# Carrying Capacity

A population of wolves was transplanted into central Idaho some years ago, and immediately came under Federal protection. Under that protection, the wolves thrived and multiplied, and now the population has grown large enough that the Feds have withdrawn the protection, and the state of Idaho plans to allow hunting of the wolves.

The problem for the state is this: Idaho has historically favored resource extraction over all other uses of the land, hence Idaho has a strong desire to help, among others, cattle ranchers and elk hunters. The cattle ranchers hate wolves passionately, and the elk hunters do not want to share their resource with wolves, hence the state wants to reduce the wolf population to the lowest level it can get away with. On the other hand, if the population drops too low, the Federal protection will kick in again, and the state's hands will be tied. Thus, the state plans to manage wolves so that the number of breeding pairs remains above 10, which is the number at which the state fears that Federal protection would be invoked.

We can study this decision using some classical differential
equations. We will model it in terms of breeding pairs *w(t)*.
It is obvious that the number of breeding pairs increases proportionately
to the current number of breeding pairs: *w'=rw*.
On the other hand, the number of breeding pairs is reduced
in proportion to the number of pairs there are over the number the
terrain can support - the carrying capacity *C*.
This results in the **logistic equation** for the wolf population:

w' = rw(C - w)

Now we have to estimate the coefficients *r* and *C*.
We know that when there is plenty of food and territory for the wolves,
there will be no downward pressure on their population, so we
could abstract the equation as *w'=rCw*. The planning
document gives numbers that indicate that there were three breeding
pairs of wolves in Idaho in 1996, and that by 2007 there were 43.

**Exercise 1:**
We will think of 1996 as year zero in the history of wolves in
Idaho.
Solve the differential equation *w' = rCw* for *w(t)*,
and then use the information that *w(0) = 3* and
*w(11) = 43* to solve for *rC*. This is an
estimate of the growth
rate for the number of breeding pairs.

The coefficient *C* is the number of breeding pairs above which
there would be downward pressure on the population of wolves - the
carrying capacity. This number is dependent on an incomprehensibly
large collection of other factors, including availability of food,
toleration of humans for the wolves, and competition among the
wolves themselves. Indeed, it would vary over time. Finding a single
such number would be difficult or impossible. On the other hand,
with the current 43 pairs there is considerable human pressure to
reduce the population. For this project
we will simply suppose that if the number of breeding pairs went over
90, then people would take matters into their own hands to reduce the
population: we will take *C = 90*.

**Exercise 2**: Solve the equation (Hint: use separation of variables. You will
need to review integration by partial fractions).

**Exercise 3**: Use the applet below to plot solutions to the differential equation.
We have initialized it with our assumed values for *r* and
*C*. Note that, since we don't really know *C*, the
solutions probably don't have much to do with reality. What
qualitative features of the plot do you notice that do have
some correlation with the way the wolf population changes over time?
Hint: change the *C* value and see the ways in which the shape
of the plot stays the same.

To use the applet, just click on a point *(t,w)* that you
want to use as an initial condition in the plot. The solution corresponding
to that initial condition will appear. If you want to change parameters,
click on the parameter you want to change in the area at right. The
current value for that parameter will appear, and you can change it.
Then click the "Set/Redraw" button followed by "Clear", and pick
some new initial conditions.

The policy developed by the state of Idaho is to have essentially
unrestricted hunting of wolves when the number of breeding pairs is
over 20, to have more careful hunting when the number of breeding pairs
is between 10 and 20, and to stop hunting if the number of
pairs drops to 10 or below. We can model this mathematically by
saying that the population of wolves is decreased by hunting
proportionally to *w(t)-10* when *w > 0*, and by zero
otherwise.
The equation is thus

w' = rw(C - w) - H(w),

where *H(w)* is a function with value *h(w-10)* whenever *w>10*,
for some constant *h*, and
and zero otherwise.

**Exercise 4**: Use the applet provided below to plot solutions to this
new equation that includes the hunting policy. There are
now two unknown parameters to vary: *C* and *h*.
What difference in qualitative behavior do you notice
when this equation is compared to the equation with no hunting?
Explain.

Idaho has always had a certain level of illegal take for the wolves they
have. A certain number of ranchers have a shoot-on-sight policy of
their own with wolves, and Idaho winks at it. If we introduce
a constant illegal take *i* into the equation, it might look like
the following.

w' = rw(C - w) - i - H(w),

**Exercise 5**: Use the applet below to
examine the qualitative behavior of this
new equation. Discuss how it differs from that for the equation
with only legal hunting. Hint: Look particularly at levels of
wolf pairs below 10.

Documentation and status for this plan may be found at the Idaho Fish and Game web site.

With the advent of HTML5, Javascript is now ready for prime time
for mathematical applications. There are
new Javascript demos
illustrating how we might use interactive web objects to
help students learn Calculus.