Chapter 9.2, Problem 29ES

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

$x_{n+1}=\frac{RK{x}_n}{K+\left(R-1\right)x_n}$

For some positive constant $R\text{ and }K\text{ with }R>1.$ These exercise explore some properties of this population model.

Let $\left\{x_n\right\}$ be the sequence of population values defined recursively by $x_1=60,\text{and}\text{for}n\ge 1,x_{n+1}$ is given by the Beverton -Holt model with $R=10\text{ and }K=300.$

(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 $0<x_n<300\text{for}n\ge 1.$

(c) Show that $\left\{x_n\right\}$ is increasing.

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