Nonlinear population modeler

This "toy model" uses a simple but robust and nonlinear way of modeling real-world population growth (and collapse). Given a population p (for mathematical purposes between 0 and 1) and a "biotic potential" r, the population in the next generation will be pnext = rp(1-p) (unless p<0, in which case we simply set p to 0). In a real-world sense, population will grow proportionally from one generation to the next but is eventually checked by dwindling food supply. (We're ignoring things like predators and disease — that's why we call it a "toy model.")

For many values of r, the population will gradually reach an equilibrium or steady state. For some it crashes. For others, it will oscillate between 2 (or more) values; and bizarrely, for some values of r, population never stabilizes at all, but jumps around quite wildly! This behavior is not just a mathematical artifact of our model but echoes what sometimes happens in real-world ecosystems.

Experiment with this phenomenon by trying different starting populations and biotic potentials. To get started, use p0=0.2. 100 generations is a good run to start with, but when you find a system that won't stabilize, you will want to extend the run to see what happens over time.

Questions

  1. Find a set of values for which population goes to zero (a) quickly and (b) gradually.
  2. Find a set of values for which population eventually (a) stabilizes to a precision of 4 decimal places, (b) oscillates between 2 values; (c) oscillates between 4 values. (It's easiest to do this by keeping p0 constant and adjusting r.) Can you find a value of r that oscillates with a period of 8? 16?
  3. Find a set of values for which population does not stabilize. (To convince yourself, you can run up to 10,000 generations with this calculator, which is quite a long time even for mayflies.) Look for periods of stability and periodicity in the chaos. Can you find a short-term oscillation with period 3? period 7?
  4. What generalizations can you make about what happens to the population as biotic potential increases? What happens at very low values and very high values of r? How would you describe in words what happens in the middle range?
  5. What kinds of real-world situations can you imagine that would produce the oscillations and chaotic behavior you see in the model?