitchill 1818 (formerly
*Stizostedion vitreum* Mitchill 1818) and white sucker] from the
Connecticut River.

Fish were transported to the flume facility in one of two truck-mounted
tanks containing either 1000 or 4000 liters. There they were measured (fork
length, *FL*), sexed, and each was fitted with an externally attached
passive integrated transponder (PIT) tag. This consisted of a
glass-encapsulated, cylindrical tag measuring 32 mm × 3 mm, length×
diameter, fastened to a fishhook, which was inserted through the
cartilage at the base of the dorsal fin (second dorsal, in the case of the
percomorphs; see Castro-Santos et al.,
1996, for a complete description of the PIT tag system). Tagged
fish were released into open, flow-through holding ponds
(Burrows and Chenoweth, 1970),
connected hydraulically to the fish passage complex and held for 24 h before
testing. The linkage between the holding ponds and the flume facility
precluded the need to handle fish the day they were tested. Instead, groups of
20–30 fish were crowded from the holding ponds into the facility at the
start of each trial, and the tailwater weir was raised to confine the fish to
the staging area. Once the velocity of flow was brought to the desired level,
the exclusion screen was opened, and fish were allowed to ascend the flume of
their own volition. Duration of trials ranged from 1 to 6 h.

Progress of individual fish up the flume was monitored both electronically and visually. The flume was electronically graduated with PIT detection antennas wired around the outside every 2.5 m, beginning at 0.5 m from the entrance. Antennas were driven by controllers that charged and read tags as they moved through, and sent the identifying codes back to a central computer at a rate of 14 Hz. The computer logged these codes, recording position to within ± 50 cm and time to the nearest 0.01 s (Castro-Santos et al., 1996; Castro-Santos, 2002).

The flume was also graduated visually, in part to verify the accuracy of
the PIT tag data. Four video cameras (NTSC i.e. a standard video format) were
positioned 5 m above the flume; high-speed video (250–500 frames
s^{-1}) provided additional detailed information on swimming mechanics
during 16 trials. The floor and one wall of the flume were covered with a
retroreflective surface (Scotchlite 6780, 3-M Corp., St Paul, MN, USA) that
was graduated into 50 cm intervals with black crosshatch marks. The other wall
of the flume was made of clear acrylic, 2.5 cm thick. Mirrors, the full height
of the flume and situated at a 45° angle to it, allowed each camera to
monitor dorsal and lateral views of the swimming fish simultaneously, thus
positioning the fish in three dimensions.

The speed at which fish moved up the flume (groundspeed,
*U*_{g}) was measured by calculating the difference in mean
times between pairs of antennas. Maximum distance of ascent
(*D*_{max}) corresponded to the highest recorded reader. Mean
groundspeed was the time between detections at the first antenna and the
*D*_{max} antenna, divided by *D*_{max}. Swim
speed (*U*_{s}) was measured by adding the measured water
velocity (*U*_{f}) to *U*_{g}.

Video was also used to determine if fish were actively seeking out low velocity zones. At least 10 individuals were tracked swimming up the flume from each species-nominal velocity combination. The proportion of time (to the nearest 5%) that each fish spent in each of 15 cross-sectional quadrants was measured using the dual perspectives provided by the camera arrangement. A correction factor (CF) was calculated for each fish by summing the proportion of time spent in each quadrant multiplied by the ratio of the water velocity in that quadrant to the mean cross-sectional velocity (Fig. 3). The resulting values were used to calculate an overall correction factor by which to multiply the mean cross-sectional flume velocity for each species-velocity combination.

### Analysis

The above data were used to calculate swim speed–fatigue time relationships for each species and mode, as described below. The associated equations were then used to evaluate whether the models described in Equations 1, 2, 3, 4, 5, 6, 7, 8 can be used to predict distance-maximizing behavior, whether these species exhibited such behaviors, and if not, how that affected distance of ascent.

