This strange surface that we’ve plotted looks a bit like a folded piece of cloth. Let’s look at a clearer drawing of it, taken from Steven Strogatz’s book on Nonlinear Dynamics and Chaos:
There’s something very interesting that happens here though if we want to move back in the other direction.
What happens if now you start to increase r again? Well, you will actually find that you stick on the upper branch. You won’t suddenly jump back. This means that if the parameter changes even a tiny bit, you may well find that you are never able to get back to the situation that you were in before.
In fact the easiest way to see what is going on here is in terms of a potential. Imagine that you are sat at the bottom of the stable minimum on the left, and you decrease r, from 2.6 down to a little less than 0 and then up again. You will find that as you pass r is 2, you will go into the situation where there is only one minimum, and then as you increase r again, you will stay in the right hand minimum. If you are then in that new minimum, it’s really hard to get out again...there’s a great big maximum sitting in your way.
OK, there’s another way of plotting all of this. Let’s look at the surface from above:
Exercise for you: One can model the population of insects by:
which is now a differential equation with just two free parameters. See how far you can get in showing that this also has a cusp catastrophe.
There is a nice explanation of this example here but I’d like you to have a go at it yourself first. Have a watch of it, and see if you can understand the behaviour as you change the different parameters. In particular look at around 25 minutes to see the change as you slowly increase the reproductive rate of the population and how an insect outbreak can suddenly occur.