Chapter 2.6, Problem 20E

Spawner-Recruit Model In fish management it is important to know the relationship between the abundance of the spawners (also called the parent stock) and the abundance of the recruits—that is, those hatchlings surviving to maturity. According to the Ricker model, the number of recruits R as a function of the number of spawners P has the form

$R=AP{e}^{-Bp}$

for some positive constants A and B. This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits.

In a study of the sockeye salmon, it was determined that $A = 4$ and $B = 0.7$. Here we measure P and R in thousands of salmon.

a. Make a graph of R versus P for the sockeye salmon. (Assume there are at most 3000 spawners.)

b. Find the maximum number of salmon recruits possible.

c. If the number of recruits R is greater than the number of spawners P, then the difference $R - P$ of the recruits can be removed by fishing, and next season there will once again be P spawners surviving to renew the cycle. What value of P gives the maximum value of $R - P$, the number of fish available for removal by fishing?