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.