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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 9.6, Problem 10E

Textbook Problem

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Populations of aphids and ladybugs are modeled by the equations

(a) Find the equilibrium solutions and explain their significance.

(b) Find an expression for *dL*/*dA.*

(c) The direction field for the differential equation in part (b) is shown. Use it to sketch a phase portrait. What do the phase trajectories have in common?

(d) Suppose that at time *t* = 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change.

(e) Use part (d) to make rough sketches of the aphid and ladybug populations as functions of *t*. How are the graphs related to each other?

(a)

To determine

**To find:** The equilibrium solution and their significance.

**Given:**

The equations are

**Calculation:**

Consider *A* and *L* be constants.

Differentiate with respect to *t*,

From given,

Solving further,

(b)

To determine

**To find:** An expression for

(c)

To determine

**To draw:** The phase trajectories.

(d)

To determine

**To draw:** The phase trajectories.

(e)

To determine

**To draw:** The rough sketch for ladubugs and aphids.

Calculus: Early Transcendentals

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