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Fishery Bulletin 92(4), 1994 



CPUE distributions by size class and depth zone 

 over seamounts B and J from the Humboldt are 

 shown in Figure 4. A preliminary examination of the 

 data revealed that they fitted portions of curves con- 

 forming to a normal distribution. It was therefore 

 assumed that for a given seamount the CPUE, in 

 terms of length and depth, was distributed over a 

 surface described by a bivariate normal distribution 

 function delimited by the maximum and minimum 

 of lengths and depths sampled. This assumption is 

 the basis of the first modelling exercise ("bivariate 

 normal model"). 



Table 1 also shows that, for a given absolute depth, 

 mean length significantly decreases as the depth of 

 the top of the seamount increases. This decrease sug- 

 gests that the length distribution depends both on 

 the absolute depth (in relation to the sea surface) 

 and on the depth of the top of the seamount. Conse- 

 quently, the bivariate normal model constructed for 

 a given seamount may not be applicable to other sea- 

 mounts whose summits lie at different depths. It is 

 therefore necessary to construct a more general 

 model (referred to as the "recursive model") which 

 would predict extrapolated estimates of CPUE over 

 any seamount by taking into account both the abso- 

 lute depth of the water column and the depth of the 

 top of the seamount. Temporal validation of these 

 two models requires data that were not used during 

 model construction but were collected in the same 

 area at different periods. Data collected on board RV 

 Alis and the fishing vessels Hokko Maru and Fukuju 

 Maru were used for model validation. 



Modelling method 



Bivariate normal model In the bivariate normal 

 model, CPUE by length and depth are calculated on 

 the basis of a bivariate normal distribution defined 

 by the density function ( 1 1 



B(x h x d ) = 



exp< 



2(l-p*) 



*/-/*/ 



'/ 



2 no, o d Vl-P 2 



{*i-Vi)(*d-Hd} 



2p- 



tf/o-rf 



Hi 



((x d -M d )) 



where x l is the length, x d is the depth, p, is the mean 

 length, o, is the standard deviation of length, p , is 

 the mean depth, CT f/ is the standard deviation of depth, 

 and p is the regression coefficient of length on depth. 

 Because sampling of the seamounts is limited up- 

 wards by their summit (D s ) and downwards by the 



maximum depth accessible with the bottom longline 

 (D a ), CPUE distributions will be modelled by a por- 

 tion of the bivariate normal distribution (2) 



CPUE est (x t ,x d ) = for x d > D a or x d < D s 



CPUE est (.r„V = XW# lt x d )for D s <x d < D a 



(2) 



where X represents the theoretical cumulative CPUE 

 estimated over the field of definition of the entire 

 bivariate normal distribution. The parameters A, p,, 

 o,, pi d , errand p were estimated by a nonlinear regres- 

 sion by using an iterative algorithm for sum of squares 

 errors (SSE) minimization (SAS, 1988). 



Recursive model The recursive model should allow 

 estimation of alfonsino CPUE by size class for sea- 

 mounts for which no data are available except the 

 depth of their summit. The size structure variation 

 shown in Table 1 should be taken into account in the 

 development of this model, i.e. 1) for a given sea- 

 mount, mean length increases with depth and 2) for 

 a given depth zone, mean length decreases as the 

 depth of the top of the seamount increases. In theory, 

 the distribution of a population of alfonsino over any 

 seamount could then be taken as the superposition 

 of the distributions of two subpopulations: one popu- 

 lation would be influenced only by the absolute depth 

 while the other would be influenced by the depth of 

 the top of the seamount. This model attempts to ex- 

 plain how the fish population of a given seamount 

 would in theory redistribute itself if it were to mi- 

 grate and settle on another seamount. Consider a 

 hypothetical population whose cumulative CPUE CK) 

 by length and depth is distributed over its seamount 

 of origin according to a bivariate normal distribu- 

 tion function of unknown ji t , o,, p d , O d , and p param- 

 eters. For the estimation of these parameters, the 

 top of the hypothetical original seamount will be as- 

 sumed to be exactly level with the sea surface in or- 

 der to include the entire depth zone that could be 

 inhabited by alfonsino. The new CPUE distribution 

 on seamounts j, 7+1, etc. . . (Fig. 5) will depend on 

 the initial parameters of the distribution over the 

 original seamount as well as on parameter p, the 

 probability that the fish will redistribute according 

 to absolute depth, and ( 1-p), the probability that the 

 fish will redistribute according to the depth of the 

 top of the seamount. At each "leap" to a deeper sea- 

 mount (from.; toj+l), the subpopulation inhabiting 

 a given depth zone D i will split into two groups: one 

 group will stay in the same depth zone D i with a prob- 

 ability p (or will migrate elsewhere if this zone is no 

 longer available on the new seamount) and the other 



