Prager: A nonequilibrium surplus-production model 



377 



The second difficulty involves a fundamental dif- 

 ference between predicting yield and predicting ef- 

 fort. For a given starting biomass and effort, one can 

 always compute the corresponding yield. For a given 

 starting biomass, however, there are some yields that 

 can never be obtained, no matter how high the ef- 

 fort. Under these circumstances, the catch equation 

 (6a or 6b) has no solution. Unless a tactic is devised 

 for such cases, it becomes impossible to compute the 

 objective function when they occur, and thus impos- 

 sible to conduct its minimization. A tactic suggested 

 by R. Methot 1 as useful in his stock-synthesis model 

 (Methot, 1989, 1990) is to place a constraint on the 

 maximum allowable value of F x (and consequently 

 of/" T ). When an estimate of F reaches this constraint, 

 it is not allowed to increase further, and the quantity 

 [log{Y r )-log(Y T )] 2 is added to the objective function 

 along with the usual squared residual in effort. This 

 allows computation of a reasonable value of the objec- 

 tive function for such regions of the solution space that 

 may be encountered during optimization. In my expe- 

 rience, however, final estimates have always come from 

 a solution in which yield is always matched exactly. 



In fitting a linear regression, observation error in 

 the predictor variables causes problems with the pa- 

 rameter estimates, including inconsistency and, in 

 the bivariate linear case, bias towards zero (Thiel, 

 1971; Kennedy,1979). The problems induced into 

 nonlinear models are less well understood, but are 

 believed to be similar. Schnute ( 1989) has illustrated 

 how the choice of dependent variable in a fisheries 

 model can affect the results substantially. In fisher- 

 ies contexts, yield is usually observed more precisely 

 than fishing effort; for that reason, it seems prefer- 

 able on statistical grounds to use the second ap- 

 proach, estimating effort from yield, rather than es- 

 timating yield from effort. 



Whichever approach is chosen, the estimation pro- 

 cess results in direct estimates of B v r, K, and q, 

 which define unique estimates of the stock biomass 

 levels B 2 , B 3 , ..., B T and the stock's production dur- 

 ing each period of time. The corresponding estimate 

 of maximum sustainable yield (MSY) under the lo- 

 gistic model is MSY = Kr 1 4 . According to the theory 

 of production modeling, MSY can be attained as a 

 sustainable yield only at one specific stock size; for 

 the logistic model this is B MSY = K/2, estimated by 

 B MSY =K/2. The instantaneous fishing mortality 

 that generates MSY at B MSY is ^msy = r/2 ; the cor- 

 responding rate of fishing effort is f MSY = rl2q, with 

 estimates given by substituting rand q for the un- 

 known true values in these two expressions. 



The logarithmic objective function assumes mul- 

 tiplicative errors with constant variance. The solu- 

 tion obtained is the maximum-likelihood solution if 

 the transformed residuals are independent, of con- 

 stant variance, and normally distributed (see Seber 

 and Wild, 1989). However, maximum-likelihood 

 methods, while generally desirable, are not neces- 

 sarily robust to outliers, nor do they necessarily have 

 desirable small-sample properties. Use of a robust- 

 regression method (such as least absolute values re- 

 gression) would be an interesting research topic. 



Another management benchmark 



An analogue of the management benchmark F Q 1 can 

 be computed for this model (or for any production 

 model). The derivative of equilibrium yield with re- 

 spect to fishing mortality rate for this model is 



dF 



(9) 



1 Methot, R. Alaska Fisheries Science Center. 7600 Sand Point 

 Way NE, Seattle, WA 98115. Personal commun., 1993. 



At F = 0, this derivative is equal to K. We define as 

 F j for this model as the value of F at which Equa- 

 tion 9 equals 0. 1 K. Substitution into Equation 9 gives 

 the following results: F , = 0.45 r, and Y Q 1 = 0.2475 rK 

 (where Y 1 is the equilibrium yield corresponding to 

 F Q j). An equivalent statement is that F Q 1 is 90% of 

 F MSY , and Y 01 is 99% of MSY. Punt (1990) used the 

 concept of F Q , for a production model but did not 

 explicitly state these relationships. 



Penalty for large estimates of 6, 



Logistic production theory implies that B x should 

 always be less than K, but the objective functions 

 used here are relatively insensitive to the estimate 

 of B y In practice, I have found that the estimate of 

 Bj obtained from some data sets tends to be much 

 larger than the estimate of K. Such results could be 

 eliminated by introducing a fixed constraint into the 

 solution, but I have used another method success- 

 fully: adding a penalty term to the objective function 

 when Bt>K- Including this term, the complete loga- 

 rithmic objective function (assuming residuals in ef- 

 fort) becomes 



L = ^[log(/- r )-log(/ r )] 2 + 0[log(B 1 )-log(A-)] 2 ,(lO) 



where <p = 1 if S x > K, and = 0, otherwise. While 

 constraining the value of Bj seems logical in accor- 

 dance with the underlying population theory, such 

 constraints can change the estimates of other param- 

 eters, compared to an unconstrained solution. The 

 amount of change can be examined by estimating with 

 and without the penalty term or fixed constraint. 



