Chapter 14.1, Problem 28E

Beverton-Holt Model Another model of population growth that has been used to model salmon is the Beverton-Holt model, given by

$f\left(x\right)=\frac{rx}{1+\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$b$}\right.},$

where $r$ and $b$ are positive constants . Source Nature.

a. Find $\underset{x\to \infty }{\lim }f\left(x\right)$ .

b. Show that $f^{\prime }\left(x\right)=\frac{r}{{\left(1+x/b\right)}^2}$.

c. Using the result of part b, for what values of $x$ is $f$ increasing? Decreasing? Does $f$ have an absolute maximum?

d. Show that $f^{″}\left(x\right)=\frac{-2r}{b{\left(1+x/b\right)}^3}$.

e. Using the result of part d, for what values of $x$ is $f$ concave upward? Concave downward? Does $f$ have an inflection point?

f. Using the results from part a-e, draw a graph of this function.

g. In your own words, describe in what ways the graph in part f is similar to the graph of the logistic growth model and the Ricker Model of Exercise $25$, and in what ways it is different.