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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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?

To determine

Part (a)

To find:

Values of

dCdt and dWdt which correspond to stable populations

Explanation

1) Given:

dCdt=aC-bCW

dWdt= -cW+dCW

2) Calculation:

When the population is stable then the growth rate is neither positive nor negative

Therefore,

To determine

Part (b)

To find: How would the statement “The caribou go extinct” be represent mathematically

To determine

Part(c)

To find:

All population pairs (C, W) that lead to stable population.

And According to this model is it possible for the two species to live in balance or will one or will both species become extinct

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