The swim speed–fatigue time relationship was calculated by regressing
speed at which the fish swam against the natural log of the time it took for
fish to reach their maximum distance of ascent. Data for this relationship
were pooled across velocities. Often, fish ascended the full length of the
flume; in this case the observation is not of fatigue, but of the failure of
the fish to fatigue. This constitutes censored data, some of the implications
of which are discussed elsewhere (Hosmer
and Lemeshow, 1999;
Castro-Santos and Haro, 2003;
Castro-Santos, 2004): using
methods of survival analysis, the observed data are included, coded for
censoring, and the likelihood of the regression model is maximized with
respect to the probability density function *f*(*T*) for
complete observations, and to the survivorship function *S*(*T*)
for censored observations. This method generates sufficient and consistent
least-biased estimates of the swim speed–fatigue time relationship
(Neter et al., 1985;
Hosmer and Lemeshow,
1999).

The model was then modified to test for evidence of a mode shift between
prolonged and sprint modes (maximum prolonged speed: *U*_{mp}).
A dummy variable and an interaction term were included:
9
such that *x*_{1}=0 for values of *U*_{s} less
than a nominal *U*_{mp} value, and 1 at greater values
(β_{i} are regression coefficients, analogous to *a* and
*b* terms in Equation 1).
This model was run iteratively, incrementing *U*_{mp} by 0.01
*BL* s^{-1}; the log-likelihood values from each iteration were
used to calculate a χ-square likelihood ratio statistic (2 d.f.;
Allison, 1995;
Hosmer and Lemeshow, 1999) to
define a confidence band around the maximum calculated value. I used this
approach to fit exponential, lognormal, Weibull and generalized gamma
distributions, and used likelihood ratios or Akaike's Information Criterion
(AIC), where appropriate, to select the most parsimonious model
(Burnham and Anderson, 1998;
Allison, 1995). The model that
provided the best fit also defined *U*_{mp}. Where significant
shifts in slope were observed (*P*<0.05), data were then divided
into observations greater and less than the estimated *U*_{mp},
and separate coefficients of Equation
1 were generated for each mode (*a*_{P},
*b*_{P,} *a*_{S}, *b*_{S} for
prolonged and sprint modes, respectively); the predicted
*U*_{gopt} from Equation
6 equals the negative inverse of the slope term, i.e.–
*b*_{P}^{-1} or–
*b*_{S}^{-1}.

Evidence for optimizing behavior was assessed by comparing observed
groundspeed with predicted values; *U*_{fcrit} was calculated
from Equation 8 and the
regression models described above. If the optimization model is correct, then
there should be costs associated with deviating from *U*_{opt};
specifically, fish that swim faster or slower than *U*_{gopt}
should swim less far. Alternatively, if fish select swim speeds that differ
from the predicted optimum, but that in fact represent true
distance-maximizing optima that this model fails to predict, then deviation
from the mean (`true optimum') will likewise yield reduced distance of ascent.
I tested for each of these conflicting hypotheses by regressing the expected
cost of deviating from the optimum groundspeed, measured as the difference in
distance of ascent predicted at the optimum minus the observed groundspeed
(values are always positive), against *D*_{max}, censoring
where fish arrived at the uppermost reader. These residuals are denoted
*R*_{P} and *R*_{S} for prolonged and sprint
predictions, respectively. To test for optimizing behavior not predicted from
the model, I regressed the absolute value of the residual groundspeed,
, against
*D*_{max}, censoring as above. Significant positive slopes
indicate greater distances achieved by deviating from
*U*_{gopt} and ;
significant negative slopes indicate costs of deviation. Either significant
positive slopes or non-significance support the null hypothesis against the
model; only a significant negative slope supports the alternative hypothesis
suggested by the model, and significance tests are correspondingly
one-tailed.

## Results

The results of these experiments suggest that the models are good
descriptors of distance-maximizing behavior. Actual behaviors were variable,
however, both within and among species. Only the anadromous clupeids
approximated the appropriate distance-maximizing behavior in both prolonged
and sprint modes. Nonclupeids selected appropriate speeds for prolonged mode,
even where *U*_{f}>*U*_{fcrit}; the data
indicate that this represents a failure to optimize, rather than an
alternative distance-maximizing strategy.

