   Chapter 11.1, Problem 93E

Chapter
Section
Textbook Problem

The size of an undisturbed fish population has been modeled by the formula p n + 1 = b p n a + p n where is p n the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p 0 > 0 .(a) Show that if { p n } is convergent, then the only possible values for its limit are 0 and b − a .(b) Show that p n + 1 < ( b / a ) p n .(c) Use part (b) to show that if a > b , then lim n → ∞ p n = 0 ; in other words, the population dies out.(d) Now assume that a < b Show that if p 0 < b − a , then { p n } is increasing and 0 < p n < b − a . Show also that if p 0 > b − a , then { p n } is decreasing and p n > b − a . Deduce that if a < b then lim n → ∞ p n = b − a .

To determine

Part (a)

To show:

If pn is convergent, then the only possible values for its limit are 0 and b-a.

Explanation

1) Concept:

Use the given information to show thatpossible values for limit of pn are 0 and b-a.

2) Given:

pn+1=bpna+pn

3) Calculation:

Suppose pn converges to p.

Then,

pn+1=bpna+pn

Taking limit,

limnpn+1=limnbpna+pn

lim<

To determine

Part (b)

To show:

pn+1<bapn

To determine

Part (c)

To show:

If a>b, then limnpn=0.

To determine

Part (d)

To show:

(i) If p0<b-a, then pn is increasing and 0<pn<b-a

(ii) If p0>b-a, then pn is decreasing and pn>b-a

(iii) Deduce that  if a<b, then limnpn=b-a

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Calculate y'. 49. y=cos(etan3x)

Single Variable Calculus: Early Transcendentals, Volume I

Convert the expressions in Exercises 6584 to power form. x3

Finite Mathematics and Applied Calculus (MindTap Course List)

Other Equations Find all real solutions of the equation. 87. x2x+100=50

Precalculus: Mathematics for Calculus (Standalone Book)

Find the derivatives of the functions in Problems 1-14. 7.

Mathematical Applications for the Management, Life, and Social Sciences

Simplify: 3642

Elementary Technical Mathematics 