   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 pn is 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 p0 > 0. (a) Show that if {pn} is convergent, then the only possible values for its limit are 0 and b − a. (b) Show that pn+1 < (b/a)pn. (c) Use part (b) to show that if a > b, then limn→∞ pn = 0; in other words, the population dies out. (d) Now assume that a < b. Show that if p0 < b − a, then {pn} is increasing and 0 < pn < b − a. Show also that if p0 > b − a, then {pn} is decreasing and pn > b − a. Deduce that if a < b, then limn→∞ pn = b − a.

(a)

To determine

To show: If {pn} convergent, then the only possible values for the limit are 0 and ba .

Explanation

Given:

The sequence {pn} is pn+1=bpna+pn .

Proof:

Obtain the limit of the sequence.

Consider the sequence, pn+1=bpna+pn .

Take limit on both sides.

limnpn+1=limn(bpna+pn)=limn(bpn)limn(a+pn)=blimnpnlimna+limnpn=blimn

(b)

To determine

To show: pn+1<(ba)pn

(c)

To determine

To show: If b>a , then limnpn=0 .

(d)

To determine

To show:

(1) If p0<ba, then {pn} is increasing and 0<pn<ba .

(2) If p0>ba, then {pn} is increasing and pn>ba .

(3) If a<b , then limnpn=ba .

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