2930 The BevertonHolt model is used to describe changesin a population from one

Chapter 9, Problem 29

(choose chapter or problem)

The Beverton-Holt model is used to describe changes in a population from one generation to the next under certain assumptions. If the population in generation \(n\) is given by \(x_{n}\), the Beverton-Holt model predicts that the population in the next generation satisfies

\(x_{n+1}=\frac{R K x_{n}}{K+(R-1) x_{n}}\)

for some positive constants \(R\) and \(K\) with \(R>1\). These exercises explore some properties of this population model.

Let \(\left\{x_{n}\right\}\) be the sequence of population values defined recursively by \(x_{1}=60\) , and for \(n \geq 1, x_{n+1}\) is given by the Beverton-Holt model with \(R=10 \text { and } K=300 \text {. }\)

(a) List the first four terms of the sequence \(\left\{x_{n}\right\}\).

(b) If \(0<x_{n}<300\) , show that \(0<x_{n+1}<300\) . Conclude that \(0<x_{n}<300 \text { for } n \geq 1\)

(c) Show that \(\left\{x_{n}\right\}\) is increasing.

(d) Show that \(\left\{x_{n}\right\}\) converges and find its limit \(L\).

Equation Transcription:

 and

Text Transcription:

n

x_n

x_n+1 = RKx_n / K + (R-1)x_n

R

K

R > 1

{x_n}

x_1 = 60

n geq 1, x_n+1

R = 100 and K = 300

0 < x_n < 300

0 < x_n+1 < 300

n leq 1

L