FISHERY BULLETIN: VOL. 83, NO. 2 



TABLE 2.— The instantaneous mortality rates of anchovy eggs 

 and larvae < 20 days (ziti ) ) by age in days (ti ) computed from 

 the daily egg and larval production estimates (P<, ) and age (<, ), 

 1980. z(t) = 0.0060 + 1.63/t is the function fitted to the data in 

 the last two columns for f > 4.5 d. 



Daily egg 

 and larval 

 production 

 ti (d) Pti 



Pti^ 



ti = 



-Pti ti-ti-^ {ti+ti--[)i2 z(tjy 



1 



2 



3 



4 



5 



6 



7 



8 



9 



10 



11 



12 



0.67 



1.67 



2.60 



3.57 



5.65 



5.91 



7.69 



8.90 



11.47 



13.83 



15.91 



17.99 



9.28 

 5.53 

 3.70 

 2.37 



1.04 

 0.99 

 0.86 

 0.49 

 0.39 

 0.26 

 021 

 0.15 



3.75 

 1.83 

 1.33 

 2.28 

 0.05 

 0.13 

 0.37 

 0.10 

 0,13 

 0.05 

 0.06 



1.00 

 0.93 

 0.97 

 2.08 

 0.26 

 1.78 

 1.21 

 2.57 

 2.36 

 2.08 

 2.08 



1.17 



2.14 



3.09 



4.61 



5.78 



6.80 



8.30 



10.19 



12.65 



14.87 



16.95 



0.40 

 0.36 

 0.37 

 0.46 

 0.18 

 0.07 

 0.36 

 0.08 

 0.14 

 0.09 

 0.14 



'Z(f,) = (Pti-^ - Pti)l(ti - ti-A)IPti. 



quite constant for egg and larvae <4.5 d old and 

 decreased thereafter. For t values >4.5 d, the 

 function z{t) = a + hit fit the data best. Based 

 upon the function relationship z{t) = bit (the 

 intercept a is not distinguishable from zero and 

 thus w^as dropped), I have the IMR z{t): 



zit) = 



\a 



^filt 



t ^ tc 



tc<t< 20. 



(6) 



Applying Equation (6) to Equation (4) leads to 



Si(0=e~"^ t^tc 



-/3 



Sit)=< 



(7) 



S2it) = 



-'-"it) 



tc<t< 20. 



Combining Equations (3) and (7) one has 



Pt = 



t ^ tc 



tc<t<20 



-Pt 



(8A) 



(8B) 



To validate both Equations (8A) and (SB), loga- 

 rithms of P^ and t were plotted: IniPt ) against t 

 should be a straight line for t '-- tc (Equation (8A) ) 

 and \r\{Pt ) against Xnit) should be a straight line 

 for tc <t <20 (Equation (8B) ) (Fig. 3). This was 

 true for egg and larval production from 1979 to 

 1981. The determination of tc , the age at which 



IMR changes, was subjective. Two values of tc 

 were used: One was the time of hatching or the 

 duration of incubation (^7) which is temperature 

 dependent and the other was the average age of 

 yolk-sac larvae (embryonic period) tc = tys : ^2.5 mm 

 = age at preserved length 2.5 mm (about 5 d old). 

 When tc was considered equivalent to the incuba- 

 tion time (tc = ti ), the egg stages were considered 

 as one group with constant IMR; and when tc was 

 equivalent to average age of the yolk-sac larvae 

 {tc = tys ), egg stages and yolk-sac length class(es) 

 were considered as one group with constant IMR. 

 In either case, Pt^ was estimated from the fitted 



curve 



Pt = Ptj(^^~''M.,Ptc = Pti(^A 



-n 



Substitution of Equation (7) in Equation (5) gives 



'/' Po e ~"^ c/^ - Pod - e-«'0/« 

 mtc^\^ «>0 (9A) 



^c  Po a = 



Ptc =Poe 



■ate 



(9B) 



where mtc is the standing stock of eggs and larvae 

 up to age tc  Equation (9A) divided by Equation 

 (9B) results in 



mf,/Ptc = (e"'"-l)/a = h(a) a>0 



tc 



a = 



(10) 



where tc = ti or tys and q is the ratio of standing 

 stock of eggs and larvae up to age tc to the larval 

 production Pt^- The estimated IMR, a, was ob- 

 tained by an iterative procedure using Equation 

 (10). The estimated egg production obtained by 

 rearranging the terms in Equation (9B): 



Po ^ Ptc-e 



ate 



The approximate variance of a and j8 were com- 

 puted in the appendix. 



TIME SERIES ESTIMATES OF 

 HISTORICAL EGG PRODUCTION (HEP) 



The HEP per 0.05 m^ (Po ) and the egg IMR (a) 

 for the central stock of northern anchovy in the 

 first 4 mo of the year, 1951-82, were estimated 

 based upon Equations (9B) and (10). For years 

 after 1978, catch data were available for CalVET 

 and bongo nets, but I chose to use samples from 



144 



