Text S1. Activity cost

Supplement to Acevedo & Urbán (2021) – Mar Ecol Prog Ser 657:223-239 – https://doi.org/10.3354/meps13543

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Text S1. Activity cost

Here, we used the approach of Watanabe et al. (2011) to estimate total metabolic power. One assumption in total metabolic power is that the cost of thermoregulation in marine mammals is negligible because the heat generated by the locomotor muscles can be used for thermoregulation (Hind & Gurney 1997, Watanabe et al. 2011).

𝑃!"#$% = BMR+ (𝑃𝑜!"#$ + 𝑃𝑜!"# + 𝑃𝑜!"#)/ɛ

where BMR is the basal metabolic rate, Podrag represents the metabolic cost of the mechanical effort required to actively propel an animal at speed V, Pobuo is the buoyancy power needed to swim against buoyancy, Poind is the induced power needed to avoid sinking or rising in the water column and, ɛ is the efficiency to convert mechanical to metabolic power.

The induced power (Poind), an aeronautical concept that was introduced into diving animals by Hansen & Ricklefs (2004), was set at zero in our biomechanical model following Watanabe et al. (2011).

The energy needed by marine mammals to propel themselves through the water (Podrag, J s-1) is a function of the drag force of the fluid (D, in Newton) exerted on the body and the speed (V, m s-1) of swimming needed to overcome this drag force (Hind & Gurney 1997). In order to maintain a constant swimming speed, an animal must exert a propulsive force to match the drag force resulting from its movement, and this amount of drag depends on the size of the animal, shape, the physical properties of the water, and the swimming speed. Follow Hind & Gurney (1997), the drag force acting on a passive body of length (L, meter) and surface (S, m2), moving at a speed V through a medium of density (ρ) is written as:

𝐷 = 12

ρ ∙ 𝑆 ∙ 𝐶! ∙ 𝑉!

where ρ is the density of seawater (1027 kg m-3), S is the body surface area and CD is a dimensionless drag coefficient.

The body surface area (S), which is the major determinant of drag force (Fish & Rohr 1999), was estimated using an allometric relationship between mass and surface area described for marine mammals (Innes et al. 1990, Fish 1993, Ryg et al. 1993):

𝑆 = 𝑎!""#𝑀! where aMeeh represents a form factor (also called Meeh's constant), b the predicted

scaling between mass and surface, and M the body mass (kg), here estimated from the growth curves. In cetaceans, the Meeh's constant has been suggested to be between 0.068 and 0.080, while the value of b between 0.65 and 0.70 (Innes et al. 1990, Fish 1993, Ryg et al. 1993). Therefore, uncertainty was incorporated into the estimation of the body surface by 10000 iterations from a uniform distribution of possible values of a and b, with each value between the limits having an equal probability of being sampled.

Drag coefficient (CD) is in turn a function of the object’s size, shape, and flow characteristics, the latter determined by the Reynolds number (Re) (Webb 1975). If the


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boundary-layer flow is laminar (Re < 2000), CD depends significantly on velocity; while if it is turbulent (Re > 4000) the dependence is much weaker and have a marginal influence on energy loss (Hoerner 1958). The Reynolds number (Re = LV/v, where v is kinematic viscosity (1.044 x 10-6 m2 s-1)). Drag coefficients (CD) have been estimated in some cetacean species (e.g., Hind & Gurney 1997, Fish 1998, Miller et al. 2004); however, these have been estimated for adult, not being useful here since we are interested in energy budgets for different demographic groups (calf and juvenile). Assuming that the drag force of a flat plate can be analogous with the drag of the body of a cetacean of equivalent surface area (Fish 1992), then the minimum theoretical CD that is dependent on Re for a turbulent type layer can be estimated as (Fish & Rohr 1999):

𝐶! (!"#$"%&'!) = 0.072 ∙ 𝑅𝑒!!.!

Although estimates of drag coefficients based on rigid body analogies, as is the case, may yield underestimated values (Fish & Rohr 1999), the drag coefficients calculated for each age class are within the ranges (2.6 x 10- 3 to 5.2 x 10-3) reported for marine mammals (Fish 1992). Now, the drag force acting on an actively swimming animal scales with velocity, so the rate of delivery of mechanical work (W, in J s) by a swimming animal (Hoerner 1965, Hind & Gurney 1997, Fish & Rohr 1999) is:

𝑊 = λ2

ρ ∙ 𝑆 ∙ 𝐶! ∙ 𝑉!

where λ is a constant of the ratio of the drag of an active swimmer to that of a passive object moving at the same speed. The data do not currently exist to calculate W for humpback whales using the methodology outlined in Hind & Gurney (1997); therefore, a value of 0.70 suggested for minke whales Balaenoptera acutorostrata (Hind & Gurney 1997) was used.

