Macewicz et al.: Fecundity, egg deposition, and mortality of Lo/igo opalscens 



311 



alive and not taken by the fishery up to time t k , and t k 

 < t max . The egg escapement rate, R eJk , up to time t k is the 

 sum of the three sources of egg escapement divided by the 

 total number of eggs that would have been spawned if no 

 fishery existed (E): 



E c + E SI + E A 



(3) 



Egg escapement rate at the maximum elapsed time U max ) is 



R e,tn 



E C + E M 



(4) 



where t k = t n 



Because there are no survivors at time t max , no eggs can 

 be deposited and E A is zero. 



Each term in Equation 3 can be expressed as functions 

 of the mean cumulative number of eggs deposited up to 

 time t k , E SP t . =E P - E YD tk = E P ( 1— e - "'* ), and total mortal- 

 ity (z) of the cohort; z includes both natural morality (m) 

 and fishing mortality if). For practicality, we considered 

 cases when t k = t max , where E A is zero. For formulas of any 

 t k , see appendix. The total number of eggs deposited by the 

 females in the catch iE c ) is 



'max 



E c = j E^ t N e- 



m+l V 



fdt 



-E P N J (l-e- l ")e-"" + '"fdt 

 o 



E~N f 



{m + f )( m+f + v) 



(5) 



(6) 



where E P - the mean number of oocytes in the ovary per 

 mature female prior to spawning; and 

 N n = the number of mature females at time 0. 



E P N m 



( m + f X m + f + v ) 



(9) 



The total eggs that would be deposited for the cohort 

 without fishing mortality is 



'max 'max 



E= \ E~^p~ t N me~ mt dt = E~pN J Tl-e" 1 '' )m e -""dt (10) 

 o o 



E = E P N  



(11) 



m + v 



where t max (days) = the maximum elapsed time; and 



time = the time at the onset of egg deposition. 



Egg escapement based on Equation 4 is 



f 



 + m- 



R.. 



( m + /')(/?? + f + v) ( w + f)(m + f + v) 



(12) 



m + f + v 



Thus, egg escapement reduces down to a simple ratio, 



involving three daily instantaneous rates: natural 



mortality (m), egg deposition (v), and fishing mortality (/"). 



R„, =1 when there is no fishing and thus i?„„<l with 



(Vmax ° '-■'I: 



fishing mortality. The lower bound of the egg escapement 

 rate for the cohort is equal to the ratio of the eggs escap- 

 ing the fishery (E r ) to the total eggs deposited if no fishery 

 existed (£): 



R=E C IE. 



(13) 



From the fishery data, we can estimate the total number 

 of eggs deposited by the females in the catch (E c ) as 



E c = n c[ e p- e yd)> 



(7) 



where E P and E YD = sample estimates from the catch; 

 and 

 N c = the total number of spawners in the 

 catch. 



The total number of eggs deposited by L. opalescens prior 

 to death due to natural mortality iE M ) is 



E M =jE SPl N e-'"-'"mdt = 



(8) 



E p N J(l- 



•")e-"' 



mdt 



Results 



Oocyte maturation and production 



Immature ovaries contain many small unyolked oocytes 

 with a pronounced peak at about 0.15 mm in size distribu- 

 tion ( Fig. 6A). As development continues and vitellogenesis 

 begins, the peak diminishes and shifts to a larger size 

 class of unyolked oocytes (Fig. 6B). Just before the onset 

 of spawning, the size distribution of oocytes becomes 

 relatively fiat without pronounced modes (Fig. 6C) and 

 remains so through the rest of the spawning period ( Fig. 6, 

 D-F). The standing stock of oocytes declines throughout 

 the spawning period. The minimum size of oocytes in 

 the ovary gradually increases after the onset of yolking, 

 indicating that new oocytes are not produced. We saw no 

 primary oogonia in our histological sections of mature ova- 

 ries, another indication that new oocytes are not produced 

 in mature ovaries. Knipe and Beeman (1978) reached the 



