Estimating Population Densities
[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?”]
In this series of posts, I develop a critique of the Malthusian argument. I think it is completely wrong and was so from the start. The internal logic is broken, and it is also at odds with the empirical evidence, even the evidence Thomas Malthus had at the time. However, this is only a negative result. One conclusion might be: The Malthusian argument is wrong, but then it is the best we have, so we have to stick with it.
To block such an escape route, I also try to develop a non-Malthusian explanation for population dynamics. Note that the two parts are independent from each other. Even if my proposal turned out to be completely wrong, that would not invalidate my critique of the Malthusian argument because I don’t rely on an alternative explanation there.
My model is still a work in progress. Although I am confident that I get the broad outlines right and also the main parts, there are some elements that are evolving. I mostly have solutions, but often they are not in the best shape yet. But then I expect that my proposal will not be perfect. It should be possible to improve upon it, and a different approach might supercede it entirely. My bar here is really low: I want to do considerably better than the Malthusian model. And what I have at this stage already wins hands down.
In this and further posts, I would like to develop certain parts of the model that can stand on their own. To show where they fit in, here is a very short sketch.
My non-Malthusian explanation is made up of three parts:
- A mechanism how a population can pursue a target size.
- A rule how a population can set such a target size in a “reasonable” way. My conclusion is that the main inputs are population density and a measure for general distress, mostly centered around nutrition.
- An explanation for population dynamics in the aggregate that work mostly via geographic expansion.
The first and the third points are in an advanced stage. As for the second, I have a solution as a “proof of concept,” but I don’t think it is yet the right one.
Noone has to believe this on faith because I say so. And I am certain that many will already balk at the very idea that something like this could be the case. The foreseeable objections should be:
- Human populations have insufficient control over their fertility (apart from modern times perhaps), and hence pursuing a target size is not possible.
- Even if it were possible, human populations could not set a target size according to the rule because they cannot get their hands on the relevant inputs.
Of course, I have to address these points. However, I think these objections are much weaker than they seem. My claim only runs against a Malthusian worldview that has become part of our culture, but is unfounded, and where all this is assumed as impossible.
As a first installment, I would like to show how a species — and I do not think here only of humans — can estimate a population density. This should be necessary to set target sizes, but also to pursue them. Of course, targeting the size itself is not possible because it is not observable. However, for a fixed area, population density is proportional to it and so it can be used instead.
My point here is that even rather “primitive” species like bacteria are able to estimate their population density. I even show how they can use it to control their population dynamics. Other species, and also humans should be even better at estimating population densities, and hence this part of the explanation is by no means absurd although it may seem extravagant to claim that species estimate their population densities.
Density Estimation by Bacteria
Bacteria are easy to underestimate because on the whole they are rather unsophisticated. They are basically single cells that mostly divide into two cells (but they also have sex, one I have read about even with three genders!). It is often stressed that bacteria can divide every so and so many minutes. The typical conclusion from this is that they must have exponential growth then. However, this rhythm for division is a potential, and it does not follow that bacteria do this on a regular basis. If they don’t, population growth can be also something other than exponential growth.
As I have explained in other posts, exponential growth is actually not possible for beings that live in two or three dimensions. This is so because it implies a population density that goes to infinity, ie. that cell sizes go to zero, or a speed of expansion that goes to infinity or both. All species can at most have quadratic or cubic growth, depending on whether they grow on a surface or in space. That alone implies that population growth for bacteria must slow down over time.
In principle, there could be pretty complicated dynamics here, but it is only necessary that bacteria stop dividing at some point, eg. those cells that are in the interior of a cell colony, and that they only begin to divide again when space becomes available because other cells die (I skip over the nutritonal situation here, which I will address elsewhere and which should, of course, also play a role.)
On the frontier of a colony you can have continuing division, but not on the interior. To achieve this behavior, bacteria only have to set a threshold for local population density where they switch further division off. To this end, they must have information on what the local population density is.
In a closely packed colony, there is a simple way. A cell need only measure its own volume. This is proportional to the inverse of population density. One way to do this, would be to produce a fixed amount of some substance per time, which is then depleted at a fast enough rate. At any time you have the constant amounts for previous times after decay, and so the total should be their integral over time. (Fast enough depletion ensures that the substance does not just accumulate.)
