Chapter 14.1, Problem 25E

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

$f\left(x\right)=x{e}^{r\left(1-x/N\right)},$

where $r$ is a positive constant measuring growth rate, and $N$ is a value of the population such that $f\left(N\right)=N$ . Source: Journal of the Fisheries Research Board of Canada. We will see the significance of such points in the next section.

a. Find $\underset{x\to \infty }{\lim }f\left(x\right)$. (Hint: See Exercise 72 in Section 3.1.)

b. Show that $f^{\prime }\left(x\right)=e^{r\left(1-x/N\right)}\left(1-rx/N\right)$.

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 $f^{″}\left(x\right)=e^{r\left(1-x/N\right)}\left(rx/N-2\right)\left(r/N\right)$.

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 in what ways it is different.