Alonzo and Mangel: Sex-change rules, stock dynamics, and the performance of spawmng-per-recruit measures in protogynous stocks 233 



other mature individuals or the frequency of smaller 

 individuals. We examine the case where sex change 

 depends on the frequency of smaller mature individu- 

 als. For each mature female, we find the frequency of 

 mature individuals at the same mating site that are 

 smaller. We let F t represent the frequency of mature 

 individuals that are smaller than the mature female 

 and F c represents the frequency at which 50% of the 

 individuals are expected to change sex. Then the prob- 

 ability of sex change is 



Pc (L) = 



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



(10) 



The probability of maturing depends on the frequency of 

 smaller individuals. We let /j represent the frequency of 

 all smaller individuals at mating site i and f M represent 

 the frequency at which there is a 50% probability of an 

 individual maturing. Then the probability of maturing 

 is 



P M (L) = 



1 



l + exp(-q(f,-f M ) 



(11) 



Rule 4: Reproductive success Finally, we consider the 

 case where sex change occurs when an individual's 

 size-dependent expected reproductive success is greater 

 as a male than as a female (Charnov, 1982). This pat- 

 tern of sex change has been proposed to explain the 

 observation that individual variation exists in size at 

 sex change and that it is not always the largest indi- 

 vidual in a group that changes sex (Munoz and Warner, 

 2003). We assume that a fish will change sex when its 

 expected egg production at its current length (E(L)=aL h 

 as given above) is exceeded by its expected paternity at 

 the mating site which is given by the total egg produc- 

 tion of all other females at the site multiplied by the 

 focal individual's relative sperm production. This value 

 is given by expected fertility S(L) divided by the total 

 sperm production (by all males at the site plus their 

 own expected fertility) at the same mating site. We 

 further assume that sex change occurs once a year in 

 rank order from the largest to smallest female at the 

 site. (For this scenario we assumed that the probability 

 of maturing depends on absolute size as in Equation 

 7.) However, we still assume that individuals can only 

 change sex once during their lifetime and only mature 

 females can change sex. Thus, mature females change 

 sex when their current expected fertilization success as 

 a male is greater than their current expected fecundity 

 as a female. 



Measures of spawning stock biomass per recruit 



We examine the same spawning-per-recruit measures 

 as in our previous paper (Alonzo and Mangel, 2004) and 

 compare the results of the patterns of sex change con- 

 sidered here with one another and with a hypothetical 

 dioecious species, where sex is determined stochastically 



at birth and the primary sex ratio is fixed. We compute 

 the total spawning stock biomass per recruit starting 

 from the beginning of fishing for the next 50 years. We 

 use the generally recognized pattern that fish wet weight 

 tends to be approximately proportional to the cube of 

 fish length (Gunderson, 1997) to convert fish length, L, 

 into relative biomass, B(L)~ZA Then we calculate total, 

 female, and male spawning stock biomass per recruit 

 (SSBR). We also keep track of the total fecundity (egg 

 production per recruit), fertility (sperm production per 

 recruit), and eggs fertilized per recruit. 



Parameter values 



We use parameters based on previous research (Warner, 

 1975; Cowen, 1985; Cowen, 1990) on California sheep- 

 head (Labridae, Semicossyphus pulcher), a commercially 

 important sex-changing fish, to provide evolutionary 

 and ecologically reasonable parameters for the model. 

 Although the growth, survival, and reproduction of 

 this species have been studied, less is known about the 

 factors that induce sex change and mating behavior. In 

 this species, sex change occurs at approximately 30 cm, 

 although the exact pattern varies among populations 

 (Warner, 1975; Cowen, 1990). It is not known whether 

 sex change is fixed or socially mediated. For the first 

 sex-change rule, we assume that individuals have a 

 50% chance of maturing (L,„) at 20 cm (the mean size 

 of maturity observed in natural populations) and of 

 changing sex at (L c ) 30 cm. This leads to a sex ratio of 

 2/3 mature females to 1/3 males on average and a mean 

 length of 20 cm in the absence of fishing as is observed in 

 the wild. For consistency, we also assume for the second 

 sex-change rule, that individuals have a 50% chance 

 of changing sex at 10cm (z\L ( ,=10) above the mean size 

 and have a 50% chance of maturing at the mean size in 

 the mating site (AL m = 0). Similarly, for the third rule, 

 the frequency of smaller mature individuals at which 

 there is a 50% of sex change is assumed to be 0.67 and 

 when 50% of all individuals are smaller, an individual 

 will have a 0.5 probability of maturing. Therefore in the 

 absence of fishing all four sex change rules lead to the 

 same maturity and sex-change patterns as a function 

 of age and size. For more information on the parameter 

 values considered here, see Table 1. 



Individual-based simulations are computationally 

 very intensive. As a result, it was not feasible to explore 

 a wide range of values for all parameters. Furthermore, 

 because growth, mortality, reproduction, maturity, and 

 sex change are coevolved characters within any spe- 

 cies, it does not make sense in this context to vary 

 them independently. Instead, we used estimates from 

 California sheephead for as many parameters as pos- 

 sible (mortality, growth, fecundity, size at maturity, and 

 sex change) and when necessary from a closely related 

 species (fertilization rate). We then focused on exploring 

 the effect, for this species, of varying the sex-change 

 rule and fishing pattern while all other parameters 

 remained the same. Our focus was on determining the 

 impact of the sex change rule on the predicted stock 



