Mathematical musings on logarithms, population growth, and sustainability
This is another post on sustainability issues inspired by the recent UN report on global population. Sometimes a little high school math can be helpful, and this is one of them. Here's what I'm talking about:
(Data are taken from here and updated from here.)
The top figure shows the estimated world population from 12000 years ago to the present; the bottom figure looks at the last 2000 years. I would not put too fine a precision on the first 8000-10000 years of that, but even uncertainties on the order of a factor of 2 or 3 would not really change anything. Populations remained on the order of a few million or so for several thousand years, until around 2500 BCE the effects of agricultural and urbanization caused a growth in population. The population steadily rose until by 1850 it reached ~1 billion, possibly a bit less. In the last 200 years it has risen to 6.9 billion. This type of behavior -- slow rise at first, then a tremendous increase looks a lot like exponential growth.
Mathematical asides
Exponential growth is described generally by the function y = nx, where n is the exponential base. Sometimes you will see it as y = nk x, where k is a scaling factor to relate the exponent to a real-world physical parameter x, such as time or distance. Essentially what we're doing is multiplying the base n x times, so clearly the number gets very large very quickly. Even if we tone it down by using a small k, eventually the value of the function shoots up to infinity, far faster than any power law (polynomial) term, which goes as y = xn. The difference is that since n is a constant, x to the n is smaller than n to the x for large x, and the difference between xn and nx gets rapidly larger and larger as x gets larger. Unless the exponent is negative, in which case nx approaches zero faster than xn.
The base n can take any value, common ones are 2 and 10, but there is a special base for which the slope of y = nx at x = 0 is equal to one. That n is a transcendental number = 2.718281828... and these days is denoted by e. The function y = ex = exp(x) comes up so often in nature that when we say "exponential function" this is the base that we usually mean. e has a number of interesting properties and representations that not only are common in nature but come up so frequently in science as we know it that it is considered to be one of 5 special numbers, all of which are involved in the equation ei π + 1 = 0.
Let's rewrite x = kt: here t is the real world parameter time, so k is the growth rate (or, if negative, the decay rate). Many, many real world systems show exponential growth or decay. This is a consequence of the fact that for such systems the future state depends on the present state. Growth that occurs independent of the population size might go as kt (linear in time); growth that depends on the population size goes as nkt. Growth in the exponential case might be slower for small t than in the linear case, but eventually it catches the linear (or power law, if it comes to that) and growth in the exponential case becomes much much faster for large t.
On the left panel we have three different functional forms of growth, each with a different growth rate k. The blue line is linear growth, with k = 2 million. The green line is the power law (or polynomial) xk, where here k is only 6. The red line is exponential growth, ek x, but here k = 1. Exponential growth is slower to take off, but inevitably far surpasses linear and power law growth (if allowed to continue), even if the growth rate k is much much smaller. We tend to think linearly: power laws are difficult to get our heads around, exponential growth more difficult still. The consequences of this are immense. Compare the red line here to the figures at the top of the page.
One other thing to keep in mind: the logarithm. The base n logarithm of y = nx is simply the value of x in that base: logn(y) = logn(nx) = x. The base e log comes up so often it is called the natural logarithm, but bases of 2 and 10 are also common. If a base is not provided, either explicitly or by context, it is likely to be e. Here's an example in base 10: if y = 1000 then what is x in nx? Since n = 10 and 103 = 1000, log10(1000) = 3. Similarly, log2(64) = 6, log3(81) = 4, you get the idea.
Logarithms of the same functions are shown in the right panel above. We see that for subexponential (power law or linear, eg) growth, the slope flattens out as x increases. For exponential growth, and only for exponential growth, we get a straight line with a slope equal to k. If you want to find the growth (or decay) rate of an exponential function, just take the logarithm.
The classic example of the yeast in the Petri dish
It's worth watching Dr Albert Bartlett's take on this.
A rough precis of the relevant section: Imagine a population of yeast -- one microbe at first -- that doubles every minute, until after one hour the population has reached maximum capacity, they've eaten themselves out of house and home, and then collapses. How long until half the carrying capacity is reached?
Well, we double the population every minute and reach maximum in 60 minutes, so we reach the halfway point at 59 minutes. How long then does it take to reach 3% capacity? Working backwards, we find that we reach 25% in 58 minutes, 12.5% in 57 minutes, 6.25% in 56 minutes, 3.125% in 55 minutes.
In 55 minutes Joe and Jane microbe find that they've collectively occupied just over 3% of the available space. If they think linearly they reckon in another 55 minutes they'll be up to 6%, so they load the kids in the SUV and move to the suburbs. But at 3% capacity they don't have 55 minutes, because in 5 minutes it's Game Over.
But wait! The microbes are intelligent and enterprising, if they can find a new Petri dish to take over they can extend the species lifetime. They can extend that lifetime indefinitely by finding more and more new Petri dishes. How many more such worlds would they have to find? They double every minute, so in one minute they'd need to find a new one. In 5 minutes they'd need to find another 31 (25 = 32, but they're already occupying one -- note that the base here is 2, not e); in 10 minutes they'd need to find another 1023(!) new worlds.
Exponential growth is unforgiving, and the only way to beat it is to turn down the exponent (the growth rate). Either that or multiply through by zero.
