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:

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:

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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

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Clearly, we have P=0, P=N, and P=M. Using the graph of tex2html_wrap_inline72,



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

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where E > 0. Clearly, the graph of the function

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can be obtained by shifting the graph of

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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 tex2html_wrap_inline96 :

Note that

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

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Author: Mohamed Amine Khamsi

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