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Behavioral Ecology Vol. 10 No. 2: 155-160
© 1999 International Society for Behavioral Ecology
The optimal allocation of time and respiratory metabolism over the dive cycle
Department of Zoology, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Address correspondence to Y. Mori, Department of Zoology, Graduate School of Science, Kyoto University, Kitashirakawa, Sakyo, Kyoto 606-8502, Japan. E-mail: yosihisa{at}zoo.zool.kyoto-u.ac.jp
Received 1 April 1998; accepted 20 July 1998.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Because anaerobic metabolism is much less efficient than aerobic metabolism in supplying energy, it is widely believed that divers rely predominantly on aerobic metabolism for diving. In this paper, a time budget model, which assumes that the diver can use either completely aerobic or partially aerobic metabolism with additional anaerobic metabolism for diving, is developed and is used to make predictions about patterns in optimal allocation of time and respiratory metabolism during the dive cycle. The results derived from the model are (1) a diver that can vary the ratio of energy supplied anaerobically to total energy spent during dive time is favored by natural selection, but the patterns of time allocation over the dive cycle by the diver do not differ from those of a diver that cannot vary the ratio. (2) Even if it is assumed that divers switch their metabolism for diving, an obvious upturn in the surface time with respect to dive time does not occur at the aerobic dive limit (ADL) but occurs beyond the ADL. (3) Use of additional anaerobic metabolism can be favored for dives shorter than the ADL. These findings provide a useful guide to understanding the factors that limit diving behavior.
Key words: anaerobic metabolism, diving, optimal foraging model.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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There are many species of air-breathing animals that forage below the water surface. These diving animals perform three activities while foraging: they spend a time, s, on the water surface; a time, t, in a foraging area, and a total time,
, in traveling from the surface to the
foraging area and back to the surface
(Houston and Carbone, 1992
).
Thus, the total time under water or dive time, u, is t +
. Energy used for diving can be supplied by both aerobic and anaerobic
metabolism, and the amount of energy supplied by each type is supposed to be
an increasing but negatively accelerated function of surface time
(Carbone and Houston, 1996).
This means that dive time should be an increasing but negatively accelerated
function of surface time.
Because anaerobic metabolism is much less efficient than aerobic metabolism
in supplying energy, it is widely believed that divers rely predominantly on
aerobic metabolism for diving (Butler,
1988). It has also been argued whether the upturn commonly
observed in the relationship between surface time and dive time is associated
directly with a shift to anaerobic metabolism (see
Carbone and Houston, 1996, for
review).
Several authors have developed theoretical models of the optimal time
allocation during dive cycle, t + + s, as a function
of , maximizing the proportion, P, of time spent in the foraging
area during dive cycle time, where
 |
;
Houston and Carbone, 1992;
Kramer, 1988;
Wilson and Wilson, 1988).
Carbone and Houston (1996)
presented two models that allow a combination of aerobic and anaerobic
respiration: the switch model and the mixed metabolism model.
The switch model assumes that a diver uses only aerobic or anaerobic
metabolism during the dive and can use the surface time to recover from only
one form or the other. The mixed metabolism model assumes that the diver
always uses both metabolisms during the dive and can use the surface time to
recover from both forms simultaneously. Based on these assumptions, Carbone
and Houston (1996) defined dive
time, u, as
 | (1a) |
 | (1b) |
 | (2) |
1 and 2 are
scaling factors for each metabolic pathways. Carbone and Houston
(1996) noted that the mixed
model is more realistic and can explain physiological and behavioral data
concerning shifts in metabolism, and Carbone et al.
(1996) tested and concluded
that the mixed model is supported experimentally.
