FISHERY BULLETIN: VOL. 84, NO. 2 



plus the average fecundity of gulf menhaden (about 

 25,000 eggs/female) places the gulf menhaden 

 among Cushing's clupeoid groups which have a 

 slightly domed spawner-recruit curve Accordingly, 

 a spawner-recruite relationship was applied of the 

 form: 



R = Se^ ~ 5 > /s « 



(7) 



where R 

 S 

 e 

 S r 



= recruitment at age 1 

 = spawning stock size 

 = base of natural logarithm 

 = maximum equilibrium stock 

 = spawning stock size yielding maxi- 

 mum absolute recruitment. 



The model, fitted by a nonlinear least squares 

 technique (Marquardt 1963), predicts an average 

 maximum recruitment of 18.4 billion individuals at 

 a spawning stock of 3.22 billion (Table 7). The curve 

 is a reasonably good fit (Fig. 5), considering the 

 variability inherent in clupeoid recruitment. Data 

 were available over a wide range of spawning stock 

 sizes and recruitment levels. Although recruitment 

 tended to fluctuate widely at lower spawning stock 

 sizes, estimates appear to converge at higher spawn- 

 ing stock levels, indicating the possibility of a strong 

 density-dependent response as spawning stock size 

 increases. The Ricker function appears to describe 

 the data, thus an estimate of spawning stock size 

 premits a general estimate of anticipated re- 

 cruitment at moderate to high numbers of 

 spawners. 



Because fecundity increases with age and because 



age structure of spawners varies from year to year, 

 estimates of the number of eggs produced should 

 provide a more accurate estimate of spawning stock 

 size than estimates of the numbers of spawners. 

 When the Ricker equation was fitted to number of 

 eggs and number of recruits, the estimate of op- 

 timum spawning stock size was similar to the 

 estimate based on the number of spawners and 

 recruits (Table 7) (S M of 39.66 trillion eggs = 2.3 

 billion spawners). The unrealistic replacement level 

 (S r ) of 283.32 trillion eggs was generated by scaling 

 factors involved in the comparison of unequal 

 spawner and recruit units (Ricker 1975). Applying 

 the function to spawning and recruitment biomass 

 also provided similar estimates of maximum recruit- 

 ment and optimum spawning stock size (Table 7, 



Table 7. — Ricker spawner-recruit estimates of maximum equilibrium 

 stock (S r ), stock size for maximum recruitment (S m ), and recruit- 

 ment at S m , for models incorporating number of spawners on 

 number of recruits, number of eggs on number of recruits, and 

 spawning biomass on recruitment biomass, 1964-76 year classes 

 of gulf menhaden. 



No. of spawners 



on no. of 



recruits 8.83 billion 3.22 billion 18.42 billion 



No. of eggs 



on no. of 



recruits 283.32 trillion 39.66 trillion 18.48 billion 



Spawning 



biomass 



on recruit 



biomass 524,172 t 336,011 t 588,236 t 



NUMBER OF SPAWNERS (billions) 



Figure 5.— Ricker spawner-recruit relationship for number of spawners and recruits at 

 age 1, estimated as of 1 January, for the 1964-76 gulf menhaden year classes. 



318 



