   # Populations of aphids and ladybugs are modeled by the equations d A d t = 2 A − 0.01 A L d L d t = − 0.5 L + 0.0001 A L (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? ### Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

#### Solutions

Chapter
Section ### Calculus: Early Transcendentals

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 d A d t = 2 A − 0.01 A L d L d t = − 0.5 L + 0.0001 A L (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.

### Explanation of Solution

Given:

The equations are dAdt=2A0.01AL and dLdt=0.5L0.0001AL .

Calculation:

Consider A and L be constants.

Differentiate with respect to t,

From given,

0=2A0.01AL

0=0.5L0.0001AL

Solving further,

0=A(20

(b)

To determine

To find: An expression for dLdA .

(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.

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