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

Chapter 9, Problem 30

(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 \(x_{n}\) 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 \(R>1\) and \(K>0\).

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

(b) Show that \(x_{n}\) is decreasing.

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

Equation Transcription:

L

Text Transcription:

n

x_n

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

R

K

R > 1

{x_n}

x_1 > K

R > 1

K > 0

x_n > K

x_n+1 > K

n geq 1

L