## Coral Reefs are 85% Shark?

In a recent TED Talk, Enric Sala says that before being sullied by people, a healthy coral reef stores 85% of its biomass in the form of sharks.

He shows this image of the “inverted pyramid” of reef biology:

I found this pretty surprising, as did the guy who organizes the talks, Chris Anderson. Anderson asked Sala after the talk:

Your inverted pyramid showing 85% of the biomass is in predators – that seems impossible. How could 85% survive on 15%?

To which Sala replied:

Imagine that you have two gears of a watch – a big one and a small one. The big one is moving very slowly and the small one is moving fast. That’s basically – the animals in the lower parts of the food chain, they reproduce really fast. They grow really fast they produce millions of eggs. And there you have sharks, and large fish that live 25 years. They live very slowly they have very slow metabolism, and basically they just maintain their biomass so basically the producion surplus of these guys down there is enough to maintain this biomass that is not moving…

Everything I know about sharks I learned from old Batman movies, but we don’t need much biological knowledge to see if this makes sense. We’ll simplify things to just two trophic levels – sharks and fish. If there are really 3, that doesn’t matter, because if fish are the entire bottom of the pyramid they’re 15% of the biomass, and if they’re the middle of the pyramid they’re maybe 12%, which is close enough.

The striking fact was the high ratio (about 6) between the sharks’ mass and the fishes’ mass, so let’s try to derive a formula for this ratio based on Sala’s idea that sharks have slow metabolism and don’t eat much compared to fish.

Suppose the biomass fraction of the sharks is $B_s$ (0.85 in the video) and of the fish $B_f$. The basal metabolic rate of the sharks is $M_s$ and of the fish $M_f$. “Basal metabolic rate” here means the number of calories per kilogram per day needed to maintain the same mass. The eating rates are $E_s$ and $E_f$. “Eating rate” means calories eaten per kilogram per day.

According to Sala, the sharks are just chillin’ at the same body mass, so

$M_s = E_s$.

The fish, on the other hand, need to grow, so that they’ll be more there for the sharks to eat. We can write this as

$B_s E_s = C(E_f - M_f)B_f$.

The left hand side represents the amount the sharks eat. The right hand side is the extra amount the fish eat, multiplied by some conversion factor $C$ that turns surplus calories eaten by the fish into calories for the sharks. These two equations give the ratio of biomass of sharks to fish.

$\frac{B_s}{B_f} = \frac{(E_f - M_f)C_f}{M_s}$

To get a high ratio of shark mass to fish mass, we need low shark metabolism (to reduce their appetite and not eat the scant fish away completely), low fish metabolism (which is wasted energy), high fish eating rates (to be converted to shark food), and a high conversion rate (to make shark food efficiently).

I think it would be helpful here to introduce the voraciousness of the fish, $V$, defined by

$V = \frac{E_f - M_f}{M_f}$.

This is a number like 2 or 6. A voraciousness of 0 would mean the fish eat just enough to survive if there were no sharks around. A voraciousness of 1 means they eat twice as much as they need, and a voraciousness of 4 means they eat 5 times their minimum diet. We’ll also introduce $R$, the ratio of shark to fish mass by

$R = \frac{B_s}{B_f}$.

With these new variables, the equation describing the aquatic eating habits is

$R = V C \frac{M_f}{M_s}$

We might expect 1 kilogram of fishy-fishy to use more energy than 1 kilogram of death shark because sharks are bigger and they keep their cool, except unless they smell blood in the water. (This is just the first search result for a shark feeding frenzy:)

I remember hearing somewhere that in general, biological organisms that are fairly similar (e.g. all mammals) will follow simple power laws when you scale them. Sharks are basically just big fish, so they should be on the same scaling law. We could try to create a heuristic argument for what this should be for the metabolic rate, but I’m not sure how to do that, and it would likely be wrong. Instead, I turned to wikipedia and found Kleiber’s Law, that total metabolism of the animal scales with the 3/4 power of the mass, or that metabolic rate per kilogram (which we are using) scales with the -1/4 power of the mass of the animal.

So let’s introduce a new variable, $S$, for the ratio of the sizes of the shark to the fish. Then Kleiber’s law states

$\frac{M_f}{M_s} = S^{1/4}$

This finally gives us a simple equation for the ratio $R$ of shark mass to fish mass.

$R = C V S^{1/4}$

Sala gave roughly $R = 6$, and a reasonable guess is $C = 0.1$ because the surplus food is getting eaten by fish, turned into new fish, and then eaten by sharks, and that takes a lot of energy.

How big is a shark compared to a fish? I googled this and found that a Caribbean reef shark is a big shark for a reef, and weighs up to 70kg. I’d think a mid-level predator fish would be at least 1kg, but let’s be nice and say just 100g. Then $S = 700$ so $S^{1/4} = 5$. That fills in enough to solve for $V$, the voraciousness of the fish.

$V = \frac{6}{0.1*5} = 10$

So the fish in Sala’s reef must be eating ten times as much daily as they need just to maintain body weight. I suppose this is a conceivable rate to get the food down the gut, but is it a reasonable rate to have the fishes’ bodies effectively processing all that food? A human base metabolism might be half a pound of dry mass, and a newborn baby is maybe 2.5 pounds of dry mass, so the rate that fish in the coral reefs are eating and growing is roughly equivalent to a woman who eats enough to grow a set of twins every day. You can find animals doing some pretty wild things if you look hard (or just turn on the Discovery Channel), so it might be possible. Nonetheless I find it dubious that coral reefs are 85% shark.