Thursday, February 11, 2016

Limits to growth, logistic vs exponential

Malthusian growth model

The Malthusian growth model sees population growth as exponential.

P(t) = Poert
P=  P(0) is the initial population size,
r = population growth rate
t = time

Growth of microbe populations are often used to illustrate this. Let's say an amoeba will grow and divide into two amoeba after an day of absorbing nutrients.

Day 1: 1 amoeba
Day 2: 2 amoeba
Day 3: 4 amoeba
Day 4: 8 amoeba

And so on. Population doubles each day. Exponential growth is famous for starting out slow and then zooming through the roof.

On the left is exponential growth in cartesian coordinates. On the right in polar coordinates, radius doubles every circuit.

Malthus imagined a rapidly growing population consuming all their available food supply and then starving to death.

Logistic growth

Sometimes populations have suffered Malthusian disaster. More often rate of growth slows as the population approaches the limit that resources can support. This is logistic growth.

P(t) = Le-rt / (L +( e-rt - 1))

Where L is the maximum population local resources can support.

At the start, logistic growth resembles exponential growth. But as the population nears the logistic ceiling, growth tapers off. Above the blue boundary represents the limit to growth. In red is the logistic growth curve, the thinner black curve is exponential growth.

What slows growth?

In Heinlein's science fiction, war limits growth. This was also the foundation idea of Niven and Pournelle's The Mote In God's Eye -- War is the inevitable result of burgeoning populations.

The Four Horsemen of Apocalypse -- plague, war, famine and death are seen as natural outcomes of uncontrolled population growth.

A declining fertility rate is a less ominous way to step on the brakes. It is my hope most people will choose to have small families. And indeed, current trends indicate people are voluntarily having fewer kids. Still, there are skirmishes as various entities compete for limited resources.

Bad vs worse

A growing population, a growing consumer appetite, a limited body of resources. It doesn't take a rocket scientist to see growth must eventually level off.

Whether it levels off via the 4 horsemen or moderation and voluntary birth control, either option sucks.  It's disaster vs stagnation.


Above is a Johnny Robinson cartoon. Used with permission.

I believe our solar system is possibly the next frontier. That has been the thrust of this blog since the start. If we do manage to break our chains to earth, it will be a huge turning point in human history, more dramatic than the settling of the Americas. The potential resources and real estate dwarf the north and south American land masses.

While settling the solar system allows expansion, it won't relieve population pressure on earth. Settlement of the Americas did not relieve population pressure in Europe, Asia and Africa. Mass emigration is impractical.

Rather, pioneers jumping boundaries starts growth within the new frontiers. I like to view the logistic growth spiral in polar form as a petri dish. When a population within a petri dish has matured to fill its boundaries, it sends spores out to neighboring petri dishes. Then populations within neighboring petri dishes grow to their limits.

The first petri dish still has a population filling the limit. They have not escaped the need to live within their means. I take issues with critics who say space enthusiasts want to escape to a new planet after earth has been trashed. Space enthusiasts know earth is fragile, more so than the average person. It is noteworthy that Elon Musk is pioneering planet preserving technologies such as electric cars and solar energy.

But even if mass emigration from Europe or Asia was not possible, the expansion into the Americas energized the economy and zeitgeist of the entire planet. It provided investment opportunities. Also an incentive to explore. This is the greatest benefit of a frontier. Curiosity is one of the noblest human qualities and I hope we will always want to see what lies over yonder hill. And that we will keep devising ways to reach the far side of the next hill. Satisfaction and contentment are for cattle. If we lose our hunger and wander lust we will no longer be human.


Eternal Expansionist said...

You mention that resources are limited. What do you really mean by that?
Life on Earth has expanded since 4,000,000,000 or so years. What resources have been "consumed" by this exponential growth process? Have Earth run out of coal or copper or what? Do you have a list of stuff we have run out of? If one copies identical bacteria, then the fixed amount of nutritions in a petri dish will run out, yes. But that is not how life works. That is not how human economy grows.

When you learn more math, do you then "consume" a finite set of "math resources" and contribute to the end of math? Of course not. And it is the same way with biological and economical evolution. Resources are endless according to cosmology and all practical biologic and economic experience since billions of years. Isn't there about 800,000,000,000 tons of physical resources per now living human, here under our feet?

Even an often claimed "peak resource" like oil is being produced in ever growing quantities at cheaper prices. Like once ships' sails, oil will be replaced by even more economical alternatives before we run out of it. Most oil reserves will be left worthless in the underground. That's why everyone who owns an oil field pumps up as much as he can while it still has some value, instead of leaving it for some future with oil shortage and much higher prices.

