Bradford Recruitment predictions from early life stages of marine fishes 



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lation between the mortality rate of adjacent intervals 

 across years. The mortality of a given interval in any 

 model year then depends partially on the mortality of 

 the previous period in that same year. With p equal to 

 the correlation between adjacent interval-specific mor- 

 tality rates, I used the following equation to calculate 

 the mortality rate of successive intervals: 



M 



k+l 



^ | SD(M,,i )] 



Mk + (l-p2)"=Mk.i. 



In this equation, the actual mortality for stage k-i- 1 is 

 a linear combination of the random variables simulating 

 the variability in stages k and k-i-1. The correlation 

 coefficient determines how much mortality in stage 

 k-H 1 is similar to that of stage k. The ratio of standard 

 deviations in the first term scales the contribution of 

 the mortality of the previous interval to the appropriate 

 variance. To simplify, I assumed throughout this paper 

 that there was no covariance between the number of 

 eggs spawned and mortality in subsequent intervals. 

 Finally, density-dependent mortality was incorpor- 

 ated in some versions of the model. Density-dependent 

 mortality was added to the juvenile period, following 

 suggestions of Houde (1987) and Smith (1985) that this 

 is the most likely interval for density effects. While a 

 number of formulations are possible, I chose a power 

 function (Peterman 1982): 



iX\ 



where in this case X and Y are the abundances of 

 juveniles and recruits, respectively. For density- 

 independent mortality, b=l; b is <1 for density- 

 dependent cases. The parameter a is thus the density- 

 independent survival rate. After taking logs, the log 

 of the abundance of recruits is now a function of 

 the log of the number of metamorphs, N^ , and the 

 density-independent mortality, Mj: 



N, 



bN„ 



M,. 



(2) 



In the stochastic simulations, this equation was used 

 to calculate recruitment with a random normal deviate 

 substituted for Mj. 



The full model was run for 1000 10-yr trials in SAS 

 (1987), and a matrix of abundances and mortality rates 

 for each stage was built up. For each 10-yr trial, cor- 

 relation coefficients were calculated between the 

 various predictors of recruitment (i.e., abundances and 

 mortality rates of each of the prerecruit stages), and 

 the numbers of recruits and summary statistics of the 

 distributions of correlation coefficients were derived. 



Table 1 



Daily mortality rates (M) and interval durations (t, in days) 

 for four species used as e.xamples in the analysis. Egg mor- 

 tality includes the yolksac period up to first feeding; larval 

 periods explained in text. Values were adapted from Houde 

 (1987; cod Gadus morhua, and herring Clupea hareiigus). 

 Smith (1985; anchovy EngrauHs mordax), and Zijlstra and 

 Witte (1985; plaice Pleuronectes platessa). 



Egg 



Early 

 larvae 



Species 



M 



M 



t 



Late 

 larvae 



M t 



Juveniles 



M 



Cod 0.061 18 0.160 10 0.063 46 0.010 291 



Herring 0.050 21 0.080 10 0.034 70 0.015 264 



Anchovy 0.250 7 0.160 10 0.050 79 0.012 269 



Plaice " 0.068 .38 0.104 10 0.045 77 0.008 245 



Model parameters 



To generalize the results, I used four fish species as 

 examples (Table 1). These were not chosen to be rep- 

 resentative of a specific stock or situation, but rather 

 to indicate the effect of different life histories on our 

 ability to forecast recruitment. To parameterize the 

 model for a specific species, the interannual variance 

 of the number of eggs laid and the mortality of each 

 prerecruit stage was required. 



I obtained estimates of the variance in the number 

 of eggs spawned from published reconstructions of 

 stock abundances (Table 2). Except for cod, I used the 

 residuals of linear regressions of log(eggs) on time to 

 estimate the variance, since time trends existed for 

 some stocks. 



Estimates of the variability in mortality rates for all 

 prerecruit stages are unavailable; I therefore sought 

 a predictive relationship between interannual variance 

 and the mean of daily mortality rates. This allowed 

 estimation of the variances of mortality rates of the 

 early life stages from mean daily rates. I surveyed the 

 literature for papers containing 2 or more years of 

 estimates of age- or stage-specific mortality for the 

 same population or stock. All stages from egg to adult 

 were used, for marine, freshwater, and anadramous 

 fish species. No screening of the data was done except 

 for estimates from adult fish, where only estimates 

 using methods independent of catch-data analysis were 

 used (i.e., tagging). Most adult estimates were from 

 lightly or unfished stocks. In some cases I estimated 

 mortality from annual estimates of abundance or from 

 regressions of log abundance on time. All estimates 

 were converted to daily values using annual estimates 

 of stage duration if available, or the long-term average 

 stage duration. Daily mortalities were then averaged 



