Chapter 9.2, Problem 30ES

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 a sequence of population values defined recursively by the Beverton-Holt model for which $x_1>K.$ Assume that the constants R and K satisfy $R>1\text{and}K>0.$

(a) If $x_n>K,$ show that $x_{n+1}>K.$ Conclude that $x_n>K$ for all $n\ge 1.$

(b) Show that $\left\{x_n\right\}$ is decreasing.

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