The slope field for a population modeled by
is shown in the
figure below.
(a)
On a print-out of the slope field, sketch three non-zero solution curves
showing different types of behavior for the population .
Give an initial condition that will produce each:
,
, and
.
(b)
Is there a stable value of the population? If so, give the value; if
not, enter none:
Stable value =
(c)
Considering the shape of solutions for the population, give any intervals
for which the following are true. If no such interval exists, enter
none, and if there are multiple intervals, give them as a
list. (Thus, if solutions are increasing when is between
1 and 3, enter (1,3) for that answer; if they are decreasing when
is between 1 and 2 or between 3 and 4, enter (1,2),(3,4). Note that
your answers may reflect the fact that is a population.)
is increasing when is in
is decreasing when is in
Think about what these conditions mean for the population, and be sure
that you are able to explain that.
In the long-run, what is the most likely outcome for the population?
(Enter infinity if the population grows without
bound.)
Are there any inflection points in the solutions for the population?
If so, give them as a comma-separated list (e.g., 1,3); if not, enter
none.
Inflection points are at
Be sure you can explain what the meaning of the inflection points is for
the population.
(d)
Sketch a graph of against . Use your graph to answer
the following questions.
When is positive?
When is in
When is negative?
When is in
(Give your answers as intervals or a list of
intervals.)
When is zero?
When
(If there is more than one answer, give a list of answers,
e.g., 1,2.)
When is at a maximum?
When
Be sure that you can see how the shape of your graph of
explains the shape of solution curves to the differential equation.