A recent article described the population (S) of cod in the North Sea next year as a function of this years population S (in thousands

A recent article described the population ƒ(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.

where a, b, and c are constants.

(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 44 of this section, f(S) = Se^r(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.

Shepherd: f(S) = Beverton-Holt: f(s) Ricker: f(S) ase-bs; as 1 + (S/b)' = as 1 + (S/b) =