### Flume tolerances and behavior

Water velocities (*U*_{f}) deviated from the target
velocities both in time and over the cross-section of the flume. These
deviations are described elsewhere
(Castro-Santos, 2002;
Haro et al., 2004), and are
summarized in Fig. 3 and
Table 1. Although flow was
turbulent at all velocities (Reynold's number >300,000) the turbulence was
disorganized, consisting of random fluctuations and microeddies with no
evident periodicity (Haro et al.,
2004). This means that, with the exception of slightly lower
velocities in the corners (Fig.
3), opportunities for fish to take advantage of hydraulic
structure were minimal. Preferred zones of ascent within the flume varied
among individuals and species, with the following general trends: (1) most
fish tended to swim within 20 cm of the bottom, this effect being least at the
1.5 and 4.5 m s^{-1} conditions; (2) most fish avoided the walls,
generally swimming more than 20 cm from either wall; and (3) white sucker
consistently swam in the corners at 1.5 and 2.5 m s^{-1}, presumably
taking advantage of the lower velocities there, but at higher velocities they
swam closer to the middle of the flume. Correction factors and adjusted mean
velocities are presented in Table
1.

The PIT detection antenna array provided a nearly continuous record of the position of fish within the flume. Read range of the antennas extended 50 cm up- and downstream of each antenna. By taking the mean value of time for each antenna, fish were located in time and space with an accuracy of ±18 cm, i.e. 95% of fish were within 18 cm of the antenna at the time their presence was logged, with no apparent bias in the error.

Once within the flume, fish tended to move steadily up the channel until
they reached *D*_{max}. Swimming behavior was mostly steady,
with burst–coast behavior observed only rarely, and then among large
individuals swimming against the lower velocity flows. Subsequent behavior
also varied with speed of flow. At the fastest flows, fish tended to fall back
passively, oriented either up- or downstream or even lateral to the flow,
maintaining at most enough velocity relative to flow to maintain equilibrium.
At intermediate water velocities, they tended to maintain greater velocity
relative to the flow (again, oriented either up- or downstream), but usually
returning rapidly to the staging area. At the slowest flows, some fish
proceeded to exit the top of the flume, or lingered near the upstream end (see
Fig. 2.5 in Castro-Santos,
2002). Some individuals made multiple ascents; in this case I used
the first ascent where a fish attained its *D*_{max} value in
my analyses. This usually occurred on the first attempt (65–95% of
individuals, by species).

### Swim speed–fatigue time curves

The relationship between swim speed and fatigue time is presented in Fig. 4 and Table 2. Whenever fish ascended to 18 m or above, observations were included as censored, i.e. the fish did not fatigue as of the last observation. This means that the ability to measure fatigue was limited by the constraints of the apparatus. The regression techniques used here, however, account for censored data and generate sufficient and consistent least-biased estimates of the swim speed–fatigue time relationship; uncertainty arising from all sources, including censoring, is reflected in the standard error of the estimates. Since censoring constitutes incomplete observation, and was more prevalent among lower swim speeds (Fig. 4), variance of the estimates are correspondingly greater at prolonged than at burst speeds. The presence of censored data also explains why regression lines in Fig. 4 do not fall in the middle of the data; they are instead adjusted upward to account for those fish that did not fatigue (Allison, 1995; Hosmer and Lemeshow, 1999; Castro-Santos and Haro, 2003).

Discrete prolonged and sprint modes were found with corresponding slopes
and intercepts for American shad, striped bass, walleye and white sucker, but
not for alewife or blueback herring. The locations of these mode shifts
represent the models with the best fit. A 95% confidence interval (based on
the likelihood statistic) around the models with the selected breakpoints
indicates that actual values may fall within about ±1 *BL*
s^{-1} of the best model. Over this range of potential models,
parameter estimates varied little: standard deviations (s.d.) of
model *b*_{P} values ranged from 0.01–0.05, except for
American shad, for which the s.d.=0.23; values of model
*b*_{S} standard deviations were even more stable, ranging from<0.01 to 0.04 *BL* s^{-1} for all species. Note that the
variance in *b*_{P} s.d. values among models matches
the proportionately larger standard error value for this estimate in the best
models (Fig. 4).

