Chapter 2.3, Problem 18P

Suppose that a certain population satisfies the initial value problem $\frac{dy}{dt}=r(t)y-k,y(0)=y_0$,

Where the growth rate $r\left(t\right)$ is given by

$r(t)=\left(1+\sin t\right)/5$, and $k$ represents the rate of predation.

Suppose that $k=\frac{1}{5}$. Plot $y$ versus $t$ for several values of $y_0$ between $\frac{1}{2}$ and $1$.

Estimate the critical initial population $y_c$ below which the population will become extinct.

Choose other values of $k$ and find the corresponding $y_c$ for each one.

Use the data you have found in parts (a), (b) and (c) to plot $y_c$ versus $k$.