4.4 Nekton & Benthos

Step 1: Empirical observation

Kleiber (1932) observed that the basal metabolic rate $R$ of animals scales with body mass $M$ not as surface area ($M^{2/3}$) but as the 3/4 power. This allometric law spans over 20 orders of magnitude from bacteria to whales:

$$R = R_0 \cdot M^{3/4}$$

where $R_0$ is a normalization constant (~10 mW for mass in grams among fish).

Step 2: Why not surface area scaling?

Rubner (1883) originally proposed $R \propto M^{2/3}$ based on heat dissipation through the body surface (area $\propto L^2 \propto M^{2/3}$). However, metabolic rate is not limited by surface heat loss alone but by the distribution network that delivers resources. The key question is: why $3/4$ instead of $2/3$?

$$\text{Rubner: } R \propto M^{2/3} \quad \text{vs.} \quad \text{Kleiber: } R \propto M^{3/4}$$

Step 3: West-Brown-Enquist fractal network model

West, Brown, and Enquist (1997) derived the 3/4 exponent from three principles of resource distribution networks (e.g., circulatory systems): (1) the network fills the entire body volume, (2) terminal units (capillaries) are size-invariant, and (3) the network minimizes energy dissipation. For a branching network with $N$ levels:

$$\text{Volume} \propto M, \quad \text{Terminal units} \propto M^{3/4}, \quad R \propto \text{Terminal units} \propto M^{3/4}$$

Step 4: Mass-specific metabolic rate

Dividing total metabolic rate by body mass gives the mass-specific rate, which decreases with body size:

$$r = \frac{R}{M} = R_0 \cdot M^{-1/4}$$

This means a 1-g copepod has a mass-specific metabolic rate ~18 times higher than a 100-kg tuna ($(10^5)^{1/4} \approx 17.8$). Large marine organisms are more energy-efficient per unit mass.

Step 5: Consequences for marine ecology

The 3/4 scaling law implies predictable relationships across marine taxa. Generation time $\tau$, population density $N$, and growth rate $g$ all scale with body mass:

$$\tau \propto M^{1/4}, \quad N \propto M^{-3/4}, \quad g \propto M^{-1/4}$$

This explains why phytoplankton ($M \sim 10^{-12}$ g) divide in days while whales ($M \sim 10^{8}$ g) have generation times of decades, and why small organisms vastly outnumber large ones.

Step 6: Temperature correction (Metabolic Theory of Ecology)

Gillooly et al. (2001) incorporated temperature effects via the Boltzmann-Arrhenius factor, yielding the Metabolic Theory of Ecology (MTE):

$$R = R_0 \cdot M^{3/4} \cdot e^{-E_a / (k_B T)}$$

where $E_a \approx 0.65$ eV is the activation energy of metabolism, $k_B$ is Boltzmann's constant, and $T$ is absolute temperature. This explains why tropical marine organisms have higher metabolic rates than polar ones at the same body size.

Step 1: Power required to overcome drag

A fish swimming at speed $U$ must exert a thrust force equal to the hydrodynamic drag. The power required is the product of drag force and speed:

$$P_{\text{swim}} = F_D \cdot U = \frac{1}{2} C_D \rho A U^2 \cdot U = \frac{1}{2} C_D \rho A U^3$$

Note the cubic dependence on speed: doubling speed increases power requirement 8-fold.

Step 2: Scale drag parameters with body size

For geometrically similar fish, the frontal area scales as $A \propto L^2$ and body mass as $M \propto L^3$. The drag coefficient depends on Reynolds number, and for turbulent flow ($Re > 10^5$) typical of most fish, $C_D$ is approximately constant ($\approx 0.01$ for streamlined bodies). Thus:

$$P_{\text{swim}} \propto L^2 U^3 \propto M^{2/3} U^3$$

Step 3: Cost of transport (COT)

The cost of transport is the energy required to move a unit mass over a unit distance. It combines the swimming power with propulsive efficiency $\eta$ (typically 0.15-0.25 for fish):

$$\text{COT} = \frac{P_{\text{swim}}}{\eta \cdot M \cdot U} = \frac{C_D \rho A U^2}{2 \eta M}$$

Step 4: Optimal swimming speed

The total metabolic cost of swimming includes the basal metabolic rate $R_{\text{basal}}$ plus the swimming cost. The total COT is minimized at an optimal speed $U_{\text{opt}}$:

$$\text{COT}_{\text{total}} = \frac{R_{\text{basal}}}{M U} + \frac{C_D \rho A U^2}{2\eta M}$$

Taking $d(\text{COT}_{\text{total}})/dU = 0$:

$$U_{\text{opt}} = \left(\frac{2\eta \cdot R_{\text{basal}}}{C_D \rho A}\right)^{1/3}$$

Step 5: Scaling of optimal speed with body size

Substituting $R_{\text{basal}} \propto M^{3/4}$ and $A \propto M^{2/3}$ into the expression for $U_{\text{opt}}$:

$$U_{\text{opt}} \propto \left(\frac{M^{3/4}}{M^{2/3}}\right)^{1/3} = M^{(3/4 - 2/3)/3} = M^{1/36} \approx M^{0.03}$$

The optimal speed is nearly size-independent! This explains why both a small herring and a large tuna cruise at roughly similar absolute speeds (~1 m/s), even though they differ by 100x in mass. However, the speed measured in body lengths per second ($U/L$) decreases strongly with size.

Step 6: Tucker's COT scaling law

Tucker (1970) showed empirically that the minimum COT for swimming organisms scales as:

$$\text{COT}_{\min} = 10^{0.6} \cdot M^{-0.3} \quad \text{(J kg}^{-1}\text{ m}^{-1}\text{)}$$

Swimming is the most efficient mode of locomotion: the COT for fish is ~5x lower than for flying birds and ~10x lower than for running mammals of the same mass. This efficiency, combined with the supportive buoyancy of water, enables marine organisms to reach the largest body sizes in the animal kingdom.