Homework #4
Dear class,
I have had a few questions come to me about the homework. I want to
clarify them for you.
A warning on problem 1: Consider exponential growth to determine the
value of r based on small populations (near equilibrium behavior).
However, you cannot keep using the exponential growth model for part
(b).
For problem 2a, I say something about d(0). Some students have
complained that this doesn't make sense because if there aren't any
individuals in the population then there shouldn't be any death. I want
to remind all of you that d(N) is the PER CAPITA death rate. The fact
that nobody dies when there is zero population comes from the TOTAL
death rate which is given by multiplying d(N) by N (which is zero).
Now, a second clarification. Even with this point, students wonder why
we would even care about d(0) if there aren't any individuals to die.
This has more to do with how we try to model the death rate. In this
problem, the death rate increases linearly with the size of the
population. So if we graphed the death rate d(N) as a function of N, we
should get a line. The statement about d(0) being nonzero indicates
that the line has a positive intercept. Alternatively, d(0) could be
considered the limit of d(N) as N approaches zero, especially if N is
measured in a quantity such as millions of individuals where N=0.000001
would mean a single individual.
For parts (b) and (c), you need to discover that this model simplifies
to the logistic growth model (where we know the carrying capacity and
the exact solution as a function of time) by observing that the
differential equations really have the same functional form. (But you
need to determine how your parameters relate to those of the logistic
growth model.) Alternatively, to find the carrying capacity, you could
look for a nonzero equilibrium.
For problem (3a) and (3c), do not overlook the request to think about
and comment on biological interpretations of mathematical results.
I will have Tuesday office hours (10:30-11:30) repeated on Thursday, as
well as office hours immediately after class on Friday. I will not be
available Thursday afternoon as I need to attend my son's
Parent/Teacher conference.
Best wishes,
D. Brian Walton
AMath 383 Instructor
Guggenheim 408C
(206) 685-9298