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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

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

d C d t = a C b C W d W d t = c W + d C W

(a) What values of dC/dt and 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)

To determine

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

Explanation

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

dWdt=dCWcW (1)

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

dCdt=aCbCW (2)

The populations of wolves’ W

(b)

To determine

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

(c)

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.

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