The cost associated with generating this mechanical propulsion work depends on both net efficiency with which muscle movement is converted to thrust (propeller efficiency, ɛP), and the efficiency with which chemical energy is transformed in muscular work (aerobic efficiency, ɛA) (Hind & Gurney 1997, Fish et al. 1988, van der Hoop et al. 2017). Propeller efficiency in marine mammals has been shown to be consistent regardless of velocity (Fish et al. 1988, Hind & Gurney 1997); therefore, this value can be assumed to be constant. Propeller efficiency in humpback whales has been estimated to be 91% (Goldbogen et al. 2012), while aerobic efficiency is set at 25% for cetaceans (Fish 1996, Ahlborn 2004). Thus, we calculated mechanical effort cost (Podrag) for swimming speeds of each behavior pattern using Equation 4 from Hind & Gurney (1997):

𝑃𝑜!"#$ =λ

2 ∙ ɛ! ∙ ɛ! ρ ∙ 𝑆 ∙ 𝐶! ∙ 𝑉!

The buoyancy power (Pobuo) was estimated using equation from Watanabe et al. (2011):

𝑃𝑜!"# =𝐵 ∙ 𝑉 ∙ sinθɛ! ∙ ɛ!

where B is the magnitude of buoyancy (in Newton), V is the swimming speed (m s-

1) and θ is the absolute angle of inclination relative to the horizontal (in degrees) during the


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dive. The magnitude of buoyancy for humpback whales was estimated as actively-ascending animals, where the buoyancy is estimated at the bottom of dives. The magnitude of buoyancy is a function of depth, since the volume of air in the respiratory system changes with depth due to hydrostatic pressure. Thus, B is given by:

𝐵 = 𝑀 ∙ 𝑔 − ρ VO!"##$% + VO!"# 𝑔

where M is the body mass (kg), g is the acceleration of gravity (m s-2), VOtissue is the volume of body tissue (m3), and VOair is the air volume (m3) in the respiratory system under the hydrostatic pressure. The first term corresponds to the gravity force acting on the object (force that pushes down), and the second term is the buoyancy force of the object (force that pushes up) established by the Archimedean principle.

Body tissue volume (VOtissues) is expressed as a quotient between body mass (M) and tissue density (ρtissue, kg m-3) (Watanabe et al. 2006). Recently, Narazaki et al. (2018) provide tissue density values for some age class and sex in humpback whales. A uniform distribution of possible values was applied through 10,000 iterations to these ranges for each demographic class. No information was provided for lactating females and male calves; therefore, the same tissue density of adult females and calves were assumed.

Given that the air space of the animal body is compressed by pressure following Boyle’s law, VOair was calculated from the equation proposed by Skrovan et al. (1999) as:

VO!"# = VO!"#$/(1+ 0.1 𝑍) where VOair0 is the air volume (m3) at the atmospheric pressure, and Z is depth (m).

VOair0 has been estimated for humpback whale with an average value of 27.7 ± 1.1 ml kg-1 (Narazaki et al. 2018).

Based on the above, VOair was estimated for two different depths, the first corresponding to shallow swimming (transit behavior) and the second to represent when the whales feed at depth (feeding behavior). Due to the lack of information on the depth of shallow swimming during transit behavior, it was assumed that the humpback whale would move horizontally to a depth of at least 10 meters to minimize the effect of additional drag induced for the waves at or near the water surface (Nousek 2010). This depth is the result of three times the maximum diameter (3.25 m in humpback) of the body (Hertel 1966), and it would consist with studies on right whales that have shown a pattern of shallow dives to depths less than 30 m to avoid high surface resistance waves (Baumgartner & Mate 2003, Nousek 2010). In the Magellan Strait feeding area, humpback whales feed on small fish, squat lobsters, and krill, which occur at different depths. Therefore, foraging activity was assumed to be carried out at an average depth of 50 m, allowing the chance to feed on both shallow fish and crustaceans at greater depths during each dive.

Finally, we used inclination angle values during the dives (θ) from Narazaki et al. (2018). These authors provide individual data of inclination angles during the descending and ascending phase of diving for adults, juveniles and two calves humpback whales, and an average value of both phases was used.


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Average fetal mass In Equation (16, main document), the average fetal mass was estimated using the

relationship between gestation time and birth weight based on the equation formulated by Huggett & Widdas (1951), where Mb is an average birth weight (in grams), af is a constant of specific fetal growth rate, tg is the gestation time (in days) from conception, and tc is a constant time (in days) from the conception prior to the phase of linear growth and has no biological significance.

𝑀!!/! = 𝑎! (𝑡! − 𝑡!)

Specific fetal growth rate (af) represents the slope of the relationship between Mb and gestation time (tg), while tc is the point where the line cuts the time axis at Mn 1/3 = 0 (intercept). The value of af was estimated as:

𝑎! =𝑀!!/!

𝑡! − 𝑡!

The values of af and tc remain uncertain since there are still doubts about which growth phase should be used to determine growth constants, and since that fetal growth presents a smooth linear phase followed by rapid exponential growth from approximately the fifth to sixth month from conception (Laws 1959, Nishiwaki 1959, Lockyer 1981). The proposal formulated by Huggett & Widdas (1951) is suggested to follow the exponential phase (Lockyer 1981). Huggett & Widdas (1951) comment that an expected value of tc for a gestation period of 100–400 days can be assumed to be 0.2 × tg. Thus, an average gestation of 12 months results in a tc value of 73.

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Text S1. Activity cost