If this substance diffuses within the cell, the concentration is the amount in circulation, which is a constant, divided by the cell volume. So the concentration is just the inverse of cell volume up to a constant. Since cell volume is the inverse of population density, the concentration of the substance is proportional to population density with an appropriate constant. Now, all it takes is some reaction that takes place beyond some critical concentration. That can then be used as a trigger to switch further divisions off.
I have no idea whether bacteria use this mechanism. But then this is only meant as a “proof of concept” that at least one such way to measure population density is available. If there are more, so much the better for bacteria. All you need here is the capability of producing a fixed amount per time in the cell, and a process that reacts beyond a critical concentration. This seems eminently doable because bacteria must use similar mechanisms also for many other purposes.
Another way in a densely packed colony would be to register pressure from outside. Again, that appears feasible. There are certainly reactions that can be triggered above a critical level for pressure.
Now, many bacteria do not organize in such a dense way. If there is a distance between them, measures via volume or pressure from outside will not work. Cells might not even touch, and there is no constraint for their size. If cells in a way collide from time to time, though, it is conceivable that bacteria could register this. Each time a reaction would go off that, for example, releases an amount of a substance within the cell that is depleted after some time. With many hits, the concentration builds up. And if the hit rate gets too high, then it exceeds a threshold and that could trigger a halt for further divisions.
But then even that may not apply because collisions are too rare. There is, however, another way to measure population density also at a distance. The mechanisms so far are just my speculation although I would by no means be surprised if they are used by bacteria, or something more sophisticated. Keep in mind that some bacteria can also communicate via electric signals and in this way relay information over a distance. Yet, as for the next mechanism it is clear that at least some species of bacteria employ it.
There has to be some method now to sense other cells also at a distance. What bacteria can do is secrete a certain substance to the outside. It will then diffuse around it and with a continuing supply, there will be a concentration that falls off with distance, perhaps also because the substance is depleted.
Now, when you have many cells around and they all do this, the respective concentrations for the substance will add up to one concentration. Again all you need now is a threshold. Once the concentration goes beyond a critical value, this can trigger a reaction that results in no further divisions. Unlike in the solutions that work for the cell itself, this even has the advantage that the behavior of different cells is coordinated, not only between adjacent cells, but also at a distance.
The reason I announced that what bacteria do here is pretty sophisticated is that this is a well-known statistical technique: kernel density estimation. That works exactly in the same way. Each cell is a data point that is drawn from an unknown distribution. Then you smear that out with a function, a kernel, over space and add the results for all data points up to get an estimate. The kernels here are the concentrations around the cells. Addition happens all by itself.
Bacteria may or may not use the exact levels as inputs. If control over population growth only needs a switch between regular division and no division, then all it takes is a threshold. But it is also conceivable that the finer information is used to modulate the time between divisions. Then you would have better control over population dynamics. The same is also possible for the other methods above. However, in a tightly packed colony, the choice might be stark: keep dividing or stop.
If you think this is sheer speculation on my part, check these articles on Wikipedia out with further information on what bacteria can do. They can use these mechanisms also for other purposes:
And if you don’t know about kernel density estimation:
Density Estimation by Locusts
Species with much lower population densities can not draw on methods that require direct contact or register pressure from without. Likewise methods that work via measuring one’s own body size will not work either if that is not really constrained by other specimens and pretty much fixed anyway.
What can work is to keep track of how often you bump into others of your species, though. In a very literal sense, this happens with locusts. They have a solitary and a gregarious phase. In the former, you would not notice them, but in the latter they form huge swarms that can devastate a region. The locusts undergo a transformation when they become gregarious, and that is triggered by how often their hindlegs are touched. Each time this happens, a certain quantity of serotonin is released. Once a critical level is reached, the locusts transform, get together, and go on a rampage.
The connection with population dynamics is evident. In the solitary phase, the locusts apparently target a low population density. I don’t know how they do that, but assume that they use feedback from population density to modulate their fertility. In a variable environment, it is silly to go for a population density no matter what. It makes sense also to incorporate information on the food that is available. When there is more of it, you can afford a higher population density than when there is less of it.
