Way back in podcast 14, we discussed the absurd consequences of exponential growth rates. If each generation is r times the size of the last, and the starting population is S, then after <n> generations the population will be S * r ^ n. This kind of growth is incredibly rapid. Assuming the Earth has a volume of about 2x10^24 rabbits, as we calculated in that podcast, then even if r=2, with each rabbit generation merely doubling, the planet would consist of nothing but rabbits within about 80 generations. And real rabbits multiply a lot faster than that. Yet it has been a few years since that podcast, and Math Mutation still has a nontrivial number of non-rabbit listeners. How did that happen?
As you have probably realized by now, exponential growth never happens for very long; there are always countervailing forces, like scarcity of food or space. There is in fact a nice relation, originally developed by Belgian mathematician Pierre Verhulst in the 19th century, that models modified exponential growth in the presence of limited resources. He was actually describing a continuous model and using calculus, but when discussing populations we can use the simplified discrete form. Assume there is some maximum population or value that can be supported by the environment. Let p1 be the proportion of this maximum tolerance we have currently reached, r be the rate of growth, and p2 be the expected proportion of the next generation. Then the "logistic difference equation" can be expressed as:
p2 = r * p1 * (1-p1)
In other words, we are simply multiplying an additional factor to the expoential growth: 1-p1. With regular exponential growth, we would just have p2 = r * p1. The additional 1-p1 factor accounts for the fact that as p1 gets close to 1, the maximum possible size of the population, this will act as a tempering force to reduce the exponential growth. When we are very far from saturation, that 1-p1 is about 1, so it looks like standard exponential growth. The closer we get to saturation, the more the growth is throttled by that extra factor.
This formula doesn't seem very complex, but if we plug in numbers some very interesting things happen. Intuitively, how would you expect this equation to act? A good guess might be that if you start with a low value for p, population growth will initially be rapid, and then will smooth out and achieve some kind of stable equilibrium. For a growth rate r of 2.9, this does indeed happen: for example, if we start with a p value of .2, indicating we have about 20% of our maximum number of rabbits, we can see it initially jump to .46, then .72, bounce back a little to .58, and eventually settle down to a consistent .66. This shows that our population will settle down at about 66% of capacity.
But for other growth rates, we see very different results. There are many growth rates for which the population never does settle. As one example, if we use r = 3.3, we can see that after a few generations, the population starts oscillating between .48 and .82. In other words, we have generations containing 48% of our maximum rabbit capacity, followed by generations at 82%, going back and forth forever. For various other growth rates, the population ends up oscillating between 3 or 4 values. There are even many values, such as r=3.57, where the value never settles down or oscillates-- it just seems to jump around forever between arbitrary totals. The show notes link some sites where you can see pretty graphs of the bevaior of this equation for various values.
What does all this mean? It means that this simple difference equation is an example of a chaotic function, a topic we also mentioned back in podcast 67. That is a function where a slight change in initial conditions can result in widely varying results. In this case, we see that small changes in the rate of growth can give us a nice stable population, boom and bust cycles, or random fluctuations. You might expect that a slight change in the rate would create a corresponding change in the equilibrium value, but this simply isn't true: you may modify the rate a little and find you have lost equilibrium entirely, changed a former equilibrium into a new boom and bust cycle, or in effect randomized the results overall.
But seeing chaos come out of such a simple model, I think we've also discovered a surprising fact: in many parts of real life, boom and bust cycles may be a natural and expected occurence. We tend to expect things we encounter on a daily basis to reach some kind of equilibrium, whether it is the rabbit population in your yard, the national economy, the planet's climate, your weight, or your spouse's mood. When we see things not in equilibrium, we usually consider it a serious problem, and try to figure out what has gone wrong. After seeing the behavior of the logistic difference equation, though, we see that boom and bust cycles are something that really does arise out of simple, natural processes. And a slight tweak to your inputs will not always get you the nice, predictable result you want.
So, next time you see world events or your daily life going out of control, take a step back before stressing about it. Maybe it is a real crisis, maybe it's totally beyond your control, or maybe you just need to modify parameters of a few of your real-life logistic difference equations.
And this has been your math mutation for today.
References:
http://www.stsci.edu/~lbradley/seminar/logdiffeqn.html : Nice site with explanation of logistic difference equation.
http://en.wikipedia.org/wiki/Logistic_Equation : Logistic equation at wikipedia.
http://www.dallaway.com/pondlife/p2.html : Nice presentation with some interactive applets to try out different values.
0 comments:
Post a Comment