Relations between morphology, buoyancy and energetics of requiem sharks

Sharks have a distinctive shape that remained practically unchanged through hundreds of millions of years of evolution. Nonetheless, there are variations of this shape that vary between and within species. We attempt to explain these variations by examining the partial derivatives of the cost of transport of a generic shark with respect to buoyancy, span and chord of its pectoral fins, length, girth and body temperature. Our analysis predicts an intricate relation between these parameters, suggesting that ectothermic species residing in cooler temperatures must either have longer pectoral fins and/or be more buoyant in order to maintain swimming performance. It also suggests that, in general, the buoyancy must increase with size, and therefore, there must be ontogenetic changes within a species, with individuals getting more buoyant as they grow. Pelagic species seem to have near optimally sized fins (which minimize the cost of transport), but the majority of reef sharks could have reduced the cost of transport by increasing the size of their fins. The fact that they do not implies negative selection, probably owing to decreased manoeuvrability in confined spaces (e.g. foraging on a reef).

1. Introduction

Within marine environments, sharks represent a wide range of upper and mid-level predators. They can be found in most marine habitats from coastal to pelagic and deep sea, and encompass a variety of feeding modes, including those specializing on marine mammals and filter-feeding [1]. These habitats also span a wide range of temperatures from arctic to tropical conditions. All sharks lack a swim bladder and therefore must generate lift either by retaining large amounts of low-density lipids (hydrostatic lift) or by generating flow of water over their fins (hydrodynamic lift). In spite of the lipids reserves, the majority of sharks are negatively buoyant and sink if they stop swimming (table 1).

Being forced to swim continuously to generate hydrodynamic lift, sharks are faced with choices regarding their swim speed. As the swim speed increases, so does the metabolic cost, and the probability of a successful encounter with prey. In all cases, sharks—as other predators—probably select the swim speed that maximizes the difference between the energy obtained from prey and the energy spent searching for it. This speed depends on morphology and buoyancy, each affecting the hydrodynamic resistance, as well as on body temperature, which affects the basic metabolic rate [2,3]. Most species of sharks are ectothermic, so variations in body temperature reflect variations in the water temperature the shark resides in.

With a few exceptions, sharks evolved having similar (fusiform) basic body shape, but with considerable differences (some of which are ontogenetic) in the relative size of fins, relative body diameter and the amount and composition of lipids retained in the body [4–9]. In this study, we suggest a unified theory (theoretical framework) that can relate some of these differences with particular lifestyles and habitats, and can explain some of the ontogenetic differences as direct consequences of allometric scaling laws of swimming performance. It is based on general predictions of energetic costs of activity in sharks and swimming speeds that minimize these costs, and specific predictions of the influences of the most conspicuous morphological parameters, buoyancy and temperature on the energetic costs and on the respective optimal speeds.

The theory is presented in §§3 and 4; its few immediate conclusions ensue the developments of §3.6, 3.7, 4.7 and 5; overviewing discussion concludes the paper in §6. The data used in the analysis are presented in §2.

2. Underlying data

The ideal dataset for this study would have included tracking data (speed, depth, body temperature, water temperature and salinity), along with the respective morphological data (length, girth, fins dimensions), and in and out of water weights, for many individuals of different species. At present, no such dataset exists. The set compiled for this study (electronic supplementary material, S1, table S2) included 58 individuals from nine species of morphologically similar requiem sharks: Carcharhinus obscurus, C. leucas, C. plumbeus, C. brevipinna, C. limbatus, C. falciformis, Negaprion brevirostris, Galeocerdo cuvier and Prionace glauca, for which in and out of water weights were reported in [7,9]. Morphological data for these individuals were estimated based on relative dimensions reported in [4,5,10]. Hydrodynamic data were estimated from morphological data, using aircraft preliminary design tools [11] (electronic supplementary material, S1). We could evaluate the accuracy of these estimates, using wind tunnel measurements at relevant Reynolds numbers (electronic supplementary material, S2); they were accurate to within a few per cent.

3. Fundamentals

3.1. Lift and drag

Consider a negatively buoyant fish swimming at constant speed along a straight path, inclined at angle γ relative to horizon (positive when ascending). ρ, v, g and m are density of water, the swimming speed, the acceleration of gravity and the displaced mass of water, respectively. The latter can be expressed as

3.1

where l and S_b are the (pre-caudal or fork) length of the fish and its maximal cross-section area and k_m is the prismatic coefficient—the ratio between the volume of a body and the volume of the minimal cylinder enclosing it; k_m ranges between 0.5 and 0.6 for most fish.

