Problem set #3

1.

In 1926, the Lotka-Volterra preditor-prey model was proposed as a means of modeling the populations of two co-existed species where one feeds upon the other.

$$\begin{eqnarray*}\frac{d p}{dt} & = & (a- b r) p \\ \frac{d r}{dt} & = & (-c + d p) r \end{eqnarray*}$$

where a, b, c and d are positive, and p and rare functions of time representing populations. Which function represents the preditor and which the prey? Why?

2.

Under what conditions (in terms of the model coefficients) if any can both species co-exist forever?

3.

Study the local behavior near any equilibrium points? Sketch the phase portrait of the system for each type of stability. Bonus: Use a Maple or Matlab to generate a phase portrait for specific examples of these cases.

4.

A person saves $200 per week into a retirement account beginning at age 25. She plans to retire at age 60. If the retirement account earns 5.125% interest quarterly, develop a mathematical model for the balance in her account. How much money will she have saved when she retires? (Hint: Review geometric series and partial sums of geometric series.)

5.

Referring back to the previous problem, suppose her account earns 5% compounded continuously. Develop a mathematical model for the balance in her account. How is this model different from your previous model with quarterly interest? What is the connection between the two? How much money will she have saved when she retires?