However, it also seems realistic to assume that the diver can (but does not
always) use both forms of metabolism during the dive but cannot simultaneously
use the surface time to recover from both forms when O2 uptake is a
function of surface time; a part of surface time, or O2, is used to
recover from aerobic respiration (e.g., accumulation of O2), and
the rest of it is used to recover from the anaerobic respiration (e.g.,
oxidation of lactate). Based on this assumption, another possible model, the
time budget model, can be developed, in which dive time, u, can be
expressed as
 | (3) |
Developing this time budget model, I illustrate the patterns in optimal allocation of time and respiratory metabolism over the dive cycle and discuss when inefficient anaerobic metabolism should be used.
The model
The currency maximized in the model is P, the proportion of time
spent in the foraging area. To simplify the model analysis, the rate of energy
use during diving (traveling and foraging) is assumed to be equal to 1 for all
calculations (see Carbone and Houston,
1996; Houston and Carbone,
1992, for the effects of the rates of energy use on the
results).
As s = s1 + s2,
s1 and s2 can be replaced to
qs and (1 - q) s, respectively, where 1 
q 0. Then, from Equations 2 and 3, dive time, u, in the
model, is assumed to be
 | (4) |
1 > 2 is assumed (see also
Houston and Carbone, 1992;
Kramer, 1988). This is a
negatively accelerated function of s, and q represents the
fraction of surface time used for recovery from the aerobic respiration during
the dive. When q = 1, for example, the diver makes a completely
aerobic dive; energy spent during dive is supplied only by aerobic
respiration. When q = 0, the diver makes a completely anaerobic dive;
energy spent during dive is supplied only by anaerobic respiration. Thus, in
this model, the aerobic dive limit (ADL;
Kooyman, 1989) is
K1 and the dive limit is K1 +
K2.
Optimal q and s (q* and
s*, respectively) that maximize P were found for
various traveling times, . Note that finding q* and
s* can be viewed as finding q*, and
optimal time in a foraging area, is t*. The computation
was performed on Apple Macintosh personal computers using Mathematica, version
2.22.
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RESULTS |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In this model, completely anaerobic metabolism is never profitable for a diver (i.e., q* is never equal to 0). Furthermore, mixed metabolism is not always profitable for the diver (i.e., q* is not always <1). In general, completely aerobic metabolism is profitable (q* = 1) when for a given
the optimal dive time is short, whereas mixed metabolism is profitable (0 <
q* < 1) when the optimal dive time is long. However,
the duration of and dive time at which the diver should use partially
aerobic metabolism with additional anaerobic metabolism for diving depends on
K2 and 2 with respect to
K1 and 1 (Figure
1a,
b). In particular, the diver
should use mixed metabolism for dives much shorter than ADL when
K2 and 2 are large
(Figure 1b).
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Figures 2 and
3 give examples of optimal
t, t*; optimal dive time, t*
+ ; optimal surface time, s*; optimal P,
P*; optimal fraction of aerobic metabolism time to dive
time, f*; and q* toward various
in the time budget model. Figure
4 shows the relationship between t* +
and s*. The general features of the results are (1) a
second peak develops in t* when the difference between
1 and 2 is large, with corresponding
adjustments in t* + , s*,
P*, f* and q*
(Figure 2); (2) increasing
K2 with respect to K1 has a similar
effect as increasing 2
(Figure 3); and (3) upturns of
surface interval against dive times occur first at the switch points around
(but smaller than) ADL, but these points are not easy to detect, whereas
second upturns of surface time occur at dive times that differ from ADL
(Figure 4).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Predicted patterns and assumptions: comparison with the previous study
The behavior of the present model shows patterns in t*, s*, t* +
,
P*, and f* versus that are
almost same as the mixed metabolism model of Carbone and Houston
(1996). This means that most of
the discussion in Carbone and Houston
(1996) is applicable to the
time budget model. For example, they noted that their mixed metabolism model
may explain why upturns in the surface time with respect to dive time may not
always reliably indicate a shift to anaerobic respiration. The results
described above show that the time budget model also may predict this; even if
it is assumed that divers use either completely aerobic or partially aerobic
metabolism with an additional anaerobic one, an obvious upturn in the surface
time with respect to dive time does not occur at the ADL but occurs beyond the
ADL (Figure 4). These findings
suggest that upturns in the surface time with respect to dive time cannot
reliably indicate a shift to anaerobic respiration in most cases.
