844 



Fishery Bulletin 98(4) 



130° E 



135° E 



140° E 



Figure 3 



Distribution of'pilchard eggs alung the Pacific coast of Japan in February and March 

 1994. Solid circles and crosses show the mean egg density (number of eggs per 0.159 

 m-) in 1° X 1° squares. Data in the area surrounded by the bold lines was used for 

 the analysis (after Zenitani et al., 19981 



E{N^,} = Nap[l- 



{k/{k + mo})} 



(1) 



where N and TV 



number of samples with a positive 

 number of eggs and total number of 

 samples in sui-vey area (A^), respec- 

 tively; 

 a and co = sampling efficiency parameters; 



p = probability that a observation station 

 is a habitat for spawning stock; 



k = the over-dispersion parameter of neg- 

 ative binomial distribution; and 



m = mean number of eggs at a habitat area 

 for spawning stock. 



To simplify, we assumed that « = 1, p = 1, and (o = 

 1. Moreover, we assumed that the spawning biomass (B) 

 and total egg abundance (m A'^) is linearly related, that 

 is B =8m N, where 5 is a constant. Actually, a linear rela- 

 tionship exists between the spawning biomass and egg 

 abundance of the Japanese pilchard (Fig. 4). We assumed 

 that a survey area (AJ is equally divided by A'' stations. 

 Because A^,^ is a random variable, the observed spawning 

 area A=A(Np )=Y^p is also a random variable, where 7 is a 

 proportional constant <=AJN). 



From Equation 1. we thus obtain 



E\A] 



yE{N,,} 



A\l-{k/{k + fB>}'\ 



where r = 1/(N5) 



If (/, = />■. u., = f, A, = 640 ( 10^ km^), and A 

 relationship V between A and B. 



E[A]. we have 



Parameters estimation, confidence interval analysis, 

 and selection of the optimal relationship 



We assumed that each relationship (I-V) had an error 

 term 2; ln(A)=ln(/('B))-i-ln(2); that 2 had a log-normal distri- 

 bution; ln(z) ~MO,CT-); and that N{0.a-) has a normal dis- 

 tribution of mean zero and variance o'-. z will take only pos- 

 itive values, so we expected the mean of 2, £(2|=exp(cT-/2). 

 to be larger than 0. In addition, the log-normal distribu- 

 tion has a long tail, which is common with ecological data 

 (Hilborn and Mangel, 1997). 



