Behavioral Ecology Vol. 10 No. 2: 155–160
The optimal allocation of time and respiratory
metabolism over the dive cycle
Yoshihisa Mori
Department of Zoology, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
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. [Behav Ecol 10:155–160 (1999)]
T
here 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,
t, 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 1 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 1 t 1 s, as a function of t, maximizing the proportion, P, of time spent in the
foraging area during dive cycle time, where
P 5 t/(t 1 t 1 s)
(Carbone and Houston, 1996; 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
Address correspondence to Y. Mori, Department of Zoology, Graduate School of Science, Kyoto University, Kitashirakawa, Sakyo, Kyoto
606-8502, Japan. E-mail: [email protected].
Received 1 April 1998; accepted 20 July 1998.
q 1999 International Society for Behavioral Ecology
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
u 5 y1(s) 5 K1[1 2 exp(2a1s)],
(1a)
u 5 y2(s) 5 K2[1 2 exp(2a2s)]
(1b)
or
in the switch model, and as
u 5 y1(s) 1 y2(s)
5 K1[1 2 exp(2a1s)] 1 K2[1 2 exp(2a2s)]
(2)
in the mixed metabolism model, setting the rates of underwater energy use equal to 1. In these equations K1 and K2 are
equal to the upper limit of aerobic and anaerobic dive time,
respectively, and a1 and a2 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
u 5 y1(s1) 1 y2(s2)
5 K1[1 2 exp(2a1s1)] 1 K2[1 2 exp(2a2s2)]
(3)
where s1 and s2 are surface times used to recover from aerobic
respiration and anaerobic respiration, respectively. Therefore,
surface time, s, is s1 1 s2 in this model.
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.
Behavioral Ecology Vol. 10 No. 2
156
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 5 s1 1 s2, s1 and s2 can be replaced to qs and (1 2 q)s,
respectively, where 1 $ q $ 0. Then, from Equations 2 and 3,
dive time, u, in the model, is assumed to be
u 5 K1[1 2 exp(2a1qs)]
1 K2{1 2 exp[2a2(1 2 q)s]}
(4)
where a1 . a2 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 5 1, for example, the diver makes a completely aerobic dive;
energy spent during dive is supplied only by aerobic respiration. When q 5 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 1 K 2.
Optimal q and s (q* and s*, respectively) that maximize P
were found for various traveling times, t. 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.
RESULTS
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* 5 1) when for a given t 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 t and dive time at which the diver should use partially
aerobic metabolism with additional anaerobic metabolism for
diving depends on K 2 and a2 with respect to K1 and a1 (Figure
1a, b). In particular, the diver should use mixed metabolism
for dives much shorter than ADL when K 2 and a2 are large
(Figure 1b).
Figures 2 and 3 give examples of optimal t, t*; optimal dive
time, t * 1 t; optimal surface time, s*; optimal P, P *; optimal
fraction of aerobic metabolism time to dive time, f *; and q*
toward various t in the time budget model. Figure 4 shows
the relationship between t * 1 t and s *. The general features
of the results are (1) a second peak develops in t * when the
difference between a1 and a2 is large, with corresponding adjustments in t* 1 t, s*, P *, f * and q* (Figure 2); (2) increasing K 2 with respect to K1 has a similar effect as increasing a2
(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).
DISCUSSION
Predicted patterns and assumptions: comparison with the
previous study
The behavior of the present model shows patterns in t *, s*,
t* 1 t, P*, and f * versus t that are almost same as the mixed
Figure 1
(a) The longest travel time, t, during which optimal metabolism is
completely aerobic, and (b) dive time at that occasion, with respect
to various K 2 and a2. K1 5 50 and a1 5 0.09. All values are
expressed in contour lines as a function of K 2 and a2.
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
Mori • Optimal allocation of time and metabolism for divers
157
Figure 2
The consequences for the time budget model of maximizing the proportion of time spent foraging: the effect of a2. The optimal policies for
(a) foraging time, t *, (b) dive time, t* 1 t, (c) surface time, s *, (d) proportion of time foraging, P*, (e) proportion of aerobic metabolism
time to dive time, f *, and (f) proportion of surface time used for recovery from aerobic metabolism, q *, with respect to travel time, t. a1 5
0.8, a2 5 0.1 and 0.01, K1 5 20, K2 5 40. The aerobic dive limit (ADL) is represented by a dashed line. Dotted lines represent the
consequences when only aerobic metabolism is available.
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 (a1 and a2 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 [a1q and a2(1 2
q) in Equation 4], that the diver can vary the ratio of energy
158
Behavioral Ecology Vol. 10 No. 2
Figure 3
The consequences for the time budget model of maximizing the proportion of time spent foraging: the effect of K 2. The optimal policies for
(a) foraging time, t *, (b) dive time, t* 1 t, (c) surface time, s *, (d) proportion of time foraging, P*, (e) proportion of aerobic metabolism
time to dive time, f *, and (f) proportion of surface time used for recovery from aerobic metabolism, q *, with respect to travel time, t. a1 5
0.8, a2 5 0.01, K1 5 20, K 2 5 35 and 15. The aerobic dive limit (ADL) is represented by a dashed line. Dotted lines represent the
consequences when only aerobic metabolism is available.
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
Mori • Optimal allocation of time and metabolism for divers
159
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
Figure 4
The optimal surface time, s * against the optimal dive time, t * 1 t.
The aerobic dive limit (ADL) is represented by a dashed line. (a)
Dive time: up to 60, (b) a small scale of view, dive time between 14
and 25. a1 5 0.8, a2 5 0.01, K1 5 20, K 2 5 35 and 15.
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 5 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
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
(t) and the physiology of the diver (K 2 and a2), 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 (5ADL), K 2, a1, and
a2. 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
160
additional anaerobic metabolism is favored. It is, therefore,
still useful to consider partially aerobic metabolism with additional anaerobic metabolism to understand diving behavior.
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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