FISHERY BULLETIN: VOL 78, NO. 1 



jf this paper. Some of the techniques they dis- 

 cussed, particularly the filtering techniques (Wal- 

 ters and Hilborn 1978), are only appropriate if the 

 model has an additive error term. While a Ricker 

 spawner-recruit curve can be transformed to an 

 additive model, many models do not have this fea- 

 ture. 



I am presenting what I feel is an improved way 

 to smooth out the fluctuations in the year-to-year 

 harvests as compared with the method suggested 

 in Walters (1975) and show that the Bayesian 

 (adaptive) model discussed in Walters and Hilborn 

 (1976) has an optimal policy with a very simple 

 form that can be readily calculated. 



Moreover, a rigorous approach is taken to define 

 the model on a grid and the effects of the grid 

 choice. None of the papers cited deal with this 

 important question; new results are presented 

 which show that the most serious effect of the grid 

 is on the estimates of the long-run (ergodic) prob- 

 abilities of the population dynamics when follow- 

 ing a given policy. Particularly the tail properties 

 of the ergodic distribution, i.e., the long-run prob- 

 ability of low harvest or low population sizes, are 

 misestimated. This is a new finding even in the 

 MDP literature, and has numerical implications, 

 particularly when calculating the trade off be- 

 tween the mean harvest of a given policy and the 

 long-run probability of undesirable events when 

 following that policy. 



THE MODEL 



The models to be analyzed were developed by 

 Mathews (1967) to describe the spawner-recruit 

 relationships of sockeye salmon, Oncorhynchus 

 nerka, populations in two rivers that run into Bris- 

 tol Bay, Alaska. Oceanographic and other factors 

 affect the number of recruits to a degree where the 

 relationships can be modeled by the random equa- 

 tions: 



Wood River: 



x,^j = expid) (4.077y,) exp(-0.800y,) 

 d-N(0, 0.2098) 



1.1a) 



Branch River: 



jc^^ = exp(c?) (4.554y,) exp( -1.845y, ) 

 d^N (0,0. 3352) 



(1.1b) 



where y, is thenumber of spawTiers in period ^ x, + 1 

 is the (random) number of recruits in period ^4-1, 

 and d~N{a, b) denotes that <i is a normally dis- 



36 



tributed random variable, with mean a and var- 

 iance b. 



For deterministic versions of Equation (1.1), the 

 primary objective of management is MSY 

 (maximum sustainable yield), which is equivalent 

 to the largest per period growth of the determinis- 

 tic model. The stochastic equivalent of this criteri- 

 on is to maximize the average per period harvest, 

 or gain optimality. Mathematically, letting E be 

 the expectation operator, this is 



max lim 



1 T 

 T ^ t=i 



(xt-y't) 



(1.2a) 



However, for many decision making situations, 

 total expected discounted harvest may be a prefer- 

 able criterion, since a discount factor can repre- 

 sent a measure of risk or uncertainty about the 

 system, over and above the variability due to the 

 random variable d. More formally, if a is a dis- 

 count factor 0^a<l, the problem is to: 



maximize E 



^ a'^ip(x,-y,) 



t=i 



(1.2b) 



subject to O^y^^Xf-, and Equation (1.1) 



where p is a weighting factor, which could be 1 or 

 could represent the average weight of the salmon 

 harvested. 



All the results in this paper are for expected 

 discounted return with « = 0.97. For a = 1, Equa- 

 tion (1.2a) must be used, since Equation (1.2b) is 

 infinite for most policies. The choice of a = 0.97 is 

 arbitrary, though numerical runs for a ranging 

 from 0.95 to 1.00 produced no significant changes 

 in the results. When actually implementing a 

 model, a careful choice of a must be made, and the 

 sensitivity of the results to changes in the value of 

 a should be tested. It should be mentioned that a = 

 1 is just as much a discount factor as any other 

 value and implies certain temporal preferences 

 and attitudes towards risk that may not 

 adequately reflect the decision maker's prefer- 

 ences. 



The shortcomings of Equation (1.1a) or (1.1b) 

 should also be noted, such as no account is taken of 

 ocean harvesting of the salmon, particularly by a 

 foreign nation. This just reinforces the idea that 

 the purpose of this analysis is not optimization per 



