Exercise 9

Let $y(t)$ be the number of thousands of mice that live on a farm; assume time $t$ is measured in years.

1. The population of the mice grows at a yearly rate that is twenty times the number of mice. Express this as a differential equation.

2. At some point, the farmer brings $C$ cats to the farm. The number of mice that the cats can eat in a year is $$\begin{gathered} M(y) = C\frac{y}{2+y} \end{gathered}$$ thousand mice per year. Explain how this modifies the differential equation that you found in part a).

3. Sketch a graph of the function $M(y)$ for a single cat $C=1$ and explain its features by looking, for instance, at the behavior of $M(y)$ when $y$ is small and when $y$ is large.

4. Suppose that $C=1$. Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice.

5. Suppose that $C=60$. Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice.

6. What is the smallest number of cats you would need to keep the mice population from growing arbitrarily large?