On global population growth
Let's go back to the figure at the top of the page. If growth is indeed exponential then taking the logarithm of the data should give us a straight line with slope k, the rate of growth. OTOH, if it rolls over and flattens out with time then growth is subexponential. Looking at just the last 2000 years (arbitrary cutoff, data quality at the earliest dates are sketchy, 2000 years ago we should have decent quality data at least from the most populous empires of the day, Han and Rome) we find
We see several regimes here, color coded somewhat arbitrarily by growth rate (I see I switched terminology on you: here ln means natural log. In fact the "log" in all of the other figures is also the natural log -- normally I use "log" but for this plot seemed to remember that most people would use "ln". Sorry about that). The black takes us to the early medieval, the leveling off around 200 CE probably reflects the collapse of Han China in the East and the beginning of the end for Rome in the West. It picks up a bit around 500, then really picks up around 900 to about 1250, where it fluctuates a bit until 1400. The 14th century decline is almost certainly bubonic plague (the Black Death in Europe, though it was a Eurasian pandemic that devastated China and the Middle East as well); the 13th century is probably an earlier outbreak of plague and perhaps Mongol depredations, which took a huge toll in North China and much of the Middle East, partly from destruction of irrigation systems in Iraq and Iran. Things pick up quickly around 1750 until around 1900, when growth increases yet more quickly.
The second panel looks at the last 200 years, where there appears to be a break around 1900. We see the effects of World War I and (especially) the Spanish flu in the second decade of the 20th century, then World War II in the 5th, then things really take off.
So the logarithm of our growth curve has given us a growth rate k that
1. increases in magnitude over time (the slope gets steeper)
2. increases more rapidly over time (the time intervals become shorter)
Hmm, looks familiar. Not just the growth curve but the growth rate k itself looks exponential, so our growth curve (the figures at the top) is in fact superexponential. The only things slowing us down are war, famine, and plague, and even the biggest and most destructive of those are barely speed bumps.
The growth rate k may look exponential, but appearances here are deceiving: taking the logarithm of k does not quite leave a straight line, actually it curves upwards. That is, even the growth rate of the growth rate is at minimum exponential. Albert Bartlett's bacteria in a bottle have absolutely nothing on us.
We've known this for at least 40 years. There is good news: what looks like a rolloff in the growth rate in recent decades is real, and is probably a consequence largely of China's One Child policy. So we can add draconian restrictions on individual liberty to war, famine, and pestilence to our repertoire of growth slowing techniques. But even with the rolloff the growth rate is still higher than at anytime before the middle of the last century. Perhaps we will track the UN best case scenario, in which the growth rate drops enough to drive a decrease in population by 2100. Then again, perhaps not.
Sustainability
Ultimately, though, the reason we're concerned about population growth is that this planet contains a finite amount of resources. If we could use resources more efficiently and reduce our per-capita consumption we could ameliorate or even (up to a point, anyway) completely mitigate the effects of population growth.
Finding a good metric for resource use is not easy: poor societies may not use directly as many resources as wealthier ones do but because they cannot afford efficiency they may be more destructive of what they have. There have been a number of attempts to calculate the ecological footprint of various societies, such as the Global Footprint Network -- it is difficult to do and easy to criticize. A typical major shortcoming is a failure to incorporate environmental degradation. At any rate, here's a typical estimate, this one taken from the Ecological Footprint Atlas (pdf alert):
This calculation does not include environmental degradation -- a description of methodology is given here. And I have questions about the very idea of "global capacity". Still, it's a useful exercise. We see that the partitioning by country for ecological footprint goes pretty much as energy consumption. In fact, the top 5 here (US, China, India, Russia, Japan) are also the top 5 in CO2 emissions (China, US, Russia, India, Japan). So for resource consumption let's use energy consumption as a proxy.
We have good data for energy consumption, at least for the last few decades. I'll use data from BP's Statistical Review of Energy, in millions of tons of oil equivalent (MTOe).
The top plot is global population for the last 2000 years again, with the last century in blue; the second is global population since 1900 in blue with energy consumption since 1965 in red. We can see the effects of OPEC, Desert Storm, the Asian crisis, and the current crisis, but those are really noise in the trend. Again, what we're looking for is more efficient use of energy per capita over time. What we find in fact is that energy consumption has gone from 1.14 MTOe per capita in 1965 to 1.63 MTOe per capita in 2005, a 43% increase. That is, energy consumption has outstripped even population growth.
It is not just energy: here are a few figures culled from the executive summary to Steffen et al., Global Change and the Earth System: A Planet Under Pressure (2004):
The night image shows the amount of land cover occupied by just the developing nations. The changes in human related effects on the Earth system in the last 50-150 years at best match and frequently outstrip population growth, and are so rapid and profound they have given rise to the term Anthropocene as a geological age.
Please understand the scale of this problem. Population growth is not the fundamental problem -- resource consumption is. Controlling population but continuing with exponential or superexponential resource consumption helps not at all. But if we cannot get population growth under control, we won't get consumption either. The scale of this problem is not just magnitude -- the sheer amount of stuff consumed -- it is also rate, the fact that we're consuming more and more faster and faster. We're not going to grow our way out of this. Exponential growth is unforgiving, and we've blown way past even that. While I believe that space exploration is necessary for other purposes, doubling, tripling, increasing 1000-fold the space exploration budget will not find us new resources on anything remotely close to the time scale required. In fact, one could argue that technology has already increased the number of "worlds" resource-wise available to us. And doing so has brought immense benefits to us: better health, greater food security, longer lives, enhanced mobility, vaccines, shelter, etc. These are all good things. Even the very poor have benefited, at least in the sense that their populations have burgeoned, which would not have been possible otherwise. But it has all come at a cost, one that we've barely begun to pay.
We have recognized that population growth is a problem for several decades, and by dint of massive effort have turned down the growth curve, perhaps enough to see a peak in population this century. But we've done little on the consumption side, despite all the writing on the wall. There has been increasing awareness of the need for decoupling consumption from population. We have no idea how to do that yet, and the timescales are grim. But I'll leave that discussion for another post. In the meantime, any ideas?