FISHERY BULLETIN: VOL. 74, NO. 4 



an added complication is provided by the fact that 

 the run in any year represents the progeny of 

 several spawning groups. 



The Ricker spawner-return relation, for a single 

 spawning group, is given by 



R^ = aE^_,,e 



-bE. 



(1) 



where R,, is the return in year n resulting from an 

 escapement E„_,, k years prior, e is the base of 

 natural logarithms, and a and b are parameters 

 assumed unique for any river system or spawning 

 group. We may generalize Equation (1) to the case 

 of multiple spawning groups to give 



K 



Rn = ^a.E^.^e-''^'-^ (2) 



k - 1 



where the relevant spawning occurs over the years 

 (w - 1) through (w - K). In Equation (2) the 

 coefficients (o^.) now reflect not only the magni- 

 tude of the run, as in Equation (1), but the 

 proportion of the run arising from each spawning 

 group. Specifically we can write a^. = a/\ where a 

 is the parameter in Equation (1) and ip), is propor- 

 tion of the run in year n arising from spawners in 



K 



year (n - k). We have the condition X ft = 1 



fc = 1 



from which it follows that 7^ (h = a. 



k = 1 



The return as given by Equation (1) or Equation 

 (2) is a deterministic function of the parameters. 

 In actual practice, however, the return, from the 

 biologist's point of view, is a random variable in 

 which case some additive or multiplicative error 

 term must be appended to Equation (1). Thus, at 

 an appropriate point in the analysis, the return 

 will be assumed to be a random variable whose 

 expected value is given by Equation (1). 



Let Xn be the catch in year n. Then 



X„ = a ^ PkE„. 



i-e 



- bE 



E„ 



k = 1 



Let Xtot be the total catch over some fixed but 

 otherwise arbitrary number, say n*, of fishing 

 seasons. Then 



-^tot — 



n' r- K 



2 I 2 



n = 1 Lk = I 



OkK-k^' 



bE 



eX 



(3) 



If we attempt to maximize Xy^^ with respect to the 

 yearly escapements E^, E2,...E„*, it turns out 

 that as n* ^ oo: a) a steady state solution exists 



and b) the optimum steady state escapement, £"0, 

 is that which maximizes the function iaEe~ ''^ - E). 

 Let L„ denote the economic loss in any year n and 

 define L^ as the difference between the optimum 

 catch, Xopx, and the actual catch J^act. i-e-, 



L„ — X, 



opt 



Xact- From Equation (3) we obtain 



L„ = {aEoe- "^u -E^ 



fc = 1 



PkE„-ke 



- bE 



" £•) 



(4) 



Eq is fixed and all of the escapements 

 {E„ _/;}(A; = 1 . . . K) have already occurred thus 

 leaving only E„ at our disposal. L„ is clearly 

 minimized by setting E„ = but since this would 

 eliminate a portion of the run in future years the 

 subsequent loss would be high indeed. Consider 

 now the combined loss for two successive years n 

 and (n + 1). Proceeding along the same lines that 

 led to Equation (4) we obtain 



L„ + L,, ^,= 2{aEoe-'>^.-Eo) 



(a 2 p, £•„_,.- "--£'„) 



k = 1 

 K 



-(a 2 thK+x-ke-^-'-'-^-E,, ^,). 



A: = 1 



If this is treated as a function of the single 

 variable E^,, an optimum value can be obtained. 

 However, this loss also depends on .£^ + 1 which has 

 not yet occurred. Let us extend this process 

 through year (n + K), which is a convenient stop- 

 ping point since it represents the completion of a 

 cycle starting at year n. The total loss over this 

 period is given by 



/>«.« + A = iK+ l)(a£'o6'-''^„-£'o) 



- 2 a2 PkE^-ke-'^'^-El (5) 



) = ,1 *- k = 1 -■ 



The loss given by Equation (5) depends not only on 

 past and present escapements but on the future 

 values E^ +1, E„ + 2, - - • -^i + a as well. Thus, when 

 formulating a policy for any particular year one 

 must take into account future policies also. From a 

 mathematical point of view what we have emerg- 

 ing here is another dynamic program, i.e., the 

 optimum year-to-year allocation as well as the 

 within-year allocation is in the form of a dynamic 

 program. This is too great an anlytical burden to 



838 



