Carrying capacity is "ugly"???

Brendan Maher is the editor of a recent piece in Nature with which I dispute. In the comments to that piece, Maher says:
The "carrying capacity" of the earth (ugly term) is not some fixed finite value, but one that shifts with human behaviour and ingenuity.
Is carrying capacity an "ugly term"?

Before taking a hard position about what is or is not ugly, it might be a good idea to know what it is that we are talking about. Perhaps, once we know a thing or two about carrying capacity we can even be able to address a few simple questions:

Carrying capacity is a term that is commonly used in population models, and other things... This is one instance where the models are sufficiently simple so that you don't have to be some sort of specialist to get the basic idea. So let us begin.

Start by considering a simple banking model. Say that your bank account contains Pn pennies in the year 2000+n. Wisely invested with some annual interest rate, the next year your account will be worth Pn+1=(1+g)*Pn

We say that g is the annual growth rate. So long as growth rate is greater than 0, you will have more pennies with each passing year. This is a nice model because each years growth also grows the growth of previous years!

We call that geometric growth (or compound interest, if that's your pleasure). The geometric equation seems so humble in its simplicity. But there is nothing humble about geometric growth. As time goes by, P becomes very large. Bankers love this equation! Governments extort a cut, so they like it too...

Prevailing economic thought says: "Geometric growth is so absolutely beautiful, why would you want any other model?"

Well, money isn't everything. The money may grow to infinity, what that money can buy does not.

The P in geometric growth could be the number of people on earth... What happens as that number grows to infinity?

Geometric growth is God's primary directive: "be fruitful and fill the earth". Or at least according to the God of Jews and Christians and Muslims. There are a lot more of them than me. They must be right?

Here's there is the catch, God's directive has a second part, "fill the earth". Thinkers agree, there must be a limit to growth. How do we get to an understanding of that limit?

One way (there are many others) is by an indirect calculation that goes like this. For a given planet, in a given solar system, for a given species, with given capabilities, in a given ecosystem, we might suggest that there is a maximum population max(P)=C which we call the carrying capacity. Thus the geometric equation is modified to become the logistic map
Pn+1 = (1 + g*(1-Pn/C))*Pn
The modification amounts to transforming the maximum growth rate g into a grow rate g*(1-Pn/C) that declines to zero as Pn approaches C and even becomes negative (a loss rate) when Pn exceeds C... This isn't the only way that growth rate can be brought to zero as population increases, just the most straightforward way. It isn't difficult to reduce the above logistic map into its standard form but the above form is easier to interpret because it is expressed using real-world quantities.

The value of mathematics is that it gives us a formal way to relate ideas. The logistic map captures the essence of many ideas and shows us how one idea is related to another.

The solution to this equation has some interesting properties:

So what the heck does all that mean for humanity? Let's mostly look at the matter on a global scale, as though we truly did live in a global village (we don't).

First we notice that the value of g depends upon the length of the time step from Pn to Pn+1. As the time step increases, so does g increase. For a small, isolated population of hunter gatherers it would make sense for the time step to be a generation, or perhaps even a life time.

When we are considering a modern global population it would make more sense to view humanity as an almost continuous sea of people. In that case we would take the limit that the time step is very small. (In the limit of infinitesimally small time steps, geometric growth becomes exponential growth and the logistic map becomes the logistic differential equation.) Thus the effective value for g is almost certainly bigger than 0 and less than 1. It follows that:

Many have suggested global disaster. Global starvation will never happen, unless it is caused by something in addition to high population. Certainly starvation will happen locally (because locally we have bigger effective values for g). Starvation always has happened locally, throughout history.

This should not be viewed as a repudiation of things that people like Paul Ehrlich have to say. A local catastrophe is still a catastrophe and Ehrlich has a caring message: In order to combate the scourges of poverty and warfare, in order to promote freedom and well being, in order to protect the environment and other species, there is one thing that must be achieved above all else. We must achieve a stable human population that is well below carrying capacity.

Regardless of overshoot, for the majority of people to live satisfying lives it is necessary to achieve a stable human population that is well below carrying capacity. It should be intuitively obvious that a maxed-out human population will not be a satisfied population.

God has a bad attitude towards humanity: He wants to "fill the earth" and that causes misery.