Among blueback herring, a small sample size resulted in poor power to
detect a mode shift that was probably present (*P*=0.08). A further
mode shift may have been present for striped bass: when data greater than the
observed breakpoint of 10.4 *BL* s^{-1} were excluded, an
additional shift was detected at 5.7 *BL* s^{-1}. To avoid
potential bias introduced by including an additional mode, data less than 5.7
*BL* s^{-1} were excluded from the regression analyses.
Interestingly, when swim speeds below *U*_{mp} were excluded
from the American shad analysis, an additional mode shift became apparent here
also at 10.2 *BL* S^{-1}.

Slopes and intercepts for each species are presented in
Table 2, along with the
predicted groundspeed optima (*U*_{gopt}) within each mode and
the estimated maximum prolonged speed (*U*_{mp}). Where no mode
shift was observed, these parameters are assumed to correspond to their values
for sprinting – otherwise they are subscripted with P or S to refer to
prolonged and sprint modes, respectively. A separate regression for striped
bass swimming at speeds <5.7 *BL* s^{-1} resulted in
coefficients of *a*=6.6 and *b*=–0.98.

The groundspeed at which the various species of fish actually swam is shown
for each nominal velocity in Fig.
5, with the predicted optima overlaid for reference. Variance in
estimates of slope lead directly to variance in predicted optimal swim speeds.
Since |*b*_{P}| was always of greater magnitude
than |*b*_{S}|, the inverse predicts a lower
optimal swim speed at prolonged than at sprinting modes. However, the inverse
of the variance also increases proportionally, thus two estimates with similar
variance, such as in striped bass, yield predictions with ranges of
substantially different magnitude.

Where data from two modes were available, values of
*U*_{fcrit} ranged from 4.28 to 5.92 *BL*
s^{-1}, or between the relative speeds for the 1.5 and 2.5 m
s^{-1} nominal velocities (from Tables
1,
2;
Equation 8). Thus, of all species
that exhibited a mode shift, most individuals should select the optimal
groundspeed for the prolonged mode at the 1.5 m s^{-1} condition and
that for the sprint mode at the higher velocities. American shad did precisely
this, and the other clupeids also appeared to follow a distance-maximizing
strategy (Fig. 5). Although
most alewife swam at groundspeeds slightly slower than the predicted optima,
this is because several outliers – fish that had unusually short fatigue
time at low swim speeds (Fig.
4) – acted to reduce the slope of the swim
speed–fatigue time curve. When these observations are removed, the mean
groundspeeds coincide with the predicted optima. Among the nonclupeids, the
actual behavior was quite different. Instead, these three species selected a
constant groundspeed that corresponded with the optimum for the prolonged
mode, regardless of *U*_{f} (with the possible exception of
striped bass at 1.5 m s^{-1}), even though most of these fish were
swimming at speeds corresponding with the sprinting mode. This consistency was
remarkable: among white suckers, for example, a 7.1 *BL* s^{-1}
range of *U*_{f} produced a *U*_{g} range of
only 0.88 *BL* s^{-1}_{.}

Among species that exhibited mode shifts, >90% of all individuals swam
at speeds corresponding with the prolonged mode when swimming against 1.5 m
s^{-1}. Although 100% of clupeids swam at sprint modes at nominal
velocities ≥2.5 m s^{-1}, behavior of non-clupeids was more
variable. Here, 10–69% swam within prolonged mode at each of the higher
velocities, except for white sucker, where all fish swimming against the 4.5 m
s^{-1} condition swam in sprint mode
(Table 3).