However, if you have very dramatic moves from very little food during a dearth to an abundance of food when things return to normal, locusts apparently overshoot considerably with their reaction to that. In normal times, they must keep this under control.
That’s when the mechanism with the hindlegs kicks in. As it seems it would be of little use for a single locust to go on a rampage. So what now happens is that a newly gregarious locust drags others in. It eats more and raises its fertility. This leads to a positive feedback where more an more locusts become gregarious as they bump into each other. Once there are enough of them, a swarm can build up that then goes beserk.
After the locusts have done that, the population apparently dies down to a low population density again. The next generation then goes back to being inconspicuous solitary locusts. Although the gregarious phase fits Malthusian preconceptions, on the whole locusts are not Malthusian. Most of the time, they do not grow their population, but apparently stay at a low population density. The gregarious phase appears to be a solution to inevitable overshooting in rare situations.
One of the ingredients to population dynamics here is density estimation where the idea of counting hits is at the root. I assume that locusts use other methods as well in their solitary phase because the frequency with which their hindlegs are touched would not yield any usable results with few of them around.
You can find more information on the topic in this article on Wikipedia:
Density Estimation by Birds and Whales
Methods that work with immediate contact are not feasible for species that have only a low population density and rarely come into contact with other specimens directly. Here is another method such a species could use. This is only my speculation, but it seems plausible that it could work this way. My point is only a “proof of concept,” whether the concept is used or not in reality.
Two such species (or rather large families of them) are birds and whales. Both have songs they sing on a regular basis. There are many explanations why they do that. But as far as I can see noone seems to think of its use for estimating population densities.
In a way, this is similar to bacteria that secrete a substance to the outside and then measure the concentration. Here it is only sound instead. If you sing a song and others do, too, you can hear the population. The amplitude of an acoustic signal fall off exponentially with distance. This is again a kernel function around the specimen that sings. And all you have to do now is add those signals up to obtain a quantity that corresponds to population density. Since songs are only intermittent, you have to take an average over time, though. But as long as the signals come with a certain regularity that should work quite well.
However, acoustic signals are even better than concentrations. In the latter case you can register the sum, but not where it comes from. But with binaural hearing (ie. with two ears), you can also locate the direction. For animals on land, eg. humans, this is geared towards two dimensions. Identifying a source above or below you is less important and also less developed. For species in the oceans, I presume that audition is also good in three dimensions.
Now, you have additional information with directions. So a bird or a whale could in principle even make estimates for varying population densities around them, not only for an average. This would be over the space of directions, a circle or a sphere. I don’t know whether this is the case, but it seems certainly possible to do this. However, if you move around, this will not help you all that much because the space of directions moves with you, and directions in one location do not correspond to those in another.
You can use acoustic signals even better, though. Different frequencies fall off at different rates. High frequencies get attenuated fast, low frequencies only slowly. When someone has a party far away, you will only hear the bass. And that can be put to use: You can not only estimate from which direction you hear another specimen, but even how far away it is. We humans can do that, too.
So, you can now localize where a sound comes from, at least to some extent. You have to be able to keep track of it as you move around, ie. you need a mental map that you place the sound sources on. In this way, you can not only estimate some average population density where you are, but even how the population is distributed in two dimensions. Again, I don’t know whether birds or whales do this on a regular basis, but it is clearly feasible. Especially under water, sound can travel very far. A hundred miles seems a plausible range. And so you could understand the population density even at a large distance.
These more sophisticated density estimations don’t seem particularly relevant as inputs for controlling population density. But they could play a role for directed behavior. You identify niches with few specimens that you could move to. Or you might do the opposite and flock together where others are. With songs, this is doable even if specimens usually don’t get close enough to each other to learn from direct contact and observation.
I assume that songs have also many other purposes, but once I had the idea, it seemed plausible to me that one use could also be to estimate population density as an input for population dynamics.
Here is some further information:
Density Estimation by Territorial Species
The idea of measuring how often you bump into others could also play a role with species that show territorial behavior, like eg. tigers, deer, lemurs, and many others. A specimen would notice how often there are intruders that have to be driven out.
