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Chapter 1: Q5E (page 1)

The logistic equation for the population (in thousands) of a certain species is given by dpdt=3p-2p2.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

猞� If p(0)=0.8, what is limt鈫�+鈭�p(t)?

猞� Can a population of 2000 ever decline to 800?

Short Answer

Expert verified

猞� The Sketch is drawn for the direction field

猞� The limiting population is 32

猞� The limiting population is 32

猞� No

Step by step solution

01

1(a): Drawing the Sketch for the direction field of the given equation

Hence, the Sketch is drawn for the direction field.

02

3(b): Applying the initial condition  p(0)=3

Hence, the limiting population is .

03

4(c): Applying the initial condition p(0)=0.8  in the solution

32-2c2=0.8c2=1-31.6c2=-0.875Now,p=3e3t2e3t+1.75limt鈫�鈭�p(t)=32

Hence, the limiting population is 32.

04

5(d): Analyzing the graph and the different initial conditions

From the above two parts (b), (c) and the graph,

the limiting value of population approaches 1.5 (i.e., 1500) as t tends to infinity.

Hence, the population of 2000 can never decline to 800.

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