I used the same distinction described above to test for the costs of
deviating from predicted optima: tests include data from only those velocities
where >10% of individuals swam within the designated mode. The results of
these tests, as well as tests of the cost of deviating from the observed mean
swim speed within each velocity, are presented in
Table 3. These tests indicate
that the distance-maximization models are correct in sprint mode, but results
for prolonged mode were equivocal. Significant reductions in
*D*_{max} were associated with deviation from
*U*_{goptS} among all species, but there was no correlation
between *D*_{max} and deviation from
*U*_{goptP}, except among white sucker, where deviation was
associated with greater distances of ascent. This is not surprising, because
most of these individuals should, under the model, have made the switch to the
sprint optimum. Similarly, there was no significant effect of deviating from
the mean groundspeed, except for striped bass and white sucker, where greater
deviation was associated with greater distance of ascent.

In addition, I separately tested for the possibility that
*U*_{goptP} was the optimizing speed at the 1.5 m
s^{-1} condition as well as at the faster nominal velocities. Only
walleye showed a significant cost of deviation under the 1.5 m s^{-1}
condition (negative correlation; *P*=0.004). Heavy censoring under the
1.5 m s^{-1} condition resulted in poor power to detect a cost here:
only among walleye did fewer than 50% of individuals successfully reach the
upper end of the flume under this condition
(Haro et al., 2004). Thus,
failure to identify a cost of deviating from the predicted optimum probably
reflects the constraints of the experimental apparatus, rather than any flaw
in the model. At the higher velocities, where all species should have been
swimming at the sprint optimum, only white sucker showed any effect, with
greater distance of ascent associated with deviation from
*U*_{goptP}. This concurs with the model hypotheses, and
indicates that fish swimming at *U*_{goptP} were not selecting
a distance-maximizing strategy at these speeds.

## Discussion

The results of this study indicate that, although distance-maximizing behaviors can be predicted from the swim speed–fatigue time relationship, species differ in the extent to which they approximate these optima. Further, the consistent failure by the nonclupeids to switch to the distance-maximizing groundspeed for sprint mode means that these species are less likely to successfully traverse velocity barriers, even though such success is within their physiological capacity.

### Assumptions and parameters

The approach to quantifying the swim speed–fatigue time relationship presented here differs substantially from the standardized approach developed by Brett (1964). Where others have produced fatigue using coercive methods such as electrified screens, prodding, or impingement avoidance to induce fish to swim against sequentially increased water velocities, we have presented fish that are innately motivated to swim upstream with an opportunity to do so volitionally, measuring fatigue as a behavioral choice to abandon the effort.

This approach is not without assumptions, however, and the following are implicit in this analyses: (1) a linear relationship adequately describes the effect of swim speed on the log of fatigue time; (2) the methods and data presented here were sufficient to identify any mode shifts; (3) the apparatus provided a realistic estimate of the slope(s) of this relationship; and (4) fish are either unaware of the length of the velocity barrier, or such knowledge does not affect their behavior.

Substantial empirical evidence exists to support the first assumption, both
in this study and elsewhere (numerous references in
Beamish, 1978;
Videler, 1993). The second
assumption is more suspect, however. Although clear mode shifts were
identified for American shad, white sucker and walleye, the phenomenon was
less clear for the other species. Because of their smaller size, alewife and
blueback herring swam at faster relative speeds against a given flow than did
the larger fish, and the absence of slow swimming speed data precluded
identification of mode shifts for both species. Conversely, the large size of
some striped bass allowed them to swim at a slower mode against the 1.5 m
s^{-1} flow condition. In both cases, these limitations may have
precluded accurate prediction of *U*_{gopt}, particularly
against low velocity flow.