Carbone and Houston (1996)
assumed, as shown by constant scaling factors (1 and
2 in Equation 2), that the ratio of energy supplied
anaerobically to total energy spent during dive time is determined by surface
time and that the diver cannot vary the ratio and the surface time independent
of each other. In contrast to this assumption, the present model assumes, as
shown by variable scaling factors [1q and
2 (1 — q) in Equation 4], that the diver can
vary the ratio of energy supplied anaerobically to total energy spent during
dive time independent of the surface time. Because O2 uptake by
divers during surface time is used for recovery from both forms of metabolism
(accumulation of O2 and oxidation of lactate) and because
O2 used for accumulation cannot be used for oxidation of lactate,
it is likely that an increase in the amount of O2 used for recovery
from one metabolism decreases the amount of O2 used for recovery
from the other metabolism. Therefore, the assumption in the present model
would be more realistic than that in the model by Carbone and Houston
(1996).
The question of which model is more likely in nature cannot be answered by
records of diving behavior because both models predict similar results, and it
is possible to say that the present model is tested and supported
qualitatively by the results of Carbone et al.
(1996). To answer this
question, lactate concentration in tissue must be measured during a short
dive; the time budget model predicts that lactate will not increase
significantly in tissue during a short dive, whereas the mixed metabolism
model predicts certain production of lactate during a short dive.
Unfortunately, as Boyd (1997)
pointed out, few studies have measured lactate concentration in tissue during
a dive apart from the study by Kooyman et al.
(1980). In this study, the
authors showed that for dives shorter than 20 min there was no significant
increase in blood lactic acid concentrations in Weddell seals,
Leptonychotes weddelli (see also
Boyd, 1997).
It should be noted that, in mathematical form, the time budget model
includes the mixed model because they are equivalent when q = 0.5.
However, optimal q is not always 0.5 (Figures
2f and
3f). This means that the diver
that can vary the ratio of energy supplied anaerobically to total energy spent
during dive time is likely to be favored by natural selection. Thus, it can be
argued that the time budget model should be applicable to animals highly
adapted to diving (e.g., pinniped) as opposed to other divers (e.g., flying
aquatic birds), although both of them would show similar behavioral patterns
in diving. It is interesting that, even if constraints on diving physiology
changed, the patterns observed in diving behavior may not change. Although I
do not deny the possibility that divers always use mixed metabolism during a
dive, it seems more natural to suppose that adapted divers use completely
aerobic metabolism during short dives because of its efficiency. This
supposition is consistent with the fact that Weddell seals seem to use
completely aerobic metabolism during short dives
(Kooyman et al., 1980).
Carbone and Houston (1996)
noted that the assumption in the mixed metabolism model would not be strictly
true if the metabolites produced by anaerobic respiration are used as a
substrate for aerobic respiration (e.g.,
Chappell et al., 1993;
Croll et al., 1992;
Stephenson, 1994). This also
applies to the present model.
Conditions in which additional inefficient anaerobic metabolism
should be used
It is important for diving physiologists to know whether anaerobic
metabolism is always or sometimes used by divers. However, from the viewpoint
of diving behavior, the important question is when and why divers use not only
aerobic but also anaerobic metabolism for diving, even though anaerobic
metabolism is very inefficient.
When a diver should use partially aerobic metabolism with additional
anaerobic metabolism depends on traveling time () and the physiology of
the diver (K2 and 2), and the threshold
of dive time at which the diver should use additional anaerobic metabolism is
always smaller than ADL (Figure
1). These suggest that the reason divers use apparently
inefficient anaerobic metabolism is not that the dive time exceeds the ADL.
