So, we have seen that bifurcations diagrams are a useful

geometric device, a type of graph that lets us see, all at once,

the different behaviors a dynamical system might have as a parameter is changed.

In this lecture, I will focus on the phenomenon of bifurcations

themselves. I think this is an important topic from dynamical

systems, particularly for the study of complex systems.

Bifurcations aren't as well-known, or flashy, or perhaps as

exciting as the butterfly effect or strange attractors. But, I think

they're important and definitely good to know about. So, I'll

begin making this argument by return to the the logistic equation with harvest.

So, here's the logistic equation with a harvest term. rP times

1 minus P over K. K is the carrying capacity and then h is the

harvest, the number of fish, or whatever, that are caught

every year. And, before we talk about the bifurcation

diagram, just a reminder that the right-hand side of this

equation is quite well-behaved. This is just an upside-down

parabola, this term here, and the h just shifts the whole graph

down. So, here is what that looks like for, say, for h equals

40, and as h gets larger this graph just slides down like this.

But, this is a smooth function of P,

a smooth function of h; it's differentiable,

very nice and well-behaved. Ok, so as we've seen, the

bifurcation diagram looks like this. And, here, I've just drawn

in different colors the two different types of equilibria, or fixed

points. So, red, this is the stable attractor and blue is the

unstable attractor. So, let's think about this in terms of fishing

in this model population, and h is the fishing rate, the number

of fish that we allow to be caught or the pounds of fish or

whatever, every year or every generation. So, when we are

down here, we are doing no harvest at all; and, this red value

up here, this is actually the carrying

capacity, the equilibrium value in the absence of this

harvesting. And, so then we begin, imagine we discover a

lake or some people arrive there, and time to start fishing and

we allow a certain amount of fishing. And, a little bit of fishing

leads to a small decrease in the steady-state population, the

equilibrium population of the lake. And, that makes sense,

it would be surprising if anything else happened. When you

start fishing, there will be less fish. But, maybe not such a

big deal, there is a growth rate. You kill some fish; the fish

grow back towards their carrying capacity. They don't

actually hit the carrying capacity, but they come pretty close.

Down here, this unstable equilibrium has gotten a little bit

larger and that's not surprising at all. If, say, we are catching

40 fish a year, but there are only 20 fish in the lake then we're

not going to be able to recover because we will have killed

all the fish, and so that population would move down. So, if

you start harvesting. If you start this fishing, one can imagine

that if you start off with very few fish you'll just kill them all.

So, that seems reasonable. So, lets continue thinking about

what's going on up here, on this curve. So, maybe we've

allowed fishing up to this rate and things are going pretty

well. And, so a bunch of people ask to fish more because

fishing is fun, or it's lucrative, or people are hungry and they

want to eat fish. And, so we allow a bunch more fishing and

then the population, the steady-state population, decreases a

little bit more, but, not that much. And, all along this curve, a

small increase in the fishing rate gives rise to a small

decrease in the equilibrium population, seems to make really

good sense. And, then we keep going, maybe we are out

here and we decide, ok, let's do a little bit more fishing.

Things have been going well. And, we allow a bit more

fishing, and all of a sudden we end up over here. So, if h, the

fishing rate is this high, then there is no steady-state

population and the fish population would just crash. It would

go right down to zero. But, the thing to note is, well, a couple

of things, the steady-state, this red value, you might think that

the equilibrium value would approach h smoothly. But,

instead, it just sort of falls off a cliff. You can have a stable

equilibrium here, but you can't have any stable equilibria for

a population less than this. So we might think that this red

curve should kinda go down and touch this, but the red curve

just blinks out of existence right here.

It's important to know that in this

fishing scenario, we have control over h. That's something

that could be set by policy and could be measured. We just

count how many fish people catch. But, what this equillibrium

population is wouldn't be observable, typically. There might

be some clues that the fish suddenly get hard to catch.

But, maybe, right, the population is still pretty large. You can

imagine being out here, close to the point where you fall off

this cliff and the fish suddenly die, with a small in increase in

h. Still a lot of fish and so they can still be pretty easy to

catch, it might not be really,

it might not be immediately noticeable.

So, the lesson here, the thing that I think is important, is that

we can have a very sudden transition, a mathematically an

instantaneous transition, and it's in a sense a discontinuous

one where the population would go from an equilibrium

here all the way down to zero. There is no equilibrium

value in between these two points. So, we have this

discontinuous behavior, even though this function is

continuous and smooth can be. So, even a continuous and

differentiable function that varies as

function of h can have this sort of potentially

disastrous and discontinuous behavior. One more possible

scenario to note. So, imagine we're moving along this curve

or somewhere along here, and we don't really, but we don't

realize it because we can't really see this curve, so we just

increase the fishing rate a little bit more and we start to

plummet down here. Right, so the population is here. We allow

allow too much fishing. The population starts to die off,

maybe suddenly people aren't catching fish.

We notice the fish stock is decreasing, and so the

logical thing to do then might be to cut back on fishing. And,

maybe we cut back on fishing a lot. So, that would move us

back here. But, even doing that, once you are down here and

population starts to crash, if you cut back on fishing

even by a factor of 50%, you still might be in this region

where the fish are still going to die. So, once you sort of

move over this edge, you might need to reduce fishing

entirely, or almost entirely in order to get back here and creep

back up to this equilibrium value. So, again, I don't think

these people actually use these models to, in a numerical

way, to understand how fish rates behave. But, this does

suggest that you can have a sudden collapse, that a stable

point can disappear suddenly and without warning. And, it

turns out that that's a fairly generic feature of differential

equations, even ones that are smooth and continuous like this one.