I care too much to be sanguine about human population stabilizing at or near the carrying capacity. Throughout the history of civilization, population has usually been close to carrying capacity. Mostly civilization has not been nice for most people. (The record of suffering is probably inadequate because history is mostly written by and for a privileged few.)

Nevertheless, there have been brief times when carrying capacity was greatly increased. Internal combustion engines mark the most dramatic example. So yes, human intentiveness does change carrying capacity. In the case machines powered by petroleum, that increase in carrying capacity will peak and then decline as oil becomes less available. (Global per capita oil availability peaked in 1980-81.) In the meantime, population has grown rapidly towards the new carrying capacity. Optimism and "boundless" growth of the 1950's has become a stalled global economy. Egalitarian ideals and trust that were fostered by the good times are now being swallowed by heavy-handed laws, fundamentalism, and the great sucking sound of global corporatization.

When Brendan Maher approvingly comments that:

Birth rates are going down throughout most of the world and that decline tracks very closely with education, prosperity and social justice.
he is confusing cause and effect. Setting matters straight requires us to answer two questions: (1) Why was population growth rate so high in the 1950's? (2) Why has it falling now? Then we might be in a position to know the condition of our condition.

In a nutshell, the answer to the first question is that the population growth rate was high in the 1950's because there was room for it to grow. The logistic equation can be used to take us through the details. The growth rate became high in the 1950's when widespread application of new technologies vastly increased carrying capacity. Thus the ratio Pn/C became small and so the growth rate g*(1-Pn/C) was close to its maximum value g.

The short answer to the second question is, because population grew to fill the room. As population grew towards the new carrying capacity, the effective growth rate fell according to g*(1-Pn/C) . The fact that the growth rate has slowed in the early part of the 21st century is simply a statement of the facts that population has grown as the logistic equation says it will grow and that carrying capacity has stalled.

Brendan Maher observes that falling growth rate coincides with increased education. (Or at least increased numbers of people are being educated. Standards are a quite different matter.) One wonders why an education would lower growth rates? After all, Brendan Maher (and Megan Scudellari and Joel Cohen and Nicholas Eberstadt) are all highly educated and they all think that overpopulation is a myth! (Was Joel Cohen hoodwinked by Scudellari?) What's more, most of the professors of business and economics propagandize for piles more population growth! They stuff nonsense into the empty heads of their mediocre students. The irony makes me wonder about unintended consequences...

There are sensible and honest intellectuals in the education system. Hardly anyone studies physics, "It's too hard". Overall, one would have to concede that people are "educated" to breed.

Nevertheless, more education (not necessarily better education) is associated with a lower rate of population growth. Again, we can use the carrying capacity concept to get a handle on why this would be so. That model relates growth rate to population and carrying capacity g*(1-Pn/C) . Parents and students and taxpayers have to invest a lot of capital in order to get a modern education. The effect is to decrease the effective value of C (which slows growth rate) because the cost of providing for each individual is increased. So yes, human behaviour can change carrying capacity and that is another two-edged sword. As Colinvaux explained: the human breeding strategy is for parents to have as many children as they think that they can afford. Education is just one of the many things that makes having children expensive. Strangely, the modern fashion is to rebadge rising costs as "affluence".

"Hang on a minute," some say, "education might help find ways to increase carrying capacity". Indeed, it might, and has. Nevertheless, everything saturates, even the utility of innovation... There comes a time when ever more education of ever more people will result in the costs outstripping any increase in carrying capacity. Most knowledge can not translate into increased carrying capacity. The laws of physics and ecology and evolution scream to us: "Beware the boundedness of knowledge that is useful for increasing carrying capacity".

Of course I have only presented a tiny fraction of what there is to say about carrying capacity. Mathematically I would call it beautiful. Frankly, I don't much care that Brendan Maher deems carrying capacity to be "ugly". I only wish that he would make the effort to know what it is that he is talking about.

I don't know Brendan Maher so I can only guess his reasons for thinking carrying capacity to be "ugly". My best guess is that he does not like the idea that people are bounded. (Perhaps Maher thinks we are exempt from the laws that apply to all the other animals in nature?) Or maybe he just holds carrying capacity in contempt because it is so hard to know what exactly it is that determines carrying capacity.

