Let’s try and make the previous example at least a little more realistic. Let’s suppose that the environment only supports a fixed number of rabbits, let’s call that fixed number . This is the maximum number of rabbits that we can stably have. It turns out that there is a very very important equation which will model this sort of behaviour very well, and it shows up all over the place. This is called the logistic equation and looks like:
Solving this equation means finding a function whose derivative and whose functional form are linked in this specific way.
We have pulled this equation out of thin air, so rather than deriving where it comes from, we will simply motivate that it seems to have the right sort of behaviour of what we want.
For very small populations (much less than the stable equilibrium population ), , so the term in brackets can be safely ignored.…