Can someone explain how to solve this problem?

Question

Rates of Change

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W(t), and caribou, given by C(t), in northern Canada. The interaction has been modeled by the equations dC/dt=aC−bCW, dW/dt = −cW + dCW.

(a) What values of dC/dt and dW/dt correspond to stable populations?

(b) Suppose a=.05, b=.01, c=.05, d=.0001. After finding population pairs (P, C) that lead to stable populations, is it possible for the species to live in balance or both species become extinct? Answer "b" for balance and "e" for extinction.

Answer 1

I assume there is a typo in the question:

"(P,C)" should be "(W,C)"

(a)

Stable populations mean populations not changing in size. That is,

dC/dt = 0 and dW/dt = 0

(b)

Rewrite the equations as

dC/dt = C(0.05 - 0.01W)

dW/dt = W(-0.05 + 0.0001C)

Suppose there is a time when the populations are stable, i.e.

0 = C(0.05 - 0.01W)

0 = W(-0.05 + 0.0001C)

Case 1: C = 0 satisfies the first equation

After substituting C = 0 into the second equation we get W = 0

Case 2: W = 0 satisfies the second equation

After substituting W = 0 into the first equation we get C = 0

Case 3: C ≠ 0 and W ≠ 0

Then

W = 5 from the first equation

C = 500 from the second equation

The conclusion is that there are two solutions to stable population:

1) W = 0 and C = 0

2) W = 5 and C = 500

The answer to the question is that it is possible for the two populations to live in balance.