The important factor in using additional anaerobic metabolism is the
relationship among K1 (=ADL), K2,
1, and 2. More precisely, the additional
use of anaerobic metabolism is preferred if the recovery time for anaerobic
metabolism does not exceed the time required for aerobic recovery. The reason
for this is that, under such circumstances, the diver can increase dive time
at no additional recovery cost (Carbone and
Houston, 1996). In this respect, anaerobic metabolism is not
inefficient compared to aerobic metabolism. Therefore, the assumption that
divers switch to anaerobic respiration on reaching the ADL, which is often
made, may not be true. The divers should use additional anaerobic respiration
before dive time reaches ADL for optimal foraging, and when the diver should
use mixed metabolism depends more on the type of recovery functions than on
the value of ADL.
Ydenberg and Clark (1989)
revealed, using the dynamic programming model, that anaerobic diving should be
practiced when prey is aggregated and hard to find. Mori
(1998) demonstrated, using the
classical patch use model, that anaerobic metabolism is favorable when prey
patch quality is high, the prey patch is situated in deep water, and the prey
patch is hard to find. These models are different from each other in currency
maximized: the Ydenberg and Clark
(1989) model maximizes gross
energy intake over a given foraging period, and the Mori
(1998) model maximizes
long-term average rate of net energy intake. These currencies are also
different from the present model. However, in spite of these differences in
the approaches and currencies among the models, all the models involving
anaerobic metabolism point out that using only aerobic metabolism does not
always take relatively less energy and shorter time than using both aerobic
and anaerobic metabolism when a diver intends to increase foraging time at a
dive.
It is a matter for argument whether divers usually use additional anaerobic
metabolism or completely aerobic metabolism with O2 consumption
rate decreased for dives longer than the theoretically estimated ADL (TADL)
(e.g., Boyd 1997). If the diver
makes use of some form of metabolic depression, actual ADL becomes longer than
TADL, and this makes K1 increase in the model. However,
general features of the results derived from the present model do not change
in this case; there are still certain conditions in which use of additional
anaerobic metabolism is favored. It is, therefore, still useful to consider
partially aerobic metabolism with additional anaerobic metabolism to
understand diving behavior.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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I thank Yutaka Watanuki and Akiko Kato for their helpful comments on drafts of the manuscript. I am indebted to Fugo Takasu for giving me every convenience to carry out computations.
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REFERENCES |
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Boyd IL, 1997. The behavioural and physiological ecology of diving. Trends Ecol Evol 12:213-217.
Butler PJ, 1988. The exercise response and `classical' diving response during natural submersion in birds and mammals. Can J Zool 66:29-40.
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Chappell MA, Shoemaker VH, Janes DN, Bucher TL, Maloney SK,1993 . Diving behavior during foraging in breeding Adelie penguins. Ecology 74:1204-1215.
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Kooyman GL, 1989. Diverse divers: physiology and behavior. Berlin: Springer-Verlag.
Kooyman GL, Wahrenbrock EA, Castellini MA, Davis RW, Sinnett EE,1980 . Aerobic and anaerobic metabolism during voluntary diving in Weddell seals: evidence of preferred pathways from blood chemistry and behaviour. J Comp Physiol 138:335-346.
Kramer DL, 1988. The behavioural ecology of air breathing by aquatic animals. Can J Zool 66:89-94.
Mori Y, 1998. The optimal patch use in divers: optimal time budget and the number of dive cycles during bout. J Theor Biol 190:187-199.
Stephenson R, 1994. Diving energetics in lesser scaup (Aythya affinis, Eyton). J Exp Biol 190:155-178.[Abstract]
Wilson RP, Wilson MT, 1988. Foraging behaviour in four sympatric cormorants. J Anim Ecol 57:943-955.
Ydenberg RC, Clark CW, 1989. Aerobiosis and anaerobiosis during diving by Western grebes: an optimal foraging approach.J Theor Biol 139:437-449.
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