Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 33

Predator - prey models are often used to study the interaction between species, according to the study of ecosystems. Consider populations of Tundra Wolves, given y $w(t)$, and Caribou, given by $c(t)$, in Northern Canada. The interaction has been modeled by the equations.

$\displaystyle \frac{dc}{dt} = ac - bcw \qquad \frac{dw}{dt} = -cw + dcw$

a.) What values of $\displaystyle \frac{dc}{dt}$ and $\displaystyle \frac{dw}{dt}$ correspond to stable populations?
b.) How would the statement "The Caribou go extinct" be represented mathematically?
c.) Suppose that $a = 0.05$, $b = 0.001$, $c = 0.05$, and $d = 0.0001$. Find all population pairs $(c,w)$ that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

a.) If the populations are stable, the rate of change are neither increasing nor dercreasing, that is, $\displaystyle \frac{dc}{dt} = 0$ and $\displaystyle \frac{dw}{dt} = 0$.

b.) "The Caribou go extinct" can be represented mathematically as $c(t) = 0$

c.) For the satbility of the Caribou,

$
\begin{equation}
\begin{aligned}
\frac{dc}{dt} = 0 &= 0.05 c - 0.001 cw\\
\\
0 & = c ( 50 - w )\\
\\
\text{so, } (c,w) &= (0,50)
\end{aligned}
\end{equation}
$

For the stability of the Tundra Wolves,


$
\begin{equation}
\begin{aligned}
\frac{dw}{dt} = 0 &= -0.05 c + 0.0001 cw\\
\\
0 & = w(c-500)\\
\\
\text{so, } (c,w) &= (500,0)
\end{aligned}
\end{equation}
$


We can conclude that the two populations can live in balance if $(c,w) = (500,50)$ or the population of the Wolves and Caribou can be represented in the ratio of 1:10.