Selected Answers to Problem Set 10

2. A simple experiment can be done as follows:

• We know that, if there are no rabbits, then there is no food supply for the foxes. Therefore, we can find the death rate of the foxes by simply observing the decline.
• We also know that, if there are no foxes, then the rabbit population will continue to increase without an upper bound. Therefore, we can find the growth rate of the rabbits by simply observing the increase.
• With these two constants known, we need three more observations to find the three remaining unknowns. One was would be to look at a local max for both the number of rabbits and the number of foxes. This will be, for the foxes, an "equilibrium" between both the interaction of the two species and the total number of foxes. The same can be applied for the rabbits--an equilibrium between both the interaction of the two species and the total number of rabbits.
• Knowing four unknown constants, we can use one more observation to find the fifth unknown. Any scheme that uses five unknown equations will work, since this will create a homogeneous system of equations.

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3. To find when x' and y' are greater than and less than zero, we simply find the parts of the graph when their equations are the same. x' < 0 when y - x²2, which will be on the outside of the parabola. x' > 0 on the inside of the parabola. y' < 0 when x - 1 < 0, or when x < 1. y' > 0 when x > 1.

Now, if x(0) = 2 and y(0) = 1, then from the analysis of the phase plane we know that x' < 0 and y' > 0, so the point will move up and to the left towards the parabola. Once it reaches the parabola, the x' turns from negative to positive, while y' remains positive. Therefore, at the point on the parabola, it begins to change direction and grows to (inf, inf).

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4. The same procedure applied as before. x' < 0 when y < 0, and x' > 0 when y > 0. y' < 0 when x² + y² - 1 > 0, or inside the unit circle. y' > 0 on the outside of the unit circle.

If x(0) = 0 and y(0) = 2, then we know that x' > 0 and y' > 0, so the point will trend towards (inf, inf).