Chapter 14.1, Problem 31E

Shepherd Model The Shepherd model, a modification of the Beverton-Holt model of Exercise $28$, given by

$f\left(x\right)=\frac{rx}{1+{\left(x/b\right)}^c},$

where $r,b$ and $c$ are positive constants. Source: Nature. For Exercise $31-33$, assume $c=2$.

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

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

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

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

e. Using the result of part d, for what values of $x$ is $f$ concave upward? Concave downward? At what value of $x$ 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 to the Ricker Model of Exercise $25$, and in what ways it is different.