in this video I will introduce
a modified version
of the logistic equation
called the logistic equation with harvest.
I will use this model to show
an example of a bifurcation
and will construct a bifurcation diagram;
a realy important type of diagram
n the study
of all sort of dynamical systems.
The way I'll do this
might seem a little bit slow or involved.
The bifurcation diagram is a bit abstract
and so I think
building it up step by step,
kind of carefully, is really helpful
for getting an understanding
of what it means.
So I'll be presenting things
perhaps a little slower
than seems necessary
but my experience has been
that doing so really helps
when it comes time
to put all this together
and look at the bifurcation diagram.
Ok, so here is
the logistic equation with harvest.
Without this h term,
this is the regular logistic equation:
r is the growth rate
and k is the carrying capacity.
And I'm going to add another parameter h
which we can interpret as the harvest rate.
So for concreteness I think
that we are modeling some fisheries
in a lake or a bay or something,
and h here is ...
it would just be a number,
like a number of fish.
So a manager might set h at 50:
catch 50 fish a year,
or 100 fish or 3 fish.
Notice that these terms here
are density dependent.
The growth rate depends on
the numbers of fish that are present (P).
The h term (how many we catch)
does not depend
on the number of fish that are present.
That's maybe not
an entirely accurate view of things
but maybe for some fish
that are dumb or slow or easy to catch,
you can catch them equally well
independent of how many there are.
And as you probably guessed
this is a model to not take too seriously
(I don't think that anybody thinks
fisheries are
really, really described by this)
but it's a model that will help us think
through some trade-offs and ideas
and I will talk more about
how we can generalize
from this model towards
the end of this video or the next one.
Ok, so we've got
logistic equation with harvest
and what I am going to do
is analyse the logistic equation
using qualitative techniques
for different values of h.
The first value I am going to do
will be a little bit borring and redundant
because we've already looked at this.
And so this is the case where h is zero
so this means
we aren't harvesting any fish at all.
And this is the same number
as I plugged in before
so the picture is the same.
If the population is in here it's increasing,
if the population is above 100 it's decreasing.
So we have a stable fixed point at 100
and then an unstable fixed point at zero.
I'll draw the phase line for that....
here it is, here's a fixed point .....
there's a fixed point ....
Now I am going to use
a scale that's designed
to sort of line up with what's on this
and I'll make a note to myself that
this was the case when h is 0.
Now I should put a few more arrows on....
Ok, so this is just
what we did in the last video.
Stable fixed point at 100,
unstable fixed point at zero
and this is the case when
we are not harvesting any fish at all.
So now we analized it for h = zero
I am going to do the same thing
but for a different value of h
So here is h = 40
So let me just say a little bit
about this before we do the phase line.
The effect of this h term - just geometrically,
when you subtract
a constant from a function
it moves it down, vertically.
So here this is the graph where h = 0
h = 40, I just subtract 40 from it
so if I took this
and shifted it down by 40
I would get this curve.
So here is the curve
and lets see what the dynamics are now.
What fixed points are there?
What are their stablity?
So now we have positive growth
if we are in between
this number and this number.
So in between here, the fish increase.
If the number of fish is greater than 83 or 84
the number of fish will decrease,
move this way.
And now, if I am
less than, say, 17 or 18 fish,
then I'll move towards zero
and then negative infinity .
So if we start, say, with 15 fish,
that is bad news for the fish
because they will all die out.
So let me draw the phase line
for this situation.
Here is my line and I have
a fixed point there, fixed point there.
In between these two fixed points
the population increases,
above it decreases, and so on,
And this is h = 40.
Note by the way that the steady population
- the carrying capacity used to be 100
and now if I harvest 40 a year
presumably that would move it down to 60
and it doesn't quite work that way
because this has a pretty large growth rate,
so yes you are harvesting
but they are also sort of growing back
part of the way towards the 100
which is what it would want to be,
what the population would go to
if h wasn't here.
So now the new steady state
for this fishery would be 82, 83,
but note also that
there is a critical number of fish
and if you get underneath this population
the population will die off
and it will go down to zero.
Ok, so, not surprising I don't know
anything at all about fish
but this seems like a
not crazy story to tell about fish.
Ok, so that's h = 40
Now I am going to try another h value
so this is h = 65
again notice that
as h gets larger this curve
(it's an upside down parabola)
moves down the axis.
So now this is moved another 25 down
and it is just kind of
barely peeking up above this axis.
So now we are harvesting
a lot of fish a year,
the carrying capacity is 100,
we are harvesting 65 a year.
You might think 'Oh, is that too many?
Is that going to make
the fish die off?' Well, let's see ....
We do have a reasonable growth in here,
if we're between these two values,
the growth rate (that's on the y axis)
is positive
so the population will increase
till around a little less than 70.
If we are above this value
it will decrease and stop at 70
and if we are to the left of here,
it will decrease towards zero:
the fish will die.
So lets draw the phase line for this.
Here is my line and now I have a fixed point here
and there - remember fixed points occur
when the growth rate is zero.
Sso here and here.
In between these two points,
the population grows.
To the right, the population decreases
(moves to the left)
and if I am underneath - if I am
below this value a little more than 30 -
the fish die off; bad news for the fish.
Ok, so I'll make a note to myself. h = 65.
So that is the phase line
for this differential equation with h = 65.
Alright, I'll do two more h values.
Next we'll do h = 75
and this is an interesting one
now the curve has been lowered enough.
I am subtracting enough fish evey year
that it just barely touches
this point here.
So what might the phase line
look like for this?
And I'll draw the line in....
Now there is only one fixed point.
It occurs right here in the middle
when have 50 fish; a population of 50.
If I'm above this, I decrease,
and if I'm below this I decrease.
So this fixed point is sort of a odd case one we have not encountered much before there is one in the intermediate homework problem on this
We will call it semi stable perhaps because if you move away here you have a lot of fish you will decrease to this you will get closer to this fixed point
But if you are a little less you have to few fish below 50 you are at 49 and the fish will start to die the population decreases
the population is always decreasing except it happenes to hold steady at this one single point ok that is h = 75. I have a nice phase line for that , and one more. Maybe you can guess what is about to happen
Now h will be 85 so I am harvesting 85 fish a year and this enough of a harvest rate that the parabola is now completely below the x axis
That means that the growth rate is always negative so no mater what the population is it is decreasing towards zero and I guess mathematicaly towards negative infinity
So this realy bad news for the fish So I can draw this phase line I put no dots on this phase line because there are no fixed points no equalibrium points
the yust arrows indicating that no mater what the population is it is always decreasing I will make a note that this h value, the parameter h, is 85
Ok so we looked at this diferental equation and I varied this parameter h and we looked at 5 different values and we saw different behaviors
Most of the time there were two fixed points here there was this special case there was one fixed point and here there were zero fixed point
And this makes sense I think from the point of view fish fisherman fisher folks in at as you harvet more and more fish the population declines.
eventually your harvesting so many fish that the fish population goes to zero
So this way of looking at things is we are analysing the equation one parameter at a time I print out a h value I tell you a h value and hopefully now it not to hard to come up with a phase line for this
What a bifurication diagram will give us is a way of picturing all the posible behaviors of this equation for all posible values for h all at once
it is a nice geometric constuction that you can get a lot of information out
so that will be the topic of the next video