Ebert et al.: Size distributions with pulsed recruitment 



239 



> 0.1 



c 



0) 



ZS 0.0 



i- 



4 8 12 4 8 12 



Size 

 Figure 2 



Integration of N, vs. S, over intervals of 1.0 size unit. Dotted line emphasizes 

 general shape of the envelope of each distribution. (A) 1 recruitment epi- 

 sode/yr; envelope of the distribution has a negative slope. (B) 2 recruitment 

 episodes/yr; envelope of the distribution has high points at smallest and 

 largest sizes. (C) 10 recruitment episodes/yr; general envelope has a positive 

 slope but has modes at small sizes. (D) 100 episodes/yr; envelope has a 

 positive slope. 



P(x)=^-r(a,t + a,t 2 + a,t 3 ) + e(x) (7) 





er* 



1+Px 



(8) 



(9) 



110) 



with a l = 0.4361836, a, = -0.1201676, a 3 = 

 0.9372980, p = 0.33267, and efxklO 5 . The 

 area under the normal curve, A, from s to 



s+As is 



A = P(xL,,-P(xL 



(11) 



Areas under the normal curve for each co- 

 hort were reduced by multiplying each 

 area, A, for a cohort by e~ Zt according to 

 Eq. 5. The size-frequency distribution was 

 produced by establishing a 1-unit size- 

 interval and summing parts of all cohorts 

 in each interval. 



With one recruitment episode/yr (Fig. 3A), 

 the distribution is polymodal. Because the 

 cluster of individuals that are >1 yr is bi- 



Simulated size distributions 

 take on an appearance much closer 

 to distributions seen in the field 

 when individual sizes are dis- 

 tributed around mean size-at-age 

 (Fig. 3). A coefficient of variation of 

 0.1 was used for simulation, so 

 rr=0.1|i. Mean sizes were calculated 

 using Eq. 1, and areas under the 

 normal curve were estimated out to 

 4ct in units of o710. Areas for each 

 size segment were determined by 

 successive subtraction of terms ob- 

 tained from a polynomial ap- 

 proximation of the area under the 

 normal curve and based on a 

 program for the normal distri- 

 bution given by Poole & Borchers 

 (1979), who used an algorithm 

 from Hastings (1955) (Function 

 26.2.16 in Abramowitz & Stegun 

 1972). P(x) is the area under the 

 normal curve from the mean, u, to 

 a size, s, given a standard devia- 

 tion of a: 



0.4 



o> 



12 



Size 



12 



Figure 3 



Results of a simulation with Z=0.5, K=1.0, S~=10.0, and s=0.1xmean. Simulations 

 differ with respect to number of recruitment episodes/yr (range 1-100/yr). (A) 

 polymodal with general envelope with negative slope; (B) polymodal with general 

 negative slope; (C) polymodality still evident but envelope has a positive slope; (D) 

 unimodal with general positive slope. 