The lesson that the logistic map tells us is that we don't need to know what exactly determines the carrying capacity. We only need to know that boundedness exists. Indubitably, the mathematics shows that human population adjusts rapidly towards carrying capacity. Further, we could use population data and the logistic map to get a fairly good estimate of carrying capacity. But what is more interesting to calculate is how close we are to carrying capacity.
Pn/C = 1 - (Pn+1/Pn-1)/g
This is the number that the human breeding strategy pushes close to a value of 1 whenever carrying capacity remains constant for sufficent time. (The closer to 1, the less satisfying life becomes for the majority of people.)

And so we see that talking about carrying capacity can only ever be a means to an end. The thing that we really need to talk about is the human breeding strategy. Maher doesn't want to talk about it.

I do want to talk about it because I'm both realistic and optimistic. Enough of a realist to accept boundedness. Enough of an optimist to think that it may be possible for a semi-intelligent primate to rise above the Darwinian breeding strategy. The outcome would be a much smaller human population living much more satisfying lives.

Such a happy outcome cannot be achieved in my lifetime. If the likes of Maher prevail, it will never happen.


The logistic map captures the essence of many ideas

We could also write the logistic map as
Pn+1 = (1+g)*Pn - D*Pn*Pn
to represent growth with a density dependent death term. The essence of many types of saturating processes can be captured using the logistic map. See how many you can think of. If it's less than 100, you're not trying.

Talking about the big picture: The logistic map lead Feignbaum to constants that are universal to a broad class of systems.

The infinitesimal limit

Undoubtably mathematicians can be a pain in the arse, sometimes. Above I keep going on about geometric growth and the logistic map and yet most readers are probably more used to hearing about exponential growth and the logistic equation. OK, perhaps not the logistic equation.

I begin with geometric growth
because it requires no calculus. Understanding exponential growth does require calculus. Of course, there is a lot to be said for just describing exponential growth, as Al Bartlett did so very well and so patiently for such a long, long time. Here we are modelling, so we need to get under the hood of the exponential function.

There is a close relationship between geometric growth and the exponential function. Remember that Pn is just the population P(t) at some time t. A short time later the population is Pn+1=P(t+dt) where we use dt to represent a very short time. Thus the equation for geometric growth can be written as
P(t+dt) - P(t) = g*P(t)
Remember how I said that g depoended upon the time step? Well, for small times it is proportional to the time step so we can make the substitution g=a*dt and the equation for geometric growth becomes
P(t+dt) - P(t) = a*dt*P(t)
If we divide both sides of the equation by dt and take the limit that dt approaches zero (becomes infinitesimal) then we obtian the differential equation:
dP/dt = a*P
where dP/dt is called the derivative of population with respect to time. The solution to this equation is the exponential function that with which everyone is so familiar
P(t) = P0 * ea*t
For our purposes, we can think of geometric growth and exponential growth being the same thing (to a pretty good approximation).

A similar thing can be done for the logistic map. In the limit that the time step becomes very small the logistic map gives us the following logistic differential equation
dP/dt = a*P(1-P/C)
The logistic equation has a solution that always goes quickly towards the carrying capacity. It behaves pretty much like the logistic map behaves when g is greater than 0 and less than 1. (That's because taking the Calculus limit is pretty much the same as requiring g to be small. No mystery here!)

So when Al Bartlett talks about exponential growth, it's pretty much the same as me talking about geometric growth. The reason I go "geometric" is because I get into modelling underpinning mechanisms and I don't want to burden my readers with Calculus. Al Bartlett mostly restricts himself to describing the exponential function (and geometric growth) so he doesn't have to worry about burdening his audience with differential equations.

Global catastrophe

An abrupt climatic event might cause global famine. The year without summer (1816) was sufficiently abrupt to reduce the carrying capacity of the planet and thereby be catastrophic. The overshoot was short lived, however. Anthropogenic climate change is nowhere near so abrupt and is in the opposite sense (warming) so this is not likely to generate an overshoot. Nevertheless, global warming is coincident with the rise of population towards carrying capacity so it is associated with increasing human misery: If the UN and CBC are to be believed?

The most likely way in which global overshoot could happen would be by a total breakdown of globalization. Presently the carrying capacity of the planet is very much dependent upon only a handful of nations that consistently produce food surplus to their requirements. That highly localized production also depends very much upon highly localized extraction and processing of fossil fuels elsewhere.

