Chaos, Frog Ponds, and Red Tide
Sometimes things are less complicated than they seem...
- Simple Equations Can Produce Complex Results
- Complexity Can Make Cause and Effect Difficult to Determine
Most of the systems that we treat in physics classrooms are simple. It may not feel this way during an exam, but the fact is the systems we look at can usually be described by a single equation, such as
where some variable is changing in a predictable way. But you may have noticed that most things that happen in life are not predictable; things like relationships, car engines, weather, etc. One way of using physics to describe complicated things is by making the math more complicated, but you may be wondering (assuming you've been thinking about this in your spare time) whether we can get complicated behaviour out of simple math. The answer is yes.
To see this, we will look at a pond with some frogs in it, and ask the question, what is the population of frogs each year? To start with we could assume that, each year, some fraction of the current frog population reproduces; that is,
where is the population in the nth year, and a is the growth rate. We would like to include a term that reflects the fact that the population won't keep growing forever but level off at some point. The easiest way to include this is to multiply the right hand side of equation (1) so that it becomes:
so that, as gets closer to , the population will decrease. If we divide out the from both sides, we get an expression for the 's where the population is between 0 and 1 (and we can always multiply later to get real population numbers). So, for , and , we get the graph below, which is what we expect: the population rises for a while and then levels off.
The interesting cases come when we raise the growth rate, a, to around 3.0. For values between 3.0 and 3.4, we something like the graph below, where the value fluctuates between two numbers every other season; in this case, between about 580 and about 750.
Beyond that, the plots get increasingly chaotic and random; the chart below gives the population over time for a=3.8. The behaviour looks chaotic and complicated, yet it came out of the simple equation above repeated over and over on a spreadsheet.
Now, the question is, as always, do we see this behaviour in the real world? The answer (as you might expect) is yes. A great many species have population graphs that look like the simple graph; bears, cougars, etc. In fact, most species have population curves that look something like that, in which they rise for a bit, then level off. The picture below is one of the population of pergrine falcons after their reintroduction to Virginia1. While, in the real world, things aren't as smooth as our graphs above, the general trend is there: constant rising, with a leveling off.
A number of species also display the kind of behaviour exhibited in the second figure, where the population fluctuated between two values, as well. As we would expect, these species are faster breeding species: rabbits, woodchucks, etc. The most famous example of this is the oscillation of the snowshoe hare population, and that of it's main predator, the lynx2.
This example is often discussed in biology classes, and the essence of the oscillations are captured in the simple model we made up earlier: the population grows, but if it gets too big, it shrinks.
Now we can look at whether any species exhibit the behaviour in the third chart, the one that showed chaotic behaviour. One example of this is the “red tide” that is seen on the US eastern seaboard every so often3. The root cause of this is runoff from farms and such; the runoff seeds the water with a large amount of fertilizer, which gives the algae that grows in the rivers a tremendous growth rate. Thus, we would expect, from the simple analysis above, a rather random population distribution, and this is indeed what we see. In some years the algae is almost non-existent, in others the algae strangles everything else in the river in an out of control burst.
An interesting point to make here is that the randomness of such population shifts can cause people (even scientists) to look for proximate causes to a event, whether the event is a particularly large growth or a down year. In our chaotic graph above, we could look at, say, the peak around season 25 and try to find proximate causes, like warmer temperatures, the absence of snakes in our frog pond, etc. But really, in our model, the peak is there entirely because of the way the system is set up, and we don't even take season to season changes into account. The literature is full of people explaining red tide events in terms of various factors that were or weren't present. While these events probably play a role in population growth, a large part of the variation from year to year can be explained as chaos driven by a large population growth4.
It gets even worse. Below is a plot like the ones above, of population vs season, but this time I went out to 500 seasons, and took a very particular value of a5. What you see here is that, for a while, the population rotates between three values, completely regularly. Then, it changes, and becomes completely chaotic. Then it goes back to regular. Then back to chaotic. If I'd gone on, it would keep doing this, without any pattern in to how long it stays regular or chaotic. And yet, I haven't done anything other than set the initial value and let my spreadsheet go. Talk about complicated behaviour from a simple system!
What we've seen here is that simple models can display complicated behaviour; this is good, since the world is complicated! But it also can make it difficult to determine why, exactly, things happened a certain way. We've also seen that, sometimes, (not all the time!) things that happen, like algae blooms, or a particularly cold winter, are cause partially by nothing that happened that year, but by long term effects and the way the system works overall.
Just for fun, I've attached the spreadsheet I used to do all this at the bottom. You can change the box labeled "Growth Rate", and see how the chart of population vs. season changes.
- 1. Taken from http://ccb-wm.org/news/2009_MayAug/peregrine_breedingSeason.htm
- 2. Taken from https://www.math.duke.edu/education/ccp/materials/engin/predprey/pred1.html
- 3. http://en.wikipedia.org/wiki/Red_tide
- 4. See the references in the wikipedia article for a variety of perspectives on the causes of red tide. Most of them discuss the causes in terms of things that happened that season, rather than systematic effects.
- 5. This value, 3.82831, is from a problem set in Classical Dynamics of Particles and Systems, by Marion and Thornton, Academic Press, New York, 1965.
© Physics and Astronomy Outreach Program at the University of British Columbia (Eric Mills 2011-02-24)