Why Are There So Few Lions?

[This is part of my series on Thomas Malthus’ “Essay on the Principle of Population,” first published in 1798. You can find an overview of all my posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?” It is also a part of my related series on the Darwinian argument where you can find an overview here: “Synopsis: What’s Wrong With the Darwinian Argument?”]

While looking for research on population dynamics in biology, I have come across an article that I found interesting: “The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes” by Hatton, McCann, Fryxell, Davies, Smerlak, Sinclair, and Loreau. You can find various versions here. It was especially interesting for me because I recognized something that I had derived also in a very different way.

What the authors do is look at biomass for predators and for their prey or rather biomass per area, which is proportional for a fixed territory. Biomass is just the total weight of all such animals or plants. Mostly this is population density, but you weight larger species more than smaller species. The authors do this for various “biomes,” communities of species, eg. carnivores like lions, hyenas, tigers or wolves versus the herbivores they hunt, but also for fish or zooplankton and what they feed on. In general, biomass for prey is larger than for predators, and much larger, roughly by a factor of 10 or even 100.

The interesting question is how the biomass for predators is related to the biomass for prey. How does the number of predators go up with more abundant food for them? There are in principle three possibilities: the relationship could be superlinear, linear or sublinear. Biomass for predators could go up more than proportionally, proportionally or less than proportionally with the biomass for prey.

Apriori, the first two possibilities might seem more plausible: With more prey per area, hunting becomes easier. Predators could take advantage of this and increase their biomass more than proportionally or at least proportionally. However, the regular result that the authors find is that the relationship is sublinear, and clearly so. What that means is that the biomass of predators goes up as there is more prey, but less than proportionally so. Predators do not take advantage of the situation as they could. As the authors ask (my highlight): “From the dry Kalahari desert to the teeming Ngorongoro Crater, there are threefold fewer predators per pound of prey, which leads to the question: where prey are abundant, why are there not more lions?”

This observation also contradicts typical theories how such biomes function. For example, the assumption in a Lotka-Volterra model is that predators will increase their numbers relatively with abundant food. Since there is a certain lag, this should lead to oscillatory behavior: more predators then reduce the prey population and that then induces a reduction for predators with another lag. With fewer predators, the prey population eventually recovers, which again starts the cycle. But then, the authors do not see evidence of such oscillations in the data, biomass for predators and for prey is rather stable.

The finding is also at odds with any other theory that ties the biomass of predators to that of their prey as in the case of Lotka-Volterra model. It could be so that the biomass of prey keeps the biomass of predators in check or it could be the other way around. However, a sublinear relationship goes exactly in the wrong direction: With more abundant food, predators do not take advantage as they could, and if at all, control on the biomass of prey relaxes. That does not look like prey is kept in check by predators.

What the authors find in a phenomenological analysis of the data is that the relationship appears to be a power law with an exponent of about 0.75. That means: as the biomass of prey p rises, the biomass of predators increases like p⁰.⁷⁵ (sorry, Medium places the dot on the bottom). An exponent greater than 1 would mean a superlinear relationship and an exponent of 1 a linear one. Interestingly, the exponent seems to be almost the same across a variety of biomes.

One possible idea would be that this is related to another scaling law: Kleiber’s law. That is the empirical finding that the metabolic rate of an animal often seems to increase as the 0.75th power of its mass (the exact exponent is contentious and whether this is a power law at all). So, one explanation could be that predators just grow bigger with more food. If that were so, you would find a similar scaling also for biomass. However, that cannot be the case. As the authors show, average size does not go up with more abundant prey. And then larger predators could also grow their population with more food around.

I am skeptical about the analysis that is supposed to establish a power law. To estimate the exponent, the authors work with logarithms for both quantities. That should turn a power of biomass for prey into a linear funtion, and its slope is then the exponent. Unfortunately, this approach is known as dangerous. There are better ways to go about this. As the authors let on, they are aware of this, and the results seem to hold up also with a more sophisticated analysis. But from a statistical standpoint that is not perfect to say the least. It is also a very weak argument that others use the same method.

But there is also another reason why I am doubtful: I have a very simple explanation that can account for the phenomenon, and also shows why it is not a power law.

