So, we have seen that bifurcation diagrams are
a useful geometric device, a type of graph that
lets us see, all at once, the different behaviours
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
particular for the study of complex systems.
Bifurcations are not as well- known,
or flashy or perhaps exciting as the butterfly effect
or strange attractors, but I think they're important and
definitely good to know about.
So I'll begin by making
this argument by returning to the logistic equation
with harvest.
So here's is the logistic equation with the harvest term.
rP times 1 minus P over K.
And 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 parable
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, its differentiable
very nice and well- behaved.
OK, so, as we've seen, the Bifurcation diagram
looks like this.
And here I have 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 lets 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
for the pounds of fish or whatever,
every year or every generation.
So when we are down here
we're 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,
so imagine we discover a lake or something
and some people arrive there,
time to start fishing and we allow a certain amount
of fishing and I 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.
Maybe not such a big deal, there is a great rate, you kill some fish
and 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 but larger
and that's not surprising at all.
Say, we are catching ... 40 fish a year,
but they're only 20 fish in the lake
then we're not got to be able to recover.
This will kill all the fish, so their population will move down.
So, if you start harvesting, if you start this fishing
one can imagine that it..
If you start off with very few fish you'll just kill them all.
So that's....seems reasonable.
So lets continu thinking what going on up here, on this curve.
So maybe we've allowed fishing up to this rate and
things are going pretty well, so a bunch ask to fish more
as fishing is fun, or lucrative or people wanna eat fish
You'll allow a bunch of more fishing and the population,
the steady- state population
decreases a little bit more.
But not that much.
And all along this curve, small increase in the fishing rate gives
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, lets do a little bit more fishing.
Things have been going well...
and we allow a little bit more fishing and
all of the sudden we end up
over here.
So 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 off falls of a cliff.
You can have a steady- state parameter over 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 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 equilibrium population is
wouldn't be observable...typically...
There might be some clues that the fish
suddenly get hard to catch.
But maybe, right there, the population is still pretty large
You can imagine being out here,
close to the point where you fall off
and the fish suddenly die, with a small increase in h.
Still a lot of fish and so they still can be pretty easy to catch...
They might not be really....might not be immediately noticable.
So, the lesson here,the thing here 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 discontinous one
where a population would go
from an equilibrium value here all the way down to zero.
There is no equilibrium value
in between these two points
So we have this discontinuous behaviour,
even though this function is continuous
and smooth as can be.
So even a continuous and
differential function that varies a function of h
can have this sort of,
potential disastrous and discontinuous behavior.
One more possible scenario for you to note.
So imagine we're moving along this curve
or somewhere along here and we don't really,
but we don't realize because we can't really see this curve
so we increase the fishing rate
a little bit more and we start to plummet down here.
Right, so the population is here, we allowed too much fishing
the population starts to die of,
maybe suddenly people aren't catching fish
we notice the fish stoch 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 the 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 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 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 continual like this one.