Recruitment Model Rickers curve describes the relationship between the size of the

Chapter 5, Problem 5

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Recruitment Model Rickers curve describes the relationship between the size of the parental stock of some fish and the number of recruits. If we denote the size of the parental stock by P and the number of recruits by R, then Rickers curve is given by R(P) = Pe P for P 0 where and are positive constants. [Note that R(0) = 0; that is, without parents there are no offspring. Furthermore, R(P) > 0 when P > 0.] We are interested in the size P of the parental stock that maximizes the number R(P) of recruits. Since R(P) is differentiable, we can use its first derivative to solve this problem. (a) Use the product rule to show that, for P > 0, R_ (P) = e P (1 P) R__ (P) = e P (2 P) (b) Show that R_ (P) = 0 if P = 1/ and that R__ (1/) < 0. This shows that R(P) has a maximum at P = 1 . Show that R(1/) = e1 > 0. (c) To find the global maximum, you need to check R(0) and limP R(P). Show that R(0) = 0 and lim P R(P) = 0 and that this implies that there is a global maximum at P = 1/. (d) Show that R(P) has an inflection point at P = 2/. (e) Sketch the graph of R(P) for = 2 and = 1.