RYTHER, JOHN H. 1969. Photosynthesis and Fish 

 Production from the Sea, Science, Vol. 166, pp. 72-76. 



of Commercial Marine Fisheries, Inter-American Tropi- 

 cal Tuna Commission, Bulletin 1(2), 27-56. 



SCHAEFER, MILNER B. 19.54. Some Aspects of the 

 Dynamics of Populations Important to the Management 



SCHAEFER, MILNER B. 19,56. Some Aspects of the 

 Dynamics of Population Important to the Management 

 of Commercial Marine Fisheries, Inter-.American Tropi- 

 cal Tuna Commission, Bulletins 1 and 2. 



APPENDIX I. METHODS OF ADJUSTING CATCH AND 

 EFFORT DATA TO REPRESENT EQUILIBRIUM OBSERVATIONS 



The Schaefer (1957) Method 



The Schaefer analysis (using his notation) is 

 based on the assumption that the rate of 

 population change can be represented by the 

 equation 



where k^ is the rate of population increase, 

 k-i is the catchability coefficient, L the maximum 

 population size, F is fishing effort, and P is 

 the current population size. Further, it is 

 assumed that at level P, in year i, equilibrium 

 yield, Ye is estimated by P + Catch, and that 



(2) AP 



_ Pr + 1-P,- 



Cf 1 1 /Fi+ I — Ct I /Pf- I 

 2 



where C is catch. To use these equations it is 

 necessary to relate P and u, catch per unit effort, 

 that is 



(3) P=k2U. 



If P in equation (1) is replaced by P, then all 

 three parameters k, k-i, and L can be estimated 

 from a series of data on catch and catch per unit 

 of effort. This 1957 procedure of Schaefer's was 

 first tried as a basis for a decision rule. 



Initially a 15-year series of data was divided 

 into three equal parts, that is, 1 to 5, 6 to 10, and 

 11 to 15 years. The three parameters were 

 estimated from the three sets of data by solving 

 the simultaneous equations of the form 



]=l n, 



where A'l, and k-z, and L are parameters, Au, is 

 the change in catch per unit effort, u, is the 

 average catch per unit effort u,' is the average 

 catch per unit effort squared, f, the number 

 of units of effort and n, the length of the period 

 in years. 



Pella and Tomlinson suggested that the series 

 of data be divided into periods with the greatest 

 differences in stock levels to avoid absurd results. 

 They also pointed out the lack of a unique 

 solution, since different partitioning of the data 

 may give different results. There is also no 

 statistical basis on which to infer properties of 

 the parameters such as bias, consistency, or 

 efficiency, etc. 



Gulland (1961. 1968b) Method 



This method involves relating the mean 

 annual catch per unit of effort in a given year 

 to the fishing effort, averaged over that year 

 and a certain number of previous years cor- 

 responding to the mean number of years that a 

 year-class contributes to the fishery. 



For example, the catch in period t would be 

 related to the average effort over the last 3 years 

 for the yellowfin tuna since a year-class con- 

 tributes to the fishery for about 3 years. We are 

 doubtful of the validity of this since it gives 

 equal weight to each year of effort in computing 

 the average effort. We feel the hypothesis ex- 

 pressed in this paper is more realistic. In addi- 

 tion, the statistical properties of a moving 



90 



