748 



Fishery Bulletin 97(4), 1999 



quantified. Although a catchabiHty-at-length model 

 may be empirically determined, its interpretation 

 must be made by considering the population biology 

 and behavior of the fish. On an annual basis, the 

 model is represented by the function 



qiOaPin-f 



(4) 



whose slope ^( ' ) represents the rate of change of 

 catchability with length for a given year. If several 

 years are considered for a stable period of the fish- 

 er}', then a general function can be estimated where 

 the slope of the length-based catchability pattern in 

 Equation 4 can be expressed by /3( ',•), where the 

 symbol • represents the average stock level over a 

 range of years in which it can be reasonably assumed 

 to be stable. 



Time-dependent catchability 



GuUand ( 1983 ) expanded the scope of the catchability 

 coefficient to a structured population as expressed 

 in the following equation: 



C{l,At) = q{l,At)s(nE{At)N{i,At)- 



(5) 



This represents the capture of fish of various sizes 

 ( ' ) during a time period Dt. Taking the mean stock 

 size ( A'^ ), and rewriting Equation 5 in terms of catch 

 per unit of fishing effort, U, we obtain 



ith 



lHi.') = q(i,»}N{i), 



c/(/,•) = C(^An/.s(/)£(An 



and then 



Ln[L^( I ,t)/U{ ',•)] = Ln[(7( i,t)N(i )/q(l.»)N{ i )] 

 = Ln[q{i .t)/q(i .•)]. 



(6) 



This can be interpreted as the departure or anomaly 

 in catchability at time t with respect to the average 

 for a given length class. Arreguin-Sanchez (1996) 

 suggested that this ratio is a linear function of size, 

 represented by the midlength class as 



where 



Ln[g( ',n/(7( /,•)] = am + /J(n( , 



(7) 



p(t) = Ln[q(i + lJ)/q(i,t}]-Ln\qn + 1,« )/(/(', •)]. 



The intercept cdt) in Equation 7 can be interpreted 

 as the relative vulnerability of small fish, and also 

 as an index of the relative abundance of recruits. The 

 slope (Xt) expresses the rate of change of catchability- 



at-length with respect to the average, or in other 

 words, the departure (anomaly) from the average 

 catchability pattern with length. 



Once /^( ' ,• ) is estimated from Equation 4, the value 

 of j3(^) can be added to pi i .*) to obtain the correspond- 

 ing catchability values for a given time period. 



Amount of fishing and density-dependent 

 catchability 



The relation between population size and catchability 

 has been most studied in pelagic clupeoid fish 

 (Murphy, 1966; MacCall, 1976; Csirke, 1988, 1989; 

 Pitcher, 1996), where catchability is inversely related 

 to stock size. Thus a density-dependent effect can be 

 approached by considering the population as a whole, 

 and the function 



qw(t) = f{N(t)} 



(8) 



can be fitted, where qwtt) is the weighted value of g 

 at time t, with U at time ^ as a weighting factor. An 

 analogous approach is proposed for each length class. 

 For a stable period of time, we can assume that the 

 amount of fishing, or the total fishing effort E, is one 

 of the main sources of variation of the population 

 density and the structure of the exploited stock. A 

 change in both of these population characteristics 

 will be reflected in the pit) coefficient. With Equa- 

 tions 1 and 8, it is possible to relate changes in qU ,t) 

 as a function of coefficient fi(t): 



pu)aP(E)i, 



(9) 



where a constant natural mortality rate is assumed 

 for all length classes over time. 



Because pit) represents the departure of the 

 catchability pattern with length at time t. Equation 

 9 measures the effect of the amount of fishing on the 

 anomaly, and it is represented by the slope /j(£). The 

 magnitude and sign of piE) will provide information 

 about changes in stock density. A negative sign means 

 that for a low level of fishing, q will increase with 

 fish length. 



Catchability differences between fleets 



Following the same rationale as that in Equation 7, 

 differences in q between fleets as a function of length 

 can be estimated. If c/ and h represent two different 

 fishing fleets, Equation 6 can now be expressed as 



Ln[lHg,iJ)/Uih,iJ)] = Ln[q{g,i.t)/q{h.i,t)\ (10 I 



and 