Anything that results in our many human ecosystems becoming more isolated from each other will increase the effective value of g and this could also be a pathway to what would effectively be a population overshoot. A breakdown of globalization may be accompanied by the breakdown of nations into splintered states and that would further exacerbate matters.

History throws up numerous examples of global-local effects. The Roman Republic depended on local food production and many defeats by determined and powerful enemies only served to refine Roman methods. The Western Roman Empire came to depend upon the granary in far off North Africa. When a rag tag bunch of Vandals captured North Africa, the Western Roman Empire was doomed.

I wonder why Rome lasted as long as it did? I expect that the way in which modern technology makes globalization more complete now than it was in Roman times will also ensure that the our modern empires will last less long. There is a paradox in globalization. It is only the linkages that become globalized. Production becomes broken into smaller parts and production of each small part actually becomes less widespread and more geographically localized. It is not cost-effective for everyone to grow their own eggs.

The two-edged sword

There are many ways in which human behaviour can change carrying capacity. At the national level, we see it most clearly in the USA. Fifty years ago the USA was on a trajectory towards stabilizing at a much lower level than its present population. What happened? In a word, Americans changed their behaviour in ways that increased C but decreased the well being of most people. The change came partly from within: like lobbying for cheap labour, which is to say more people living on less. All the time the adverse part of GDP grows (uneconomic growth) giving an illusion of wealth. "The American Dream" has become "The American Farce".

Similarly, moving humanity to a vegan diet might increase carrying capacity. I say might because there is much more to carrying capacity than the food we eat. This changes nothing so long as we do not change human breeding behaviour. The population rapidly climbs to the new carrying capacity. Nothing gained. A lot will be lost because we are omnivorous. Diet, like everything, has a sweet spot when it comes to human well-being. We would all do well to prevent diet-extremists from imposing upon us.

I did observe, above, that population overshoot would not happen unless there was something else going on. Well, one of those "something else" things is human behaviour. Presently carrying capacity is greatly augmented by the combustion of fossil fuels. If the international community imposed a cultural shift that abruptly turns off the supply of fossil fuels then that would dramatically decrease carrying capacity. That would put us into a position of missive population overshoot!

On the other hand, if carrying capacity declines slowly as the supply of readily available oil slowly runs down, then the overshoot will be far less dramatic. Those versed in calculus might like to do the calculation themselves.

Defining overpopulation

If we define overpopulation as being higher than carrying capacity, perhaps it is a myth. But that would also be a stupid definition.

My definition of overpopulation is a population higher than that which optimizes well being for the majority of people.

Given that the human animal is a small-group animal, we can safely say that when we impinge significantly upon the well being of other species, we are well past being overpopulated.

Others, like Megan Scudellari, define overpopulation as:

"Fears about overpopulation began with Reverend Thomas Malthus in 1798, who predicted that unchecked exponential population growth would lead to famine and poverty."
Actually, Malthus wasn't just predicting some poverty and famine, rather he was also explaining how the poverty and famine of his time was caused by overpopulation. In the true spirit of Malthus (not the Scudellari distortion) I would say that a good functional definition of overpopulation would be the existence of poverty. (Famine being a type of poverty.) There is plenty of poverty today. We can safely say that the world is overpopulated.

Was Joel Cohen hoodwinked by Scudellari?

In the comments on the Scudellari article, Joe Bish of the Population Media Center wonders:
One is left to wonder if Mr Cohen was aware which 'myth' Ms Scudellari's article purported to bust when he agreed to be quoted as supportive of her view. There were only one billion of us a mere 220 years ago; by the end of this century, there will be some 11 or 12 billion people struggling for survival in a planet ravaged by climate change and resource depletion. Yet even today the issue of overpopulation is not openly discussed and as such does not feature in the decision-making of couples when considering how many children to have."
If he has been misrepresented, Joel Cohen needs to pipe up.

The wonder of unintended consequences

Are unintended consequences a law of nature? For example, while education can help to generate knowledge, it also empowers those who market anti-knowledge. From the trends that I have observed over my lifetime, I gather an impression that knowledge will saturate before anti-knowledge. Perhaps that is also something that might be viewed in the context of carrying capacity?