232 



Fishery Bulletin 103(2) 



adult mortality and the recruitment function. As before 

 (Alonzo and Mangel, 2004), we examine the following 

 three cases: 1) the entire population mates at one site 

 (1 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). We assume that within a 

 mating site, individuals mate in proportion to their 

 fertility and fecundity and that males that are large 

 enough to change sex have a chance of reproducing that 

 is proportional to their fertility and thus a large male 

 reproductive advantage exists. This is equivalent to 

 assuming that females exhibit a mate choice threshold 

 (Janetos, 1980) that has evolved with the size at pat- 

 tern of sex change and that male fertilization success 

 is proportional to fertility. 



Reproduction 



We assume female fecundity E(L) and male sperm pro- 

 duction SiL) can be represented by the allometric rela- 

 tionships EiL)=aL h and SiL)=cL b respectively where a, b 

 and c are constants. We assume that at any body length 

 males produce 1000 times more sperm than females 

 produce eggs. This leads to the realistic pattern that 

 (in the absence of fishing) fertilization rates are high 

 and that multiple males are needed to fertilize all the 

 eggs produced by females. We calculate the average 

 expected fertilization rate per mating site based on the 

 total production of sperm and eggs at the site, where S 

 represents the number of sperm released (in millions) 

 and E the number of eggs released at each mating site. 

 The proportion of eggs fertilized per mating site p F is 

 given by 



P F = 



1 + IkE + x)S 



(5) 



examples represent four plausible patterns that differ 

 in the cues or mechanisms that induce sex change, 

 the degree of compensation or plasticity assumed, and 

 encompass the diversity that has been observed and 

 hypothesized for a variety of sex-changing fish popula- 

 tions (Helfman, 1997). 



Rule 1 : Fixed For the first sex-change rule, we assume 

 that the probability of sex change p c (L) is determined 

 by the absolute length of the individual and is 



p c (L)-- 



1 



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



(6) 



where L c represents the size at which 50% of mature 

 females change sex and p is a constant that determines 

 the steepness of the probability function. With this sex 

 change rule, we also assume that the probability an 

 individual matures p^L) is determined by absolute size. 

 Once an individual matures, she remains female until 

 sex change. L M represents the length at which 50% of 

 juveniles are expected to mature. 



PM&) 



l + exp(-<j(L-L M )) 



(7) 



where q determines the steepness of the probability 

 function and where L C >L M . 



Rule 2: Relative size For the second sex change rule, the 

 mean size of all individuals in the mating group deter- 

 mines the probability of sex change for an individual. 

 First, we find the mean size of all individuals at each 

 mating site. We let L t represent the mean size in the 

 mating site i. Then the probability of sex change for an 

 individual of length L is 



where k and % are constants fitted to data. The pro- 

 portion of eggs fertilized (p F ) depends on both total 

 sperm production (S) and egg production (E). If sperm 

 production is very high in relation to egg production, 

 fertilization rates will be at or near 100%. However, if 

 total sperm production (S) decreases and egg production 

 remains the same, fertilization rates will decrease. Simi- 

 larly, as egg production (E) increases in relation to total 

 sperm production (S) fertilization rates will decrease 

 (see Fig. 2, Alonzo and Mangel, 2004). The number of 

 eggs fertilized per group is p F E and the total production 

 of fertilized eggs Pit) is the sum of the number of eggs 

 fertilized in all mating groups. For more details on the 

 fertilization function and individual sperm production 

 see Alonzo and Mangel ( 2004). 



Patterns of sex change 



We examine four possible patterns of sex change, deter- 

 mined by absolute or relative size of the individual. 

 Although a variety of other possibilities exist, these 



Pc (L) = 



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



(8) 



where AL C represents the difference from the mean at 

 which the probability of sex change is 0.5. For these 

 analyses, we also assumed that the probability an 

 individual matures also depends on the mean size of 

 individuals at the mating site. Then the probability of 

 maturity is 



Pm 



<L) 



1 



1 + exp(-q(L - ( L, + AL M ))) 



(9) 



where AL M represents the difference from the popula- 

 tion mean at which the probability an individual will 

 mature is 0.5. 



Rule 3: Relative frequency Sex change may also be 

 induced by the social conditions at the mating site. For 

 example, sex change may depend on the frequency of 



