Population Dynamics: Answer to Example 2

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

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.
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 function

can 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:
  
  
  
  

Note that

Clearly, the bifurcation is happening when E=f(h).

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