Alonzo and Mangel: Sex-change rules, stock dynamics, and the performance of spawning-per-recruit measures in protogynous stocks 231 



would experience sperm limitation and would affect the 

 performance of traditional spawning-per-recruit mea- 

 sures. However, we did not consider the possibility that 

 size at sex change may be plastic and depend on local 

 social conditions or relative rather than absolute size. 

 Plastic sex change may allow a protogynous species to 

 compensate for any effect of size-selective fishing on the 

 sex ratio of the population, rendering its dynamics iden- 

 tical to the dynamics of a dioecious species. However, 

 as described above, a wide variety of patterns of sex 

 change have been observed in the wild and have been 

 proposed to occur. Therefore, the exact pattern of sex 

 change and cue driving phenotypic changes may lead to 

 unique stock dynamics. In this study we apply the same 

 general method we used previously (Alonzo and Mangel, 

 2004) to examine the effect of four different patterns 

 of sex change (one fixed and three plastic) on the stock 

 dynamics of a protogynous species. 



Methods 



We applied the same general method and individual- 

 based population dynamic model as our previous study 

 (Alonzo and Mangel, 2004). However, we now included 

 the effect of four different patterns of sex change on the 

 stock dynamics and performance of spawning-per-recruit 

 measures in a protogynous species. Individuals vary in 

 age, size, sex, and location (i.e. mating site). We assumed 

 annual time periods and determined individual survival, 

 size, and reproduction as described below. We simulated 

 100 years prior to examining the impact of fishing on 

 stock dynamics and then simulated 100 more years in 

 the presence of fishing with a constant mean fishing- 

 induced mortality. This allowed the population to reach 

 a stable age, sex, and size distribution prior to fishing 

 which is independent of initial conditions. Because a 

 number of elements of the model are stochastic, we 

 examined 20 simulations for each scenario and set of 

 parameter values, which was more than sufficient in 

 all cases to lead to low variability in the key measures 

 of interest. 



Fishing and adult survival 



We assume age and size do not affect natural adult 

 mortality, i.i A and that adult mortality is density-inde- 

 pendent. The fishery is size selective; if L represents fish 

 size, F annual fishing mortality, L f the size at which 

 there is 50% chance an individual of that size will be 

 taken, and r the steepness of the selectivity pattern, the 

 fishing selectivity per size class s(L) is given by 



11) 



SiL) = - — r 



l+eiq)y-r(L-L f )\ 



and adult annual survival is 



a(L) = exp(-fi A - Fs(D). 



Population dynamics 



The number of larvae that enter the population is deter- 

 mined by larval survival and the total production of fer- 

 tilized eggs Pit), which is determined by total fecundity 

 and fertility within each mating site as described below. 

 Larval survival is assumed to have both density-inde- 

 pendent and density-dependent components (e.g., Cowen 

 et al., 2000; Sale, 2002), and we use a Beverton-Holt 

 recruitment function (Quinn and Deriso, 1999; Jennings 

 et al., 2001) to calculate larval survival . The number 

 of larvae surviving to recruit in any year t, N (t), is 

 given by 



N Q (t) = (aPit))/(l + pP(t)) 



if(aP(t))/(l + l3P(t))+'£N n {t)<N n 



(3) 



N (t) = max 



0,N max -^N n U 



o=l J 



if(aP(t))/(l + liP(t))+^Njt)>N n 



where a gives density-independent survival, ft deter- 

 mines the strength of the density-dependence in the 

 larval phase, and N max represents the maximum popula- 

 tion size. We assume that the population is open between 

 mating sites, a single larval pool exists, larval recruit- 

 ment is random among mating sites, and there is no emi- 

 gration to or immigration from outside populations. 



Growth dynamics 



Larvae that survive to recruit begin at size L and 

 growth is assumed to be deterministic and indepen- 

 dent of sex or reproductive status. We calculate growth 

 between age classes using a discrete time version of the 

 von Bertalanffy growth equation (Beverton, 1987, 1992) 

 where L lnf represents the asymptotic size and k is the 

 growth rate. Then an individual of length Lit) at time t 

 will grow in the next time period to size LU+1): 



LU + 1) = L inf (1 - exp(-&)) + LU)exp(-£). 



(4) 



Mating system 



(2) 



As in our previous model, we assume that reproduction 

 occurs at the level of the mating group, and we examine 

 the effect of varying mating group size and the number 

 of mating sites. Juveniles and adults are assumed to 

 exhibit site fidelity and larvae settle randomly among 

 mating sites. The carrying capacity of the population 

 is split equally among the mating sites and the total 

 capacity of all mating sites exceeds the maximum popu- 

 lation size in the absence of fishing as determined by 



