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:
- 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
- Find the equilibrium (critical) points. Classify them as :
source, sink or node. Justify your answers.
- Sketch the slope-field.
- Assume N=100 and M=1 and k = 1. Sketch the graph of the
solution which satisfies the initial condition y(0)=20.
- 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?
- The equilibrium (or critical) points are the roots of the
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.
- The Slope-Field is given by the following graph:
- From the Slope-Field, we get the graph of the particular
solution satisfying the condition P(0) = 20.
- 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:
Clearly, the bifurcation is happening when E=f(h).
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