Problem 3. The fox squirrel is a small
mammal native to the Rocky Mountains. These squirrels are very
territorial:

if the population is large, their rate of growth
decreases or even becomes negative;

if the population is too small, fertile adults run the risk of
not being able to find suitable mates, so again the rate of growth is negative

The carrying capacity N indicates what population is too big, and
the sparsity parameter M indicates when the population is too small.
A mathematical model which will agree with the above assumptions is
the modified logistic model:

1.

Find the equilibrium (critical) points. Classify them as :
source, sink or node. Justify your answers.

2.

Sketch the slope-field.

3.

Assume N=100 and M=1 and k = 1. Sketch the graph of the
solution which satisfies the initial condition y(0)=20.

4.

Assume that squirrels are emigrating (from a certain region)
with a fixed rate E. Write down the new differential equation.
Discuss the equilibrium (critical) points under the parameter E.
When do you observe a bifurcation?