Resources are being produced, not consumed. They are produced by being reorganized in ever more valuable ways. There is no imaginable end to this. The more humans who participate in industrialized civilizations, the more resources per individual is being produced.

Hop David said...

Eternal Expansionist, I disagree with some of your comments.

"Life on Earth has expanded since 4,000,000,000 or so years."

Earth's biomass has remained the same over the past 4 billion years.

"Isn't there about 800,000,000,000 tons of physical resources per now living human, here under our feet?"

Heat and pressure climb as we burrow deeper. More than a few kilometers down and mine shafts become impractical. Only the thin outer shell of our planet is accessible.

And presently our mining and manufacturing are having an impact on the biosphere. Increased activity will further damage our ecosphere.

"Even an often claimed "peak resource" like oil is being produced in ever growing quantities at cheaper prices."

The recent drop in oil prices is due to fracking which opened up new reserves. These aren't easier bodies than the earlier wells. Part of the recent drop in prices is willingness of oil producers to cut profits in order to keep market share.

But I am hoping solar and nuclear will provide more energy in the future. For the sake of argument, let's say energy is unlimited. Even with this optimistic assumption, there's only so much our earth's surface can sustain. If energy use grows too high, there'd be too much waste heat per square meter and we would cook ourselves.

For unlimited growth we would need to break into new frontiers.

"Resources are being produced, not consumed. They are produced by being reorganized in ever more valuable ways. There is no imaginable end to this."

Yes, human ingenuity enables us to get more from less. But there are limits. Laws of thermodynamics puts limits on efficiency of vehicles, refrigerators, heaters, etc. Moore's Law can only continue so far before transistor sizes reach the atomic level.

Chris Wolfe said...

An example of scarcity in this context is sand. One would think we have so much sand that there could hardly be a problem getting enough of it. Unfortunately, desert sand grains are not suitable for current concrete technology. Beach or river sand is required, and there are places where the supply has been exhausted. India in particular is becoming famous for illegal mines, rival gangs and violent shootouts. Over sand.

People are killing people, right now, today.

For sand.

With a rock crusher and energy you can make building-grade sand out of most rocks (referred to as crusher run, or it can be screened and graded). Just because a solution exists does not mean it will be applied. In many parts of the world the startup cost for a tech-based solution is often too high, and the long-run profit potential is too low for outside financing. Builders buy sand at the lowest price they can get, and sand gangs can use a variety of tactics (intimidation, violence and what amounts to slave labor) to ensure they get the cash.

As for Hop's underlying argument, this is well-supported by current economics research (keyword: secular stagnation). Larry Summers and Paul Krugman among others have brought the idea into the public discourse.

(part 1 of 2)

Chris Wolfe said...

(part 2 of 2)
When biologists model populations they are usually looking at competing groups. The classic example is rabbits and wolves. Even if they start out in balance, random perturbations will push the system in one direction. Maybe a bad winter kills off most of the wolves. This drives up the growth rate of rabbits until they either starve, die of disease or cause a wolf population explosion. The rabbit population peaks, then declines as the wolf population races to its peak. The cycle continues back and forth in a dynamic equilibrium based on the available resources.

Humans compete with each other. We're not all one happy spiral in the petri dish. We are hundreds of countries, thousands of cities, millions of tribes. Some of these groups are doing well, so their fertility rate is low and their competition tends to be social or economic. Others are struggling to survive, so their fertility rate is high and their competition tends to be literal and physical. Humanity covers the entire spectrum from hunter-gatherer to trust fund brat.

We already invest resources in improving the lives of various groups. This has mixed results; generally good but sometimes local forces act to reserve these outside resources for themselves. As a species we are making progress, but we still have a long tail of poverty. This is unfortunate; the one resource locked up in these groups that could benefit us all is human ingenuity. It benefits all of us to raise up those in poverty, to bring their groups to a level of self-sufficiency that allows them the freedom to interact on a larger scale. None of us can know where the next Einstein will come from; better to be sure that person reaches their potential no matter where they happen to be born.

A second danger is the long tail of wealth. Those at the uppermost end of the spectrum control vast quantities of resources. It is often to their short-term benefit to use their power to take from other groups, maintaining their relative position on top of the pile. Some use their power to benefit others, but many reached their current position through force of one kind or another. Moral arguments are pointless; the ruthless among this group will respond only to profit and power. This is one area where space exploration can benefit those still on earth, by providing the most competitive among us with a new arena to compete in and new prizes to win. I personally am not happy about the idea of space being controlled from the outset by a small number of ruthless billionaires, but tyrants eventually die. Exploration/exploitation is a rough and dangerous game, but it leads to settling and a more normal form of growth with time.