When swimming at constant speed, the hydrodynamic lift L and thrust Tcounterbalance drag¹ D and weight W

3.2

and

3.3

Hydrodynamic lift and drag are commonly expressed as

3.4

and

3.5

where S is an arbitrary reference area, and C_L and C_D are the lift and drag coefficients. We assume that the lift is contributed mainly by the pectoral fins,²and therefore, the lift coefficient depends mainly on the angle between the lifting surfaces (pectoral fins) and the flow (figure 1a). We also assume that the drag coefficient depends mainly on the lift coefficient with

3.6

where C_D0 is the parasite (zero lift) drag coefficient, and

3.7

is the induced drag coefficient. Here, b is the span of the pectoral fins, is a numerical factor accounting for increased flow separation from the surface of the fin with increasing angle of attack, for non-elliptical distribution of lift along the span and, to some extent, for the lift generated by other fins. C_D0 depends on the geometry of the shark and (weakly) on the Reynolds number.³ When the reference area is chosen as the cross-section area of the body, typical value of C_D0 for a 3 m shark swimming at 0.7 m s⁻¹ is 0.17 (figure 1b); typical value of is 1.5. The product

3.8

is referred to as the ‘drag area’. It has the advantage of being independent of the choice of the reference area. A particular case of (3.8) is .

Submerged weight of the shark, W, can be expressed in terms of the excess density parameter, β

3.9

For most sharks, β varies between 0% and 6% (figure 2).

In combination with (3.4), the balance of forces in the direction normal to the direction of swimming (3.3) can be used to define either the lift coefficient

3.10

needed to counteract weight at a given swimming speed, or the swimming speed

3.11

needed to counteract weight at a given lift coefficient. Similarly, in combination with (3.5) and (3.10), the balance of forces in the direction of swimming (3.2) can be used to define either the thrust needed to sustain speed or the sustained speed for a given thrust. Note that when descending idle (T = 0),

3.12

by (3.2) and (3.5).

3.2. Active metabolic rate

Active metabolic rate is defined here as the total amount of ATP used by the fish per unit time

3.13

It comprises the standard metabolic rate, P₀, and the cost of activity, . η_m is the chemomechanical efficiency of the muscles (the mechanical work done per mole ATP) and η is the hydrodynamic propulsion efficiency. Both efficiencies are assumed independent of the shark's morphology and swimming conditions; their typical values are 24 J per mmol ATP [18] and 0.7 [19], respectively.^4,5,6 P₀ is approximated with

3.14

where τ is the absolute body temperature, and k_P, α and k_τ are certain phenomenological parameters. Their typical values are 127 mol ATP per s·kg^α, 0.8, and 5020°K, respectively [23], but there can be interspecific differences in these parameters [24].

In a glide, T = 0, and the active metabolic rate equals the standard metabolic rate, P₀. In what follows, however, we assume that the shark swims at constant depth and speed; consequently T = D by (3.2), and

3.15

by (3.6), (3.5), (3.10) and (3.11). Equation~(3.15) can be rewritten as

3.16

where

3.17

and

3.18

are a pair of characteristic velocity scales; their physical meaning becomes clear in §3.3. The ratio u/w is a variable parameter but, in general, can be considered an order 1 quantity (figure 3).

If there were no constraints, then the minimal active metabolic rate,

3.19

would have been obtained at

3.20

the solution of the equation

3.21

For a neutrally buoyant fish (β = 0), v₊ = u = 0 and P₊ = P₀. For a non-neutrally buoyant fish (), the minimal speed is typically limited by stall of the pectoral fins, and minimal active metabolic rate is obtained at the lowest swimming speed, v_min rather than at v₊ (see section 3.6).

3.3. Cost of transport

The cost of transport C is defined as the energy used per distance travelled

3.22

Substituting (3.16) it takes on the form

3.23

Minimal cost is obtained at the swimming speed, say v_*, at which

3.24

This condition leads to the equation

3.25

Its solution is

3.26

when u → 0, and when w → 0. It lacks a closed-form analytical solution for in between these two extremes, but interpolating formulae in the last two rows of table 2 offer very good fits (figure 4). For future reference, we note that . This conjecture is apparent in figure 4a; it can be obtained formally by rearranging (3.25) as or as . Thus, w and u are the speeds that minimize the cost of transport in the limits when the buoyancy (and hence the energetic cost of generating hydrodynamic lift) is very small and when the basic metabolic rate (the energetic cost of living) is very small, respectively. Typical values of can be found in electronic supplementary material, S1, table S2b.

The minimal cost of transport and the respective metabolic rate are

3.27

and

3.28

by (3.22) and (3.23). Because lacks a closed-form expression relating it with uand w, so does C_* and P_*. They can be approximated with

3.29

and

3.30

when say, u/w < 0.6; more elaborate approximations can be found in table 2.