The third assumption is also suspect: because fish were able to abandon
their effort at will, estimates of fatigue time at a given speed will
inevitably be low. This is true for two reasons: (1) fish probably do not
voluntarily swim to exhaustion; and (2) some individuals may exert less effort
than others, i.e. abandon their effort at a reduced level of fatigue. Both
behavioral characteristics can be expected to reduce the intercept value of
the swim speed–fatigue time relationship. The slope, in contrast, may
have remained unaffected if fish abandoned their ascent at a similar level of
fatigue. However, at faster flows (*U*_{f}) and swim speeds
(*U*_{s}), the range of times at which fish could abandon their
effort was reduced, i.e. skewness was constrained at zero time. Greater
skewness at smaller values of *U*_{s} led to reduced magnitude
of the slope, as well as greater variance of the estimate. For these reasons,
estimates of both slope and intercept values are conservative, and one can
expect estimates of *U*_{gopt} to be correspondingly high. The
presence of outliers had this effect on the alewife models; among other
species, the magnitude of this error appears to be relatively small.

A further limitation of this approach that calls into question the validity of assumptions 2 and 3 is the absence of the cross-sectional uniformity of flow and consistency of velocity that characterizes most controlled laboratory studies. Future modifications to the flume apparatus may address this limitation; however, such nonuniformity of flow is a feature of natural rivers, culverts and fishways, and may provide a realistic context for fish behavior (Haro et al., 2004). In any case, the disorganized, microturbulent character of the flow in this flume can be expected to have acted to decrease performance (Enders et al., 2003); opportunities for fish to take advantage of eddies (e.g. Hinch and Rand, 2000; Liao et al., 2003) were minimal or nonexistent here. By continuously monitoring hydraulic parameters, changes in water velocity were accounted for; combining this with the correction factors (Table 1) removed any bias from values of water velocity assigned to each ascending fish. In this way, performance measures described here can be considered accurate, but conservative relative to actual performance in a natural setting.

The adequacy of the methods for identifying mode shifts and slopes was also limited by the finite length of the flume and the resulting censored observations, particularly at low water velocities. The statistical methods applied here, though novel in this application, are well-known to be robust in the presence of censoring. Any uncertainty arising from the censored data is adequately accounted for by and included in the standard error estimates; the large sample sizes presented here should be more than sufficient to eliminate any systematic bias introduced using these methods (Allison, 1995; Hosmer and Lemeshow, 1999). Heavy censoring at the lowest water velocities did limit the power of these estimates, however, especially among American shad swimming in prolonged mode.

The fourth assumption is more reasonable. Since all fish were naïve,
they clearly had no previous knowledge of the length of the barrier. They
were, however, able to stage multiple attempts, and it is possible that some
knowledge was acquired in this way
(Castro-Santos, 2004). With
knowledge of the length of the barrier, fish could select either a
time-minimizing (i.e. speed-maximizing) strategy (swim faster than
*U*_{gopt}, and thus minimize, for example, exposure to
predators), or a time-maximizing strategy (swim slower than
*U*_{gopt}, and thus reduce instantaneous energetic costs). By
matching fatigue time and swim speed to barrier length (Figs
1,
2A), fish could potentially
reduce energetic costs or other risks associated with the distance-maximizing
swim speeds. While these strategies may make sense in some circumstances, they
are unlikely in this context. Moreover, because most individuals achieved the
greatest distance on the first attempt, there is no reason to expect the fish
to adopt any strategy other than distance-maximization, i.e. by swimming at
the appropriate *U*_{gopt}.

Although this approach will probably tend to underestimate the physiological limits to performance, it may provide a more realistic measure of the behaviors that fish actually exhibit in the wild, and may therefore be more meaningful from ecological and evolutionary perspectives. Moreover, because of the limitations of the coercive approach, along with those of the machines within which fish are usually swum, few studies exist that describe swimming performance at such high speeds. Indeed, many of the observed swim speeds far exceeded predicted maxima for fish of this size and morphology (Videler and Wardle, 1991).

### Swim speed optimization

Of the six species tested, only the anadromous clupeids fully adopted the
predicted distance-maximizing behavior. This was most evident with American
shad, which switched from *U*_{goptP} to
*U*_{goptS} at the predicted flow velocities, and maintained a
relatively constant groundspeed in sprint mode against a
*U*_{f} range >4 *BL* S^{-1}.