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Unlike the authors, I was not surprised by the result that there is a sublinear relationship between the biomass for prey and for predators. This is only suprising if you expect species to always max out on any opportunities they have. Basically, that is the Malthusian assumption that forms the basis for Darwin’s “On the Origin of Species.” I guess that’s why you are baffled when the claim is roundly refuted because it also means something is wrong there.

But then my conclusion for humans is that we don’t do that. It is a common observation that population density and real incomes go up together. This is quite similar to predators and prey, only that we often produce the prey ourselves. And it can only mean that while we humans increase our numbers with opportunity, we do it less than proportionally.

In this context, I also derived a relationship that I would expect in such a situation, which goes along these lines:

For humans, but also other species, the relevant part is not the average amount of food we have. There is only a narrow range where we can make use of it. If we fall short only by a little over time, we starve to death, and if we have somewhat more, we can at best gain some weight, but that is limited. To observe a population over the long run means that it had enough food on average. That never really changes. People in developed countries today basically eat as much as did hunter-gatherers 100,000 years ago, at least if you correct for relevant features like height, build, sex, age, and the activity level. Any comparisons that find no increase start from a silly assumption as if you were expecting an increase for body temperatures over the millennia.

What matters, though, is the volatility of the food supply. If there are severe shortfalls that last for too long, we are in serious trouble, and that would even be so if we had enough food on average over time. Volatility makes a huge difference here. People in developed countries these days do not have more food on average than people in poor countries or in the past, but they have practically complete food security, ie. zero volatility for their food supply. The average amount of food may be the same, but never having to starve at all is much better than going through a famine every so and so many years or in a less severe version: suffering from undernourishment from time to time.

Now, if you were a rational species, you would not only want your average food from an environment, but also a safe buffer against the volatility on top. That will keep you away from extreme problems. Basically, this also means that you have more food per capita, but only in principle. You would not try to get you hands on more than you can use, that would be silly because you would have to let it rot away. What you really want is a reserve you can tap into when things get bad.

Gregory Clark and other authors are baffled why hunter-gatherers work so little. They can do with perhaps five hours a day to feed themselves. However, when you think about it in a different way, this is totally rational. If you hunt and gather only five hours a day in normal times, you can ramp that up to ten hours when only half as much food is around, or even fifteen hours, if it falls to a third. What you see here is a reserve for bad times. And the only thing that is baffling is why people like Clark do not get the point. Hunter-gatherers are certainly much smarter than he is because they understand this intuitively. Actually, lions get the point, too.

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Now, here is the connection with population density. The inverse of population density is area per capita. So if you double your population, that means you have only half as much land per capita to feed yourself from. You can think of the land as so many food sources. On half as much land, you have half as many of them. Hence, doubling the population also halves the food sources per capita. If there is no volatility, that does not matter. With twice as much food, you can double the population. However, with volatility, that is not true.

Suppose all food sources have the same volatility and are independent of each other. Then with twice as many food sources, you have a volatility that is lower by a factor of one over the square root of two. And with half as many, volatility goes up by a factor of square root of two. Hence if there is twice as much food around, the decision to increase the population density is not only about the average. Since you have to do with only half as much area and half as many food sources per capita, you would also want a higher buffer and hence you should increase population density less than proportionally.

Actually, that works pretty well to explain the findings in the paper although I don’t think that’s the whole story. My suspicion is that populations use more abundance even better. If you have more food around, you can not only increase your population density and have a buffer that leads to the same probability of really bad events, you can also increase your population density somewhat less and lower the probability, too. And as far as I can tell, that is what humans do. We have used progress exactly in this way to move from high food insecurity to practically none, and have not only kept the frequency of famines constant.

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Yet, even with such drastic simplifications, I am able to reproduce the results in the article quite well. It is best to start with the biomass of predators and think of them as all of the same size, so this is just population density for a fixed territory. Area per capita is the inverse of population density and is assumed to be proportional to the number of independent food sources. The volatility of the food supply is proportional to 1 / root(number of food sources), which is proportional to 1 / root(area per capita for predators). And hence the volatility of the food supply is proportional to root(population density for predators) assuming a fixed and equal volatility for the individual sources.

