Population Dynamics: 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 when the population is too big, and the sparsity parameter M indicates when the population is too small. A mathematical model which agrees 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?

Solution.

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