Fishery Bulletin 102(1) 



that juveniles and adults exhibit site fidelity but that larvae 

 settle randomly among mating sites. We also assumed that 

 the population carrying capacity is split equally among the 

 mating sites and that the total capacity of all mating sites 

 exceeds the maximum population size in the absence of fish- 

 ing as determined by adult mortality and the recruitment 

 function. Therefore, mating sites do not limit recruitment 

 but may affect reproductive rates. We examined three cases: 

 1 ) the entire population mates at one site (one mating site 

 with up to 1000 individuals); 2) a few large mating groups 

 exist ( 10 sites with a maximum of 100 individuals per site); 

 and 3) many small mating aggregations exist (20 mating 

 sites with a maximum of 50 individuals per site). For sim- 

 plicity, we assumed that within a mating site, individuals 

 mate in proportion to their fertility and fecundity. Therefore, 

 large males and females have higher expected reproductive 

 success. However, we assumed that all males that are large 

 enough to change sex have a chance of reproducing propor- 

 tional to their fertility. This is equivalent to assuming that 

 females exhibit a mate choice threshold I Janetos, 1980) that 

 has evolved with the size-at-sex change and that females 

 have an equal probability of mating with males above this 

 size threshold. However, a large male mating advantage 

 clearly still exists. We also assumed that fishing mortality 

 remains constant as mating aggregation size varies. Thus, 

 we assumed that fishing effort per site does not increase as 

 the number of mating sites decreases. An alternative would 

 be to assume that total fishing mortality increases as the 

 number of mating aggregations decreases. 



Maturity 



The probability that an individual matures p m (L) is deter- 

 mined by size. Once an individual matures, she remains 

 female until sex change (see below). We let L m represent 

 the length at which 50% of the individuals will have 

 matured. 



EiL)=aL h , 



(7) 



P,JL)- 



1 



where a and b are constants. 



Once an individual has changed sex (as determined by 

 the sex change rule described above) sperm production (in 

 millions) S(L) is given by 



S{L)=cL d , 



(8) 



l + exp(-q(L- L m 



(5) 



where c and d are constants. 



Size-dependent fecundity has been measured in many 

 fish species (e.g. Gunderson, 1997). A general allometric 

 relationship between sperm production and size has not 

 been established. Therefore, we assumed that male gamete 

 production increases with size at the same rate as that for 

 females ib=d). We also assumed that males produce many 

 more sperm at any body length than females produce 

 eggs. Clearly, other possible patterns exist. We examined 

 the case where males produce from 10 2 to 10 6 sperm for 

 every egg produced by a female. In the pelagic spawning 

 wrasse (Thalassoma bifasciatum ), large males release ap- 

 proximately 1000 times more sperm than females release 

 eggs (Schultz and Warner, 1991; Warner et al., 1995). 



We used recently published data on sperm production 

 and fertilization rates in the bluehead wrasse (Thalas- 

 soma bifasciatum) to generate a biologically appropriate 

 fertilization function for our model (Warner et al., 1995; 

 Petersen et al., 2001). It is critical to consider a biologically 

 appropriate form for the function to express fertilization 

 rates when considering the potential for sperm limitation. 

 The probability an egg will be fertilized is an increasing 

 function of the number of sperm available for that mat- 

 ing (Fig. 2). The number of eggs released per mating also 

 affects the fertilization rate (Fig. 2). For simplicity, we cal- 

 culated the average expected fertilization rate per mating 

 site based on the total production of sperm and eggs at the 

 site. We let S represent the number of sperm released (in 

 millions) and £ the number of eggs released at each mating 

 site. We assumed that the proportion of eggs fertilized per 

 mating site p F is given by 



where q determines the steepness of the probability 

 function. 



Sex change 



The probability of sex change, p c iL), is a logistic function 

 of absolute size L 



P,.(L) = 



l + exp(-p(L-L, )) 



(6) 



where L r represents the size at which 50% of the indi- 

 viduals will change sex from female to male and p is a 

 constant. 



Reproduction 



We assumed that female fecundity E(L) depends on indi- 

 vidual size according to the allometric relationship 



Pf 



l + iisE + X )S 



(9) 



where k and % are constants fitted to the data. 



The number of eggs fertilized per group is p h -E and the 

 total production of fertilized eggs. Pit), is the sum of the 

 number of eggs fertilized in all mating groups. 



Measures of spawning stock biomass per recruit 



To measure the impact of fishing on stock dynamics, we 

 computed the total spawning stock biomass per recruit 

 starting from the beginning of fishing for the next 50 

 years. We used the generally recognized pattern that 

 fish wet weight tends to be approximately proportional 

 to the cube offish length (Gunderson, 1997) to convert 

 fish length, L, into relative biomass, B(L)~L\ Then we 

 calculated total female and male spawning stock biomass 



