#### To determine

**To find:** The value of dCdt and dWdt that corresponds to a stable population.

#### Answer

The value of dCdt and dWdt that corresponds to a stable population is 0.

#### Explanation

The given equation for the rate of change of wolves’ population is as below.

dWdt=dCW−cW (1)

The given equation for the rate of change of caribou population is as below.

dCdt=aC−bCW (2)

The populations of wolves’ W and caribou *C* amount remains stable, will indicate that there is no change in the value of populations of wolf and caribou with respect to time. No change indicates that the population amount neither increases nor decreases. The first derivative of populations W and C with respect to time will be zero.

dWdt=0

dCdt=0

#### To determine

**To find:** The mathematical representation of the statement “The caribou go extinct”.

#### Answer

The mathematical representation of the statement “The caribou go extinct” is limt→∞C(t)=0.

#### Explanation

The statement “The caribou go extinct” is implying that the population of caribou will be becoming zero, after certain period *t*. The period *t* is not known, but the fact that the caribou will become extinct is true. It might take a very long period, but it will happen.

Apply limit x→∞ for C(t) and equate it to zero.

limt→∞C(t)=0

The mathematical expression for statement “The caribou go extinct” is limt→∞C(t)=0.

#### To determine

**To find:** The stable population levels for the Wolves, *W* and caribou, *C*, and check out whether the two species can live in balance, or will they go extinct.

#### Answer

The population pairs that lead to stable populations of wolf and caribou is (0,0) and (50,500).

#### Explanation

Substitute 0.05 for c and 0.0001 for d in equation (1).

dWdt=0.0001CW−0.05W

For a stable population condition, substitute 0 for dWdt.

0=0.0001CW−0.05W0.0001CW=0.05WC=500

Substitute 0.05 for a, and 0.001 for b in equation (1).

dCdt=0.05C−0.001CW

For a stable population condition, substitute 0 for dCdt.

0=0.05C−0.001CW0.001CW=0.05CW=50

The population pairs possible are (0,0), (50,500), (50,0), and (0,500).

The population pairs of (0,0), (50,500) will lead to stable populations of wolves and caribou.

It is possible for both the species to live in harmony at population level of (50,500).