Alonzo and Mangel: The effects of size-selective fisheries on the stock dynamics of and sperm limitation in sex-changing fish 3 



group at different reproductive sites. Individual sur- 

 vival, maturation, sex change, and mating site were 

 determined stochastically as described below. 



Fishing and adult survival 



We assumed that adult survival is density indepen- 

 dent but depends on fishing selectivity, fishing mor- 

 tality, and baseline adult mortality in the absence of 

 fishing. For simplicity, we assumed that age and size do 

 not affect nonfishing adult mortality p. A . We assumed 

 that the fishery is size selective; we let L represent fish 

 size, F represent annual fishing mortality, L f represent 

 the size at which there is 50% chance an individual of 

 that size will be taken, and r represent the steepness 

 of the selectivity pattern. Then fishing selectivity per 

 size class siL) is given by 



siD- 



l + exp-HL-L,)) 



and adult annual survival becomes 



cr(L) = exp(-/i A -Fs(L)) 



(1) 



(2) 



We assumed that fishing does not differentially 

 affect the sexes independent of size. We recognize, 

 however, that for some species this may not be the 

 case. We also assumed that fishing occurs each year 

 prior to reproduction and can represent either pulse or 

 continuous fishing with an annual mortality F. We let 

 N it) represent the number of individuals in age class 

 a at time t so that population size N(t)= S a N a (t). 



Population dynamics 



We assumed that the number of larvae that enter the popu- 

 lation is determined by the production of fertilized eggs Pit) 

 and the probability that those larvae will survive to recruit. 

 Pit) is determined by the adult fecundity and fertilization 

 rates described below. For computational tractability, we 

 also assumed that a population ceiling N max exists (Mangel 

 and Tier, 1993, 1994 ). However, we chose N mBX large enough 

 that the stable population size was below the ceiling. Larval 

 survival has both density-independent and density-depen- 

 dent components (e.g. Cowen et al., 2000; Sale, 2002). We 

 used a Beverton-Holt recruitment function to determine 

 larval survival to the next age class (Quinn and Deriso, 

 1999; Jennings et al., 2001). Larvae represented the zero- 

 age class N (t) and thus the number of larvae surviving to 

 recruit in any year t is given by 



N n it) = (oPit))/(l+pPit)) if (ctP(t))/(l+pP(t)) 



+J j Njt)<N max 



(3) 



AT (*) = max| 0,N max -^N a (t) | if (aPlt))/(l+ pPit)) 



MATING SITES 



Adult survival determined by baseline mortality and fishing pattern 



Reproduction determined by group fecundity and fertility 



No migration between mating sites 



Density-dependent and 

 density-independent 

 larval survival 



Recruitment random 

 across mating sites 



Figure 1 



Structure and population dynamics of the individual-based 

 model. We assumed that all mating sites contribute to a single 

 larval pool. 



where a gives density-independent survival; and /3 deter- 

 mines the strength of the density-dependence in the larval 

 phase. In this function, we used the number of fertilized 

 eggs produced, Pit), rather than spawning stock size. We 

 selected parameter values for larval survival that allowed 

 the mean population size to be stationary near the ceiling 

 in the absence of fishing. We assumed a single larval pool 

 and that larvae recruit to mating sites at random (Fig. 1). 

 The population was open between mating sites and we 

 were simulating the entire stock. Thus, there was no emi- 

 gration to or immigration from outside populations. 



Growth dynamics 



We assumed that all larvae enter the population at the 

 same size, L . We assumed that growth is deterministic 

 and independent of sex or reproductive status. We used a 

 discrete time version of the von Bertalanffy growth equa- 

 tion (Beverton, 1987, 1992) to determine growth between 

 age classes of surviving adults in which L mf represents the 

 asymptotic size and k is the growth rate. Then an indi- 

 vidual of length Lit) at time t will grow in the next time 

 period to size Lit+1) as follows: 



Z,(f+l) = L lnf (l + exp(-fc)) + L(f>exp(-A). 



(4) 



Mating system 



We assumed that reproduction occurs at the level of the 

 mating group, and we examined the effect of varying mating 

 group size and the number of mating sites. We assumed 