Common explanations focus on the struggle and how only the strongest survive. But they have a hard time with ritualistic behavior. Why does the winning side not just kill the losing side and maybe even devour it as another food source? If the main use is to learn about population density in the region, though, having others wander around can provide you with valuable information about population density although a specimen stays put. That would not be so if those who cannot secure a territory of their own were killed off right away.
With shifting sizes for territories, specimens can also draw conclusions from the area they can hold on to. This would be a first stab for population density, which is inversely proportional to area per capita. And then territoriality is also a direct way to target a certain population size. There are only so many slots there.
Density Estimation with Humans
I can only speculate about how we do it, but then there are so many ways that even a very precise estimation is within reach. My intuition is that observing how many people are around us plays a role. Maybe we use chatter from others like birds or whales might do with their songs. However we do it, we certainly register whether it is crowded in a place or whether there are only few people or even none. Obviously, we also track when people come too close, and that results in a reaction, even more so if we bump into them. However, population density for us is not like for locusts or bacteria where we have hardly enough space around us.
Since we can work with mental maps, we can also put all this information together when we roam around. That could then be used to calculate a rather sophisticated estimate for population density in two dimensions. With language, we can go even beyond direct observation. We listen to what others tells us and infer the population densities they are talking about. That could then go into a mental map of how the population is distributed farther away. We may not have been over the mountain ridge, but can still have an idea how densely populated it is there. And we may never have been to the city, but we have an idea about its population density. I have never visited Hongkong, but from what I have heard, I still entertain a rather clear idea of how crowded it is there compared to where I am now.
That we track population densities on a regular basis is evident from how we form judgments like that there are “too many” or “to few” people in a location. Think of someone from the countryside who visits the big city. At least the cliché is that they are totally overwhelmed by the mass of people and even disturbed by it. In turn, the city dweller who spends time on the countryside may be gripped by the horror that there is “no soul” here. And then there are extreme reactions like claustrophobia or agoraphobia where people panic because they perceive a place as too crowded or too devoid of humans.
Territorial behavior may play into this as well, which is not unknown for humans. We have a sense that some piece of land belongs to us, and we can also have a sense of how large it is. And area per capita is just the inverse of population density. With knowledge about what other people have, we can also average this out and get a better idea over a larger ambit.
Contrary to common intuitions, hunter-gatherers are not loners that roam around in small bands. They are typically organized as tribes with a size of about 1,000 people. Now, if this is pretty fixed and you understand how much territory your tribe occupies, then you also have an estimate for population density even if there is no private property in land. The number of encounters with others, be they peaceful or not, will also provide information on population density over a larger region. While accounts focus on the fighting, letting the messengers live would be of use also here.
If you bring all this information together, it should yield a pretty precise estimate, not only at one point, but even over two dimensions and at a distance. You will certainly have some estimation error, but it seems as if this could be kept at a pretty low level.
Conclusion
My point in none of these cases is to divine how exactly a method for density estimation works, only to show that there is abundant information available that can be put to use. Already supposedly “primitive” species like bacteria have means to tackle this problem and apparently do it with remarkable sophistication.
Other species display behavior that can also be understood as a means for density estimation. Songs or territoriality may serve such a purpose along with others. As for humans we clearly have a sense for population density and apparently keep track of it. Language can even help us gather information from far away.
My focus is on humans. It is hard to see how we could not do density estimation. This should not be a conscious operation where we do some math mentally, but an automatic procedure that we can use as we go along. So one of the relevant inputs for population dynamics seems to be readily available. And that looks also feasible for many other species, down to bacteria.
I will turn to estimating nutritional status in later posts. Here it is even more obvious that species can do this. Lack of food induces strong feelings of hunger and affects us in many ways. Inputs for estimation force themselves on us. We as humans have also a strong sense for when others suffer. Think of how you feel when you see only the picture (!) of someone who is emaciated to their bones. This is very disturbing and certainly something that we register.
But then the problem might not only be to know what your current nutritional status, but also what it is over time, and even more importantly how often you will suffer from critical situations. Especially when you are doing fine, you might have little information to work with. I will, therefore, develop ideas how humans and also other species can solve this problem, eg. by using proxies that provide information already before distress strikes.