The nonclupeids also adopted constant groundspeeds, but these were
appropriate only for prolonged mode (*U*_{goptP}), and no
apparent benefit accrued to any species for swimming at
*U*_{goptP} at
*U*_{f}*>U*_{fcrit}. This is evident from the
general absence of significant negative correlations with the
*R*_{U} and *R*_{P} residuals, and is consistent
with the hypothesis that this was not a distance-maximizing strategy. On the
contrary, positive coefficient values for white sucker and striped bass
suggest that there was a cost associated with the observed speed, and fish
that deviated from the mean swam greater distances. Conversely, fish did
maximize distance by swimming at *U*_{goptS} at flows>
*U*_{fcrit}, as indicated by the strongly significant
negative values of coefficients of the *R*_{S} statistic.

These results, while supporting the hypotheses indicated by the model, may instead be an artifact of varying condition of individual fish: fish in better condition may swim faster and farther than the others (Brett et al., 1958; Hochachka, 1961). Furthermore, inaccuracies arising from the scaling of swim and flow speeds to body lengths may cause spurious results (Drucker, 1996; see also Packard and Boardman, 1999 for a more mathematical treatment of this issue). This is a concern primarily for the striped bass models, owing to the large size range; scaling errors should be minimal among the other species (Table 1). Nevertheless, these data do support the idea that an optimum groundspeed exists for each mode, and that failure to swim at the correct speed results in reduced distance of ascent.

The same logic used above can be applied to mode shifts in the absence of
flow. The distance-maximizing critical swim speed at which fish should switch
from prolonged to sprint modes (*U*_{scrit}) occurs when
10
Thus, if the switch between modes is a facultative behavioral response
(Drucker, 1996;
Peake and Farrell, 2004), it
should always be a discrete effect, whether in still or moving water. This may
help explain why Brett (1964)
and others have described modes as categorical shifts, with transition zones
characterized by few data.

Trump and Leggett (1980)
explored the effect of currents on optimal swim speeds, and produced
predictions that are superficially similar to those presented here. Where a
velocity challenge is encountered that is constant in time but finite in
space, their model predicts an optimal groundspeed of *m*^{-1},
where *m* is the exponent of the energy equation
*E*_{s}*=ae*^{mU}s [J kg^{-1}
s^{-1}] (Brett, 1965;
Webb, 1975), much like the
model presented here. However, because the slope of the metabolism–swim
speed relationship should increase as fish recruit anaerobic processes, the
optimal groundspeed should decrease as fish switch from prolonged to burst
modes – exactly the opposite of what my model predicts, and what these
data suggest.

Likewise, the predictions of models generated by Weihs (1973) and Videler (1993) are not supported by these data. When Weihs' equations 7 and 8 (Weihs, 1973), and Videler's equations 9.1 and 9.2 (Videler, 1993) are adjusted for flow (similar to Equations 2, 3, 4 here), both sets of models predict distance-maximizing groundspeeds that accelerate with increasing flow. Again, this is not in accordance with the observed behaviors. None of these other models was developed for fish swimming in nonsustainable modes, however, and recruitment of anaerobic metabolism, alternative gaits, etc., may alter the relationship between cost of transport and swim speed on which they are based.

Other strategies may optimize for conditions unlike those present in this study. For example, fish may approach velocity barriers by swimming at maximum possible speed. This could be appropriate for leaping species like salmon that may want to maximize the likelihood of traversing a falls of unknown height. None of the species tested here employs leaping behavior in migration, so it is not surprising that maximum speed was not employed. Another strategy might be to employ alternate gait patterns, thereby improving energetic efficiency (Weihs, 1974; Videler and Weihs, 1982), or to capitalize on low-velocity zones, as white sucker did against the lower velocities here. Any such kinematic or behavioral strategy will still have an associated swim speed–fatigue time relationship, however, and so will be intrinsically included in the models presented here. As such, these models may be considered robust in the presence of behavioral, kinematic, and physiological diversity.