For a certain population density, predators want a fixed minimum of prey plus a buffer against volatility per capita. This multiplier per capita is also the ratio of population density for prey to population density for predators. Hence we have with some constants C and a:

C * density prey * ( 1 — a*root(density predators) ) = density predators

Or:

C * density prey = density predators / ( 1 — a*root(density predators) )

The lefthand side in the first equation is what predators care about, namely a minimum of prey even under unfortunate circumstances. If volatility did not play a role, ie. a=0, then the density of predators would always be proportional. But with volatility, you have to correct that for the volatility of the population density of prey (which I assume as fixed and equal for all food sources) times something that is proportional to the root of the population density of predators because they have fewer food sources per capita with a higher population density.

With the formula, you can calculate the population density for prey for a given population density for predators. For small values of a approaching zero, ie. a small buffer for predators or low volatility for the population density of prey, the equation approaches a linear function:

C * density prey = density predators

Hence, the “exponent” would be close to 1. Here is what that looks like for a small value of a with logarithms for both densities over three orders of magnitude as mostly in the article (horizontal axis for prey, vertical axis for predators, German settings: points and commas the other way around):

But for larger values of a, we get this, which is pretty much the result in the article:

It is easily conceivable that you could mistake this functional form for a power law with some noise on top. But that would be mistaken. For still larger values of a, you namely obtain this:

As there are so few food sources that ( 1 — a*root(density predators) ) approaches zero, the multiplier for the density of prey escalates to infinity because it is the inverse of this. That means even with more food available, the population density of predators will no longer grow because you would be stuck with too few food sources, that make things dangerous for you even with a lot of prey around in normal times. (In addition, there is no restriction in my calculation against fewer than one food source, which is absurd in reality.)

The three graphs are actually for the same function: on the left, in the middle, and on the right. I only focus in on different parts by shifting the parameter a around and that also expands or contracts the scale on the horizontal axis. But the main story here is the second case. My hunch is that you will find actual populations in a stable biome mostly in the middle of the curve, especially if you average out a lot. Then you should end up with an “exponent” close to 0.75. And as is obvious, we don’t have a power law here. It only looks that way over a certain range, but that is not a general relationship.

I have no idea whether my explanation is correct, but it is quite simple, has a stringent logic, and leads to results that are otherwise inexplicable. If so, then the conclusion has to be that predators have control over their population size and manage it in a very sensible way. Their population targets are responsive to the availability of food, but not in the way that Malthusians think it works. Predators do not take advantage of more abundant prey to the max. Instead they pursue a reasonable population size that tackles the really important part: not average food per capita, but the risk of falling short because of volatility. And since less area per capita with higher population density increases risk, you get a sublinear relationship, and even with almost the exact “exponent” as in the data.

Since the setup is the same in all such situations, you should also find the same result across very different biomes. It does not depend on the specifics, only on the general relationship between the number of independent food sources and the effect on volatility on the one hand, and on area per capita and hence population density for predators on the other. This should be a general observation because behind it there is the general statistical relationship that volatility goes down with the number of food sources like one over the square root.

My assumptions in the derivation were certainly unrealistic: food sources all fluctuate with the same volatility and are also independent. Spatial autocorrelation or correlation in time are very plausible here. That would decrease the effective number of food sources. There should also be different food sources that contribute more or less to the food supply, and all have their own distributions. However, my guess would be that you can wiggle around quite a bit and still obtain a similar result. And then with a whole biome a lot of averaging goes on.

As noted, my impression from human populations is also that there might be even more to it: More prey not only goes into the same average plus a higher buffer, but an even higher buffer that lowers the risk of shortfalls. Human populations have not grown to a level where they always have the same risk of famine, but have used improvements to lower that risk, too, by now practically to zero. Yet, I guess the result would survive that as well.

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PS

In the first version, I had a stupid mistake. Instead of multiplying with ( 1 — a*root(density predators) ), I effectively divided by it and multiplied the other side with the factor. I hope I get it right now. Somewhat unsure, this is a pretty fresh insight.