**Example:** The fox squirrel is a small
mammal native to the Rocky Mountains. These squirrels are very
territorial. Note the following observations:

- 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

.

**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.

Also, discuss the equilibrium (critical) points under the parameter*E*. When do you observe a bifurcation?

**Answer:**

**1.**- The equilibrium (or critical) points are the roots of the
equation
Clearly, we have

*P*=0,*P*=*N*, and*P*=*M*. Using the graph of ,

we get the phase-line of the equilibrium points,

We conclude that

*P*=0 and*P*=*N*are sinks, while*P*=*M*is a source. **2.**- The Slope-Field is given by the following graph:

**3.**- From the Slope-Field, we get the graph of the particular
solution satisfying the condition
*P*(0) = 20.

**4.**- First, the new equation is
,

where

*E*> 0. Clearly, the graph of the functioncan be obtained by shifting the graph of

down

*E*-unit along the vertical axis. Clearly we have three cases according to the value of*E*and the value of*f*at the local maximum :- If 0 <
*E*<*f*(*h*), then we have similar behavior as for*E*=0:

- If
*E*=*f*(*h*), we have two critical points:

- If
*E*>*f*(*h*), we have only one critical point:

Clearly, the bifurcation is happening when

*E*=*f*(*h*). - If 0 <

**
**

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