The petri dish expansion into space will start small, but once we hit our technological stride we will be deep into exponential growth for decades if not centuries. Individual colonies will level off and stabilize but the rate of new colony construction will offset this for a long time. We will eventually transition back to log growth as the demand for mass begins to squeeze the easily available supply. Along the way, our technologies for harvesting energy will continue to improve. It is not yet clear which will be the limiting factor, the available mass in our solar system or the available energy from our Sun; workable fusion power would tip the balance toward mass as the ultimate limit for our species in this solar system.
In the meantime, innovation and trade will continue to drive economic growth as it has for centuries, set free from Earth's limits of economically extractable resources.

Chris Wolfe said...

In regards to peak oil, the reasons for low prices today are several. They can be reduced to competition and supply vs. demand, but read on for more detail.

First, fracking has released incredible volumes of natural gas in the US, China and other leading economies; cheap and relatively clean methane is used for home heating, industrial cogeneration and peaking plants so the demand for heating oil is much reduced. A reduction in demand tends to cause lower prices.

Second, most oil exporters (the US included) are currently involved in one or more wars. Oil provides a major revenue source for Daesh and other terrorist factions. It also provides funding for rebel or resistance factions as well as pro-government factions (Assad, Putin). Some of these groups are so politically toxic that only a few countries are willing to buy their oil and only at deep discounts. There is some evidence that Russia is buying oil from Assad and may also be buying it from Daesh, simply because they can get it for less than it costs them to produce back home. In turn, Russia has few buyers for its enormous reserves thanks to their invasion of the Ukraine and the resulting international sanctions. Unfortunately for them, their economy fundamentally depends on oil revenues; they must sell even if the profit is low. The end result of these political realities is that oil is cheaper today if you are willing to ignore where it came from.

Third, we are in the grip of a global lost decade, a crushing recession that has already toppled European and Middle East / North African governments. China, the world's leading industrializing juggernaut, has slipped into single-digit growth with long-term projections much lower than before the US financial sector raped humanity. Their demand for concrete and steel has precipitously collapsed, which in turn slashes demand for coal and oil. Advanced economies are less affected because they are already post-industrial, but reduced demand leads to reduced capital investment which lowers demand for energy one way or another. (I don't mean to imply that China is not advanced, but the truth is the country still has a lot of room to grow their industrial capacity; these capital investments have been the driver of China's incredible economic growth over the last few decades.) Again, the net effect is reduced demand.

Producers are scrambling to slash prices so they can stave off bankruptcy. Consumers are cutting back and switching to alternatives. Soon oil will not be a factor in heating and power generation. Over the medium term biological alternatives will replace petroleum as a chemical feedstock. Eventually petroleum will be a niche market, still in use but about as economically important as saddles. As a species we are deciding that the cost of oil is much higher than its purchase price and we are no longer willing to pay. This is what peak oil looks like.

DevHyfes said...

When in history has there ever been a Malthusian Population Disaster? By Malthusian, I mean, a population increases in size to the point it outstrips its food production and collapses. Neo-Malthusian population disasters are the same thing except instead of Food being outstripped, it is all the resources needed to sustain the population.

Many people confuse the Great Famine in Europe and related instances in Asia to be Malthusian. However in these cases, massive climate shifts caused food production to Plummet. The populations didn't outstrip their production, they were blindsided by a natural cataclysm. Perhaps the closest we can get to a Malthusian collapse was on Easter Island- but it is very debatable as to whether the chronic mismanagement of resources by a religious cult on an isolated island with very limited resources is at all transferable to the world at large.

The Malthusian population growth formula is demonstrably false when compared to humans. Because we use price to signal the availability of resources, as those resources get more expensive, humans moderate their behavior. They have fewer kids, they switch to alternatives and they conserve.

Your general point that we need to get off this rock is true, but I think it is less about humans using up the earth's resources. Instead it is about humanity avoiding major natural cataclysms.

Hop David said...

DevHyfes wrote "When in history has there ever been a Malthusian Population Disaster?"

I don't know of any. So? Is that supposed to be a rebuttal of some sort? Reread the blog, this time for comprehension.

Law Wong said...

"Mass migration is impractical."
Probably true (especially considering the high cost of space travel and the harsh, unfriendly? environment), but...
The world's population is growing at a rate of ~80 million people per year (just over 0.1%/year).
But at the same time, ~8 million people fly on airplanes every day (including short-haul flights, yes).
And approximately 1 million people move to the USA every year.
If space travel becomes as cheap as air travel (unlikely in the short-medium term), the capability may exist for transporting sufficient numbers of people off-world to curtail population growth.
Whether the will to do so will exist is another matter entirely.