FISHERY BULLETIN: VOL. 74, NO. 4 



gramming to determine an optimum set of such 

 allocations. Such fine-scale is not practical here so 

 that the individual tj, are irrelevant here except 

 that they must satisfy the condition 



m 



I = 1 



Let{TJj} {i= 1, ... m) be the actual allocations 

 where each tjj will be assumed to be a random 

 function of the management decision 5j taken 

 during the /th period. It will be assumed that the 

 {tj,} {i = 1, . . .m) have independent beta distribu- 

 tions where the beta parameters ( t; , ju^ ) are 

 uniquely determined by the management decision 

 S, that is taken during period /. Thus we have 



m 



giVi^Vz ...n™ 181,82, ...5„) = llgdn,\8,) (9a) 



i = 1 



Vi 



Ml-^) 



tf.-l 



(9b) 



This is a reasonable probability density to assume 

 since it confines tj; to the interval (0,1) and the 

 parameter choice permits, within appropriate 

 limits, the specification of the mean and variance^ 

 of r}i . 



Return now to the central feature of the analy- 

 sis which is to take into account the dynamics of 

 the fisher}'. Equation (7) is the probability density 

 of N appropriate for the first period of the fishery 

 during which only preseason conditioning infor- 

 mation, denoted symbolically by yo, is available. 

 Assume now that, during the first and subsequent 

 time periods, additional population data, z/i, 2/2, .• . 

 become successively available. This data may then 

 be used to condition the probability density of N, 

 hopefully in such a manner that our knowledge of 

 the true value of A'^, as measured by its variance, 

 improves as more data are gathered. At each stage 

 of the fishing season we compute the Bayes risk 

 with respect to the then current probability den- 

 sity of N and adopt a strategy that takes into 

 account all available data and all previous man- 

 agement decision. This will be formalized analyt- 



■•The conditioning of tj by 5 only is probably an 

 oversimplification. There is evidence to indicate that ^ also 



depends on the number of fish that enter the fishery during any 

 fishing period. 



ically upon the specification of an appropriate 

 sampling distribution for the {?/; ) {i = I, 2, . . . 

 m - 1). t/^ is irrelevant since it is obtained after the 

 final decision 8„, will have been made. 



Assume that during each stage of the run some 

 fixed fraction e of the total number of fish entering 

 the fishery is vulnerable to sampling. For example, 

 if the sampling is done by gill nets e may be 

 determined from knowledge of the length, the 

 time of soak, and the efficiency of the net. With 

 such a sampling scheme, it is reasonable to assume 

 that the samples ?/,, 2/2, ... Vk -\ will have in- 

 dependent Poisson densities with parameters Aj, 

 ^2. • • • K -1 where Aj = tpjN, i.e., ep^N is the 

 expected sample size for the ith period and 



PiYi =yi\N)^e 



_ c- f A' 





(10) 



where y^ = 0, 1, .... Equation (10) and Bayes 

 theorem may be utilized to modify or update 

 Equation (7) to reflect the additional information 

 that is assumed to have become available. Assume 

 that the system is now at the start of the second 

 stage and that the sample y^ is now available. 

 Bayes theorem gives 



..xn ^ PiYr = y.\N)fmyo) .... 



XP(Y,=y,\N')AiN'\yo)dN' 



Substituting Equations (7) and (10) in Equation 

 (11) gives, after dividing common factors. 



fiiNlyo.yi) = 





(12) 



The integral in the denominator of Equation (12) 

 is a standard form expressible in terms of gamma 

 functions which gives 



r(ai) 



(13) 



where oj = ao + ?/i and (i^ = /3q + epj. The updat- 

 ed probability density for A^ given by Equation 

 (13) is, like the prior density given by Equation (7), 

 a gamma density but with modified parameters a^ 

 and /?!. The process by which Equation (13) was 

 obtained may be repeated indefinitely to give 



fkiN\yo.yi 



2/fc-i) 



_ (^fc-lT'-xrV,-! -^.-.'V 



n^k-i) 



N 



(14) 



840 