One optimizing strategy that these models may not adequately account for is
the staging of repeated attempts. By increasing the rate at which they stage
successive attempts fish can increase the likelihood of passage (Castro-Santos
2002,
2004). Fish may reduce
recovery time by swimming at slower speeds, thereby increasing attempt rate
and possibly offsetting the costs incurred by deviating from
*U*_{gopt}. This strategy still does not account for the
consistency in groundspeed observed here, particularly among the non-clupeids,
nor for the fact that most individuals reached their maximum distance of
ascent on the first attempt. It seems likely that some other factor is at
work.

The apparent presence of distance-maximizing behavior among the anadromous clupeids, and its partial absence among the potomodromous non-clupeids, suggests the presence of underlying selective processes. Webb (1994) points out that the range of gaits available to fishes can have profound evolutionary consequences; perhaps the relationship between swim speed and fatigue time is shaped in part by the hydraulic conditions fish need to traverse in order to maximize fitness. Anadromous clupeids need to ascend rivers during spring freshets to spawn, when high flows and cold temperatures place strong demands on swimming capacity. Thus sprinting among these fish constitutes a fitness-critical migratory mode. Potomodromous species, in contrast, have greater choice in where they spawn, and the striped bass used here are amphidromous, entering the river to feed. The fastest modes for these species may therefore not be associated with migration, but rather with other fitness-crucial behaviors, like predation and predator avoidance. By selecting the appropriate groundspeed for the prolonged mode, these species may be optimizing for different habitats, a behavior that could help explain observed limits to their distributions.

In addition to their ecological context, these results also have important implications with respect to the design of fish passage structures. To maintain such consistent groundspeeds, fish must use some means of detecting their progress relative to the ground (presumably vision). This may help explain why anadromous fish often follow structure when migrating up rivers, and also points to the potential harmful effect of entrained bubbles and turbulence on passage success, and even on willingness to attempt to traverse zones of difficult passage.

Bainbridge (1960) observed that maximum distance of ascent through fishways is governed by the swim speed–fatigue time relationship, and such data have been used extensively to determine the location and size of resting pools within fishways (Beamish, 1978). The distance of ascent predicted from this relationship, however, assumes that fish swim at their optimum speed which, as in this study, may often not be the case. Any recommendations for fishway designs based on the swim speed–fatigue time relationship should therefore take into account the expected variability around the optimum, and the costs of such variability in terms of distance of ascent when predicting passage success.

**List of symbols**

- a,b
- coefficients
- BL
- body length
- CF
- correction factor
*D*_{g}- ground distance
*D*_{max}- maximum distance
*D*_{maxP}- maximum distance in prolonged mode
*D*_{maxS}- maximum distance in sprint mode
*D*_{s}- maximum distance a fish can swim
- f(T)
- probability density function
- FL
- fork length
- R
- residual
- S(T)
- survivorship function
- T
- fatigue time
*U*_{f}- speed of flow
*U*_{fa}- mean
*U*_{f}adjusted for species- and*U*_{nom}-specific ascent routes *U*_{fcrit}- critical speed of flow
*U*_{g}- ground speed
*U*_{gopt}- optimal ground speed
*U*_{mp}- maximum prolonged speed
*U*_{ms}- maximum sustained speed
*U*_{nom}- nominal velocity
*U*_{opt}- optimum swim speed
*U*_{s}- relative swim speed

## ACKNOWLEDGEMENTS

This work was conducted at and funded by the S.O. Conte Anadromous Fish Research Center (USGS-Biological Resources Division). A number of individuals have made important contributions to this work, for which I am very grateful. David Hosmer provided guidance on the application of survival analysis methods and their interpretation. Alex Haro and Mufeed Odeh made extensive resources available to support this work. Alex Haro, Steve McCormick, William Bemis, Beth Brainerd, Mike Sutherland, and George Lauder also provided careful reviews of this manuscript and made many helpful suggestions.

- © The Company of Biologists Limited 2005