658 



Fishery Bulletin 93(4). 1995 



Methods and analysis 



Variability of mortality 



To calculate correlations between stage-specific mor- 

 talities (or abundances) and recruitment, we required 

 estimates of variability of mortality for each stage. 

 Bradford ( 1992) compiled from the literature a large 

 set of data on mortality rates and their interannual 

 variabilities for the prerecruit stages of marine fishes. 

 From this consolidation of data, Bradford regressed 

 the interannual variance of daily mortality on its 

 mean (averaged over years). We adopt the following 

 notation: M represents an estimate of M from a single 

 year's survey (often only two abundance estimates 

 are used to calculate M); M represents the average 

 over a number of years of M values; Var(M) is the 

 estimate of the variance of the mortality calculated 

 from a number of years of M data. Note that Var(M) 

 is not equal to the true variance, Var(M), an issue 

 dealt with below. Bradford found the following fit, 

 holding across both stages and species: ln[Var(M)] = 

 2.231 In M - 1.893 (r 2 = 0.90; P<0.0001 ). We can re- 

 write this relation as 



Var(M) = 0.15M 



2.2 



(1) 



This very appealing relationship specifies an almost 

 constant CV for mortality; however, it is unclear if it 

 is affected by measurement error. 



When mortality is calculated from the difference 

 of two field estimates of log abundance each with 

 error e, the following relationships hold: 



M = (VT)[\nN(t 1 ) - In N(t 2 ) + e(t 2 ) - e(* a )], (2) 



where t t is the time of the ith observation. This re- 

 duces to the right-hand side of Equation 3 when n=2, 

 and decreases asymptotically as 1/rc for large n. For 10 

 evenly spaced observations, the estimation error vari- 

 ance will be approximately reduced by one-half, com- 

 pared with the case of two observations. To a good ap- 

 proximation, Equation 3 will provide a good estimate 

 of the estimation error variance because only a few per- 

 cent of the data used by Bradford had n larger than 10. 



Predictability of recruitment: no density 

 dependence 



We can write recruitment as 



R(t) = E(t)exp[-(C 1 (t) + C 2 U) + ...)], 



(5) 



where t refers to a specific year, E is the total num- 

 ber of eggs produced, and C, is the cumulative mor- 

 tality in stage /'. To be specific, we designate i - 1 for 

 the egg stage, i = 2 for early larvae, i = 3 for late larvae, 

 and i - 4 for juveniles. In accord with Equation 5, the 

 abundance of prerecruits, iV,, at the end of stage i, is 



N i U) = E(t)exp[-(C l (t) + C 2 (t) + ... + C l (t))]. (6) 



These equations form the basis of the forthcoming 

 analysis. 



Let C l (t) = C l + AC, (t), and 



\nEit) = \nE + MnE(t), 



then \nR(t) = \nR + AlnE-(AC 1 (t) + ... + AC 4 (t)).a) 



It follows from Equation 6 that 



Var(M) = Var(M) + (21 T 2 )ai 



(3) 



where N represents the true abundance, o £ is the 

 standard deviation of the estimation error e, and T = 

 t 2 -i v Approximately 70% of the mortality estimates 

 in Bradford (1992) were obtained as the difference 

 of two abundance estimates. 



When mortality is estimated from a regression 

 equation, by using a slope of log numbers versus time 

 with n observations equally spread over time inter- 

 val T, then we can use the standard formula in re- 

 gression for the variance of the estimate of a slope to 

 obtain 



(4) 



I>-^ )2 



T'n{n + \) 

 (n-1) 2 



(2n±l_n±l\ 

 I 6 ' 4 J 



lnN,(t) = \nN,+A\nE-(AC 1 (t) + ... + AC 4 (t)). (8) 



Equations 7 and 8 are general, they hold whether or 

 not correlations are present between stages. We make 

 immediate use of these equations to examine the 

 predictability of recruitment in the absence of inter- 

 stage correlations, a simple case which serves well 

 to illustrate the technique. 



In the following calculations we concentrate on 

 environmentally induced recruitment variations and 

 neglect the contribution of interannual variations in 

 egg production. Accordingly, we remove the stock ef- 

 fect from data-based estimates of recruitment vari- 

 ability before comparison with model-based values. 

 Only trivial modifications are necessary to include 

 the egg production factor should this be desired. In 

 the absence of interstage correlations of mortality, it 

 is easily shown that 



