So there are a few more things I'd like to mention about bifurcations.

First, I should briefly mention that bifurcations are a topic that are covered pretty extensively...

much more extensively than I have here...

...in most mathematical courses on dynamical systems, especially those that focus on differential equations.

And the conclusions or realizations that emerge from this body of work are that there's only a small...

...set of-- just a handful-- of different types of bifurcations.

So you might think, well, there are so many different types of differential equations...

that each one might have a different type of bifurcation, and that turns out to not be the case.

That there's just a small set-- you can classify them in a very precise and analytic way-- what these different types are.

And, surprisingly, to me at least, this classifcation holds by-and-large-- it gets a little more complicated-- but it holds in higher dimensions as well.

So there's a really nice theory behind a lot of this empirical stuff I've been presenting about bifurcations.

It's beyond the scope of this course but I can point you to some references if you'd like to dig into those further.

The second point that I want to raise is to just underscore the

basic phenomenon of bifurcations, which is that you can have a very large change in equilibrium behavior associated with a tiny change in a system parameter.

Now, bifurcations don't occur all the time-- most of the time, in these differential equations, you change your parameter, and the system changes just a little bit.

A small change in one thing gives rise to a small change in another.

But every now and again, if you're at or near one of these bifurcations, you can make a small change in a parameter, and this time it will make a huge change in the system behavior.

In the case of the logistic equation with harvest, once you go past that critical harvest point, it's disaster for the fishery.

The equilibrium point that existed just disappears.

So it might be worth thinking about this in a somewhat inverted way, so, starting not with equations, but with observations.

So suppose we're monitoring some fishery, and we notice that the-- you know, we change the harvest rate every now and again, or we just notice that sometimes the fisherfolk catch more, sometimes less...

...and, then, all of a sudden, one day or one year, the population just disappears. And we might wonder, "What happened? What did that?"

And it might be natural to look for some outside source-- pollution, or some horrible fish disease. I don't know enough about fish to know what an outside source would be.

But if we see some sudden collapse, some sudden change in a system, we might think that there must be something external that caused that.

And that could be the case, but what the phenomenon of bifurcations shows is that you can also have sudden changes that arise not from an external source...

but from the intrinsic dynamics of the system. So, these sudden changes, you don't need to appeal to something outside; the cause can be internal.

And that's very similar to one of the realizations from the butterfly effect and chaos, from the last unit.

If you see a population behaving erratically, that's not evidence that it is not following a rule.

It's not evidence that there's some external source that's making that behavior appear random or be random.

You can have internal or intrinsic sources of randomness in a simple model, just like you can have jumps or sudden transitions be intrinsic to a simple model.

Another point I want to make is that bifurcations don't have to be bad things.

I've been presenting them as bad things because, well, in the logistic model with harvest, they are kinda a bad thing.

We have a fish population that's kinda doing alright, and then just disappears. So that's bad news for the fish, and bad news for people that like fish.

But I could also imagine cases where bifurcations are not catastrophic, or bad news.

Bifurcations let one, sort of, shift, between two different types of behavior or two different regimes, and that could be a useful thing for an organism, in some settings.

And just in general, bifurcation doesn't automatically mean something that's catastrophically terrible or bad.

Okay.

The last point that I want to make is to think about what the phenomena of bifurcations might have to tell us about the study of complex systems.

So by complex systems, in this context, I'm thinking of a situation where we're concerned not just with one species of fish, but many many species of fish.

And these species of fish might interact! Some, help each other out, some are antagonistic; we'll need to think about what fish eat.

Algae, or sometimes other fish, or fish f... I don't know what fish eat, but we'll need to think about what fish eat.

And then we'll need to think about fisherfolk, and what they prefer to harvest, and maybe markets, and what types of fish people want to eat, and so on.

And the point is, there's a lot of things going on, a lot of variables, things we want to keep track of, and they're all interacting with each other.

One thing will influence another, and vice-versa.

So, okay.

For the simple logistic equation with harvest of this unit, we've seen that we can get these sudden changes...

...these jumps, these gaps, in behavior.

Would we expect to see that in a complex system that was built up out of many, many logistic-equation-like things.

I think the answer to this question is not so clear, and it gets to maybe the heart of what is different and interesting about complex systems.

So, one could think, well, alright, if simple equations, one-dimensional equations, have these jumps in them...

You put a whole bunch of those equations together, and kind of tie them together with interactions, then, probably, the whole system would have jumps in it.

Could happen.

Or, perhaps, when we put all these equations together and tie them together, make them interrelated...

maybe that interrelation and the diversity of interactions actually serves to stabilize things, and so we tend to not see these jumps.

I don't know that either of those scenarios is generically true.

I think in some settings, we might still see jumps, and in other settings, we might not.

So I think that's sort of an open area in the study of complex systems.

We have these generic properties for simple systems; to what extent to they carry over in the-- when we study more complex systems?