Answer
(a)
If the populations are stable, then the growth rates are neither positive nor negative; that is, $\frac{dC}{dt}$ = $0$ and $\frac{dW}{dt}$ = $0$
(b)
“The caribou go extinct” means that the population is zero, or mathematically, $C$ = $0$
(c)
we have the equations $\frac{dC}{dt}$ = $aC-bCW$ and
$\frac{dW}{dt}$ = $-cW+dCW$
Let $\frac{dC}{dt}$ = $\frac{dW}{dt}$
$a$ = $0.05$, $b$ = $0.001$, $c$ = $0.05$ and $d$ = $0.0001$
$0.05C-0.001CW$ = $0$ is 1st equation
$-0.05W+0.0001CW$ = $0$ is 2nd equation
so
$W$ = $0$ or $50$ and
$C$ = $0$ or $500$
Thus, the population pairs $(C,W)$ that lead to
stable populations are $(0,0)$ and $(500,50)$. So it is possible for the two species to live in harmony.
Work Step by Step
(a)
If the populations are stable, then the growth rates are neither positive nor negative; that is, $\frac{dC}{dt}$ = $0$ and $\frac{dW}{dt}$ = $0$
(b)
“The caribou go extinct” means that the population is zero, or mathematically, $C$ = $0$
(c)
we have the equations $\frac{dC}{dt}$ = $aC-bCW$ and
$\frac{dW}{dt}$ = $-cW+dCW$
Let $\frac{dC}{dt}$ = $\frac{dW}{dt}$
$a$ = $0.05$, $b$ = $0.001$, $c$ = $0.05$ and $d$ = $0.0001$
$0.05C-0.001CW$ = $0$ is 1st equation
$-0.05W+0.0001CW$ = $0$ is 2nd equation
so
$W$ = $0$ or $50$ and
$C$ = $0$ or $500$
Thus, the population pairs $(C,W)$ that lead to
stable populations are $(0,0)$ and $(500,50)$. So it is possible for the two species to live in harmony.