The terms in the parentheses on the right-hand side of (3.26), (3.29) and (3.30) manifest the difference between negatively and neutrally buoyant fishes (for which u = 0). Negatively buoyant fish have to swim faster than similarly shaped neutrally buoyant ones, and their cost of transport and active metabolic rate is higher. In fact, estimated optimal swimming speeds of C. leucas, C. limbatus, C. brevipinna and N. breviostris are up to 30% higher than what they would have been if these sharks were neutrally buoyant; respective costs of transport are up to 40% higher (see electronic supplementary material, S1, table S2b).

3.4. The speed ratio

Choosing w as a unit of speed, and the basic metabolic rate P₀ as a unit of power, all reduced performance characteristics—the minimal active metabolic rate P₊/P₀, the minimal cost of transport and the swimming speeds, and , at which they are obtained—become dependent on a single parameter, . All four increase with and hence, in many cases, energy expenditure of a shark can be reduced by making small.

Expression for ,

3.31

follows by (3.17), (3.18) and (3.14). Because , (see footnote 2) and , we may expect or, what is equivalent, ; the powers with l and m are positive (0.36 and 0.12, respectively). Typical values of u/w can be found in figure 3; they do not exceed 1.1 for all individuals on our list, and do not exceed 0.8 for the two pelagic (P. glauca and C. falciformis) and the two ‘cosmopolitan’ (C. obscurus and G. cuvier) species included thereat. can be reduced mainly by decreasing the negative buoyancy β, by increasing the span of the pectoral fins b/l, and by decreasing the mass. It can also be reduced by increasing the body temperature, but the resulting increase in the standard metabolic rate more than offsets the beneficial effect of reducing the value of .

3.5. Energy balance

If prey is uniformly distributed along the swimming path, and the energy intake of the shark is directly proportional to the amount of prey encountered en route, the energy balance of a shark—the difference between energy gained E_in and the energy spent E_out—can be expressed (with help of (3.22)) as

3.32

where X is the distance swum at speed v, T is the swimming duration and e is a certain coefficient reflecting the prey density and the probability of its capture. Minimizing the cost of transport, C, maximizes the energy gain, irrespective of e[25].

If, however, the amount of food encountered by the shark is independent of the volume of water searched during swimming, but depends only on time, then the energy balance becomes

3.33

where e′ reflects the prey encounter rate and the probability of its capture. Minimizing the active metabolic rate, P, maximizes the energy gain, irrespective of e′.

Realistic scenarios are bounded between these two extremes, suggesting that a shark probably swims between v₊, the speed at which its active metabolic rate is minimal, and v_*, the speed at which its cost of transport is minimal (this conjecture is assessed in §3.7); its active metabolic rate varies between P₊ and P_* (figure 4). The prerequisite to this analysis is that v_* and v₊ exceed a certain minimal swimming speed.

3.6. Minimal swimming speed

From a hydrodynamic perspective, the minimal swimming speed is the lowest speed at which the forces acting on the shark can be balanced. It is an immediate consequence of the existence of the upper bound on the lift coefficient (figure 1a); in fact

3.34

at powered ascent or descent, and

3.35

when gliding at angle with zero thrust; C_D,max is the drag coefficient at . Equation (3.34) follows from (3.11) and (3.17); equation (3.35) follows from (3.12). Because is invariably small relative to (figure 1), the minimal glide speed and the minimal speed at constant depth are hardly different. At the same time, the minimal speed in vertical ascent is identically zero. Typical values of range between 1 and 1.4 for all practical combinations of morphological parameters.⁷

To exploit the minimal active metabolic rate when swimming at constant depth, v₊ should exceed . It implies

3.36

by (3.34) and (3.20). This condition cannot be satisfied with any admissible set of morphological parameters (see the preceding paragraph), and no shark on our list can exploit the minimal active metabolic rate when swimming at constant depth (figure 4). Consequently, the lowest active metabolic rate when swimming at constant depth, , is achieved at the minimal swimming speed, . Given that the difference between and v₊ is small, the difference between and ,

3.37

is also small (figure 4c). Nonetheless, because , minimizing the active metabolic rate was not the evolutionary objective with any of these species.

