Arreguin-Sanchez and Pitcher: Catchability estimates for the Epinephelus mono fishery 



747 



Materials and methods 



We assume that catchability 1 ) is length dependent; 

 2) depends on fish behavior; 3) and is density depen- 

 dent. For fishing effort, we also assume that 1 ) these 

 assumptions can be valid for individual fishing fleets, 

 such that we can explicitly consider the individual 

 contribution of the several fleets participating in the 

 fishing process and 2) that units of effort have static 

 properties, that is to say, a consistent fishing power 

 over time. 



where G = the effect of growth in absence of mor- 

 tality; and 

 S = theeffectofmortality (survival) and also 

 of selection of the sampling gear. 



Matrix G( i ,k) can be easily estimated by following 

 the criteria defined by Shepherd (1987) to assign 

 growth probabilities to each length class. The ele- 

 ments of S(/f)can be defined in terms of mortality as 



S(k} : 



-Zi/ri( -lM+ol*./ls(*l£'(n| 



e = e ' ', 



Length-dependent catchability 



The catchability coefficient must be estimated for 

 each length class in a given time, qi i ,t). A convenient 

 form to represent the transformation of one length- 

 frequency distribution (representing the structure of 

 the stock) into another is by means of a transition 

 matrix (Shepherd, 1987; Caswell, 1988), so that 



N(i.f + l) = A{i,k)N{i,t), 



(2) 



where k and ' = successive length intervals; 



N( I ,t) = is the stock size in numbers at time 

 t; and 

 A = the transition matrix that depends 

 on growth and mortality. 



Although these processes occur concurrently. Shep- 

 herd ( 1987) assumed that they can be separated as 

 the product of two terms: 



A{i,k) = Gii,k)S{k), 



where S(k) represents the values of the elements in 

 the main diagonal of the survivorship matrix (other 

 elements are zero; see Caswell, 1988); Z{k)t is the in- 

 stantaneous rate of total mortality for the /c"^ length 

 group at time t; M is the instantaneous rate of natural 

 mortality (assumed constant); s(k) is the probability of 

 gear selection for length k\ and E(t) is the fishing effort 

 at time / ( which is assumed nontargeted to specific sizes 

 within a range of sizes captured by the gear). Then fish- 

 ing mortality is given as Fik. t) = q(k, t) sik)E(t). 

 Equation 2 can therefore be represented as 



N(i ,t + h = Y^G(i ,k)e-^"^'"'--"""""^N(k,t). 



(3) 



If A'^( (t-i-1), NikJ) and Gi i ,k) are known, as well as 

 M,s(k). and E(t), then g(/e,/) can be estimated. Equa- 

 tion 2 can then be solved iteratively for qik.t) by us- 

 ing an algorithm to minimize differences between 

 observed and calculated values o^ Ni i . t + 1). 



Once q(k.t) values are obtained, the catchability 

 pattern with length can be observed and a tendency 



