I promise I\'ll award all the points to whoever helps me the most. a or A=alpha
ID: 2986644 • Letter: I
Question
I promise I'll award all the points to whoever helps me the most.
a or A=alpha
In this laboratory exercise, you will study a one-parameter family of nonlinear, firstorder systems consisting of predator-prey equations. The family is
dx
dt =9x%u2212 a*x^2%u22123xy
dy
dt =-2y+xy,
where a%u22650 is a parameter. In other words, for different values of a we have different
systems. The variablexis the population (in some scaled units) of prey, and yis the
population of predators. For a given value of a, we want to understand what happens to
these populations as t approaches infinity. .
You should investigate the phase portraits of these equations for various values
of A in the interval 0 %u2264 A %u22645. To get started, you might want to try A =0, 1, 2, 3,
4, and 5. Think about what the phase portrait means in terms of the evolution of the
xandypopulations. Where are the equilibrium points? What does linearization tell
you about their types? What happens to a typical solution curve? Also, consider the
behavior of the special solutions where either x=0 or y=0 (solution curves lying on
the x-or y-axes).
Determine the bifurcation values of A %u2014that is, the values of A where nearby A%u2019s
lead to %u201Cdifferent%u201D behaviors in the phase portrait. For example,A=0 is a bifurcation
value because for A =0, the long-term behavior of the populations is dramatically
different than the long-term behavior of the populations if A is slightly positive. The
technique of linearization suggests bifurcation values.
Your report:After you have determined all of the bifurcation values for A in the interval 0%u2264A%u22645, study enough specific values of A to be able to discuss all of the various
population evolution scenarios for these systems. In your report, you should describe
these scenarios using the phase portraits and x(t)-andy(t)-graphs. Your report should
include:
1.A brief discussion of the significance of the various terms in the system. For example, what does the 9xrepresent? What does the 3xyterm represent?
2.A discussion of all bifurcations including the bifurcation at A =0. For example, a
bifurcation occurs between A =3 and A=5. What does this bifurcation mean for
the predator population?
Address the questions above in the form of a short essay, and support your assertions
with selected illustrations. (Please remember that although one good illustration may
be worth 1000 words, 1000 illustrations are usually worth nothing).
Explanation / Answer
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