To exploit the minimal cost of transport when swimming at constant depth, v_*should exceed . It implies

3.38

by (3.25), which after some rearrangement, can be recast as

3.39

At the same time

3.40

by (3.34); whence (3.39) sets un upper bound on the speed ratio

3.41

For the same combinations of morphological parameters as those listed in footnote 7, the right-hand side of (3.41) ranges between 0.8 and 2. The left-hand side varies with buoyancy and body temperature, as well as with basic morphological parameters (see above), and is, in general, an order 1 quantity. Consequently, (3.41) is not automatically satisfied, and buoyancy and body temperature have to be coordinated with morphological parameters to allow a shark to exploit its minimal cost of transport. In particular, because (see the paragraph following (3.31)), inequality (3.41) implies that large sharks (large l) and/or ectothermic sharks residing in cold water (small τ) must also have small negative buoyancy (small β) and/or large pectoral fins. Examples include the basking shark Cetorhinus maximus [9], the Portuguese dogfish Centroscymnus coelolepis [26], and the six-gill shark Hexanchus griseus [27].⁸ Some sharks exhibit ontogenetic increase in hepatosomatic index (proportional mass of the liver), which is inversely correlated with the value of β. Examples include the oceanic whitetip shark C. longimanus [6], the dusky shark C. obscurus [28] and the tiger shark G. cuvier [7]. Some sharks exhibit an ontogenetic increase in the span of the pectoral fins; examples include the bull shark C. leucas and dusky shark [5].

3.7. Optimal swimming speed

It was predicted in §§3.5 and 3.6 that under most circumstances, the optimal swimming speed of the shark is bounded between the larger of v₊ and , and v_*. Reliable corroboration of this conjecture is complicated by the fact that average speed measurements are commonly cited without the necessary complementary data, which includes length, mass (or girth), temperature, span of the pectoral fins and buoyancy. Moreover, many of these measurements were made immediately after having released the shark, and hence may not reflect its natural behaviour [29]. Notwithstanding these caveats, reference [30] cites voluntary swimming speeds of two bull sharks and one sandbar shark C. plumbeus in a large water tank. The bull sharks were 2 and 2.3 m long, the sandbar shark was 2.1 m long (total length). They swam in 26°C water with average speeds of 0.72, 0.62 and 0.64 m s⁻¹, respectively, accelerating and decelerating a few hundredth m s⁻¹ about these values. Referring to electronic supplementary material, S1, tables S2a and S2b, v_* for comparably sized bull sharks (sharks 5,8,13 and 15) is between 0.67 and 0.78 m s⁻¹, depending on buoyancy and morphology; v_* for comparably sized sandbar sharks (sharks 1–4,6 and 7) is between 0.62 and 0.74 m s⁻¹. For both species, v₊ is roughly 0.28 m s⁻¹ smaller than v_*.

Reference [31] cites average swimming speeds of three blue sharks P. glauca, tracked over the period of a few days (sharks 16, 22 and 23). With body temperatures of about 18°C, the three sharks, measuring 2.2, 2.7 and 2.6 m (fork length) averaged 0.48, 0.4 and 0.44 m s⁻¹. There are no comparably sized sharks on our list, but the optimal speeds can be estimated based on the same formulae that underlay table S2 in electronic supplementary material, S1. With β = 0.02, and depending on the length of the pectoral fins and body mass, they yield v_* between 0.55 and 0.6 m s⁻¹ for the two larger sharks, and between 0.52 and 0.56 for the smaller one; v₊ is 0.27 m s⁻¹ smaller than v_*.

For the two bull sharks and the three blue sharks, we predict v_min (0) between 0.13 and a few hundredth m s⁻¹ smaller than the respective v_*, whereas for the sandbar shark, we predict it is between 0.2 and 0.08 m s⁻¹ smaller. In other words, there are possible combinations of morphological parameters and buoyancy for which v_min (0) exceeds the observed swimming speed. v_min (0) is extremely sensitive to buoyancy (it vanishes with β), and hence obtaining unrealistic v_min (0) demonstrates the importance of having the ideal dataset mentioned in §2, as well as the importance of coordination between morphological parameters and buoyancy.

4. Derivatives

4.1. Preliminaries

Sustained performance of a shark is characterized mainly by the active metabolic rate, the cost of transport and the speed at which the minimal cost of transport is achieved. Essentially, there are six major morphological parameters affecting the sustained performance: length, l; span and chord of the pectoral fins, b and c₀; body diameter, d; buoyancy, β and body temperature, τ. The first three can be considered an evolutionary adaptation; the next two also depend on an individual's body condition; the last two also depend on the habitat the animal uses. Sensitivity of the sustained performance to variations in these parameters is manifested in the partial derivatives computed below.

Quite generally, if x denotes one of the independent parameters, namely β, b, c₀, d, l and τ, we can write a series of logarithmic derivatives

4.1

and

4.2

and, given v,

4.3

They follow from (3.25), (3.22) and (3.16). Being inherently dimensionless, logarithmic derivatives offer both simplicity of the final expressions and a convenient interpretation of the result. For example, the relative change in the cost of transport, ΔC_*/C_*, owing to a (small) relative change Δx/x in an independent parameter, is