Chapter 14.2, Problem 45E

Harvesting Cod A recent article described the population f(S) of cod in the North Sea next year as a function of this year’s population S (in thousands of tons) by various mathematical models.

      $\begin{array}{cc}\text{Shepherd:}& f\left(S\right)=\frac{aS}{1+{\left(S/b\right)}^c};\\ \text{Ricker:}& f\left(S\right)=aS{e}^{-bS};\\ \text{Beverton-Holt:}& f\left(S\right)=\frac{aS}{1+\left(S/b\right)},\end{array}$

where a, b, and c are constants. Source: Nature.

(a)    Find a replacement of variables in the Ricker model above that will make it the same as another form of the Ricker model described in Exercise 40 of this section, f(S) = Se^(1–S/P).

(b)    Find f′(S) for all three models.

(c)    Find f′(0) for all three models. From your answer, describe in words the geometric meaning of the constant a.

(d)    The values of a, b, and c reported in the article for the Shepherd model are 3.026, 248.72, and 3.24, respectively. Find the value of this year’s population that maximizes next year’s population using the Shepherd model.

(e)    The values of a and b reported in the article for the Ricker model are 4.151 and 0.0039, respectively. Find the value of this year’s population that maximizes next year’s population using the Ricker model.

(f)    Explain why, for the Beverton-Holt model, there is no value of this year’s population that maximizes next year’s population.