BEARDSLEY and CONSER: AN ANALYSIS OF CATCH AND EFFORT DATA 



Catch Model 



A basic catch model was used for dock, tourna- 

 ment, and longhne fishing in the northern Gulf of 

 Mexico: 



where u^ = a,^ + /3 



sk' 



^Uk='i.kf,j^jk%k 



(1) 



where C,,;^ is the catch of species k from the j'th 

 type of fishing during months , g,^ is the catchabil- 

 ity coefficient of the iih. type of fishing on species k, 

 f,j is the amount of fishing effort expended by the 

 ith type of fishing during month 7, N^^ is the aver- 

 age population size of species k in monthj, and e^^ 

 is a log-normally distributed error term. Note that 

 no k index appears on the f since effort is not 

 generally directed at any particular billfish 

 species in the recreational fishery in the northern 

 Gulf of Mexico. This basic catch model has been 

 used by many authors to study fisheries in which 

 more than one type of fishing was employed (Gul- 

 land 1956; Beverton and Holt 1957; Robson 1966). 

 Equation (1) can be written as a linear, two factor 

 analysis of variance (ANOVA) model by dividing 

 through by /]^, taking natural logarithms, and ob- 

 taining 



Uk 



«,, + Pjk + ^'ijk 



(2) 



where 7^, = In (C,;^,//;^), 

 l^jk =lnA^,^,and 



The ANOVA can be used to test for significant 

 differences in the catchability of various types of 

 fishing and in population density over time. The 

 ANOVA model cannot, however, be used to esti- 

 mate catchability or population densities, but it 

 can be reparameterized to obtain estimates of rel- 

 ative catchability (P^^) and relative population 

 density (Z)^^), where 



P.k = <lj^ 



sk 



Djk=^,JNs, 



(3) 



(4) 



in which s refers to the type of fishing and month 

 designated as standard. After reparameterization. 

 Equation (2) becomes 



b 



Y.jk ="A +<^',k + f^'jk +^\ 



Ijk 



(5) 



Oi 



(«,> - ««J> and 



^',. =(/3.. -/3,s.*) 



The parameters u^ , a '^^ , and ^ '^^ can be estimated 

 by solving the usual normal equations and esti- 

 mates of relative fishing power and relative popu- 

 lation density can be obtained from [P^^ = expCa.^) 

 andD^ =exp(i3V^)]. 



To apply the basic catch model to the billfish 

 fishery in the northern gulf, it is necessary to 

 assume that for each type of fishing (i.e., dock, 

 tournament, and longline) catchability is constant 

 throughout the analysis period, there is no in- 

 teraction between catchability and density, and 

 units of effort operate independently. The first two 

 assumptions may be tenuous for this fishery and 

 will be investigated in the analysis. The third as- 

 sumption appears to be reasonable. 



The basic catch model was used initially to de- 

 termine what relationship existed between catch 

 and effort data from dock and tournament data. 

 Figure 7 presents a flow diagram for the determi- 



THE BASIC CATCH MODEL 



DOES NOT ADEQUATELY 

 REPRESENT THE DATA 



Phase 

 three 



Figure 7. — Flow diagram for determining the appropriateness 

 of using the basic catch model to represent data from the two 

 different kinds of fishing methods. 



59 



