Williams and Shertzer: Effects of fishing on growth traits: a simulation analysis 



393 



that is due to the genotypes of the parents. It 

 can range from zero to one, with a higher value 

 potentially speeding the evolutionary response 

 to selection. Field estimates of heritability in 

 fish size are uncommon because in nature it is 

 difficult (although not impossible; McAllister 

 et al., 1992) to separate genetic and environ- 

 mental effects on phenotypes. Almost all esti- 

 mates come from laboratory experiments (e.g., 

 Hadley et al., 1991; Conover and Munch, 2002; 

 Vandeputte et al., 2002), mostly on populations 

 from aquaculture breeding programs (e.g., Gje- 

 drem, 1983; Jarayabhand and Thavornyutikarn, 

 1995; Henryon et al., 2002). One might expect 

 laboratory experiments to over-estimate natural 

 heritabilities, because experiments tend to re- 

 duce environmental effects on total phenotypic 

 variance, but estimates from the laboratory 

 have been similar to those from the field (Wei- 

 gensberg and Roff, 1996). The laboratory exper- 

 iments indicate that heritabilities in fish growth 

 traits may vary widely among populations but Repeat 



are high enough to allow rapid evolution, given over 



a large enough selection differential. time 



Models of evolutionary response to selec- steps (t) 



tive harvest have usually taken one of two for one 



approaches: quantitative genetics (e.g., Law, year 



1991; Ratner and Lande, 2001) or life-history 

 optimization (e.g., Blythe and Stokes, 1999). In 

 the present study, we take a different approach. 

 Rather than attempt to predict evolution ex- 

 plicitly, we focus on selection differentials, a 

 necessary (but not sufficient) condition for an 

 evolutionary response. 



We use simulation analyses to compute selec- 

 tion differentials caused by fishing. The simula- 

 tion model is one common in fisheries. It con- 

 sists of an age-structured population following 

 von Bertalanffy growth, with fishing and repro- 

 duction modeled as continuous processes. 



Our goal is to compare selection differentials 

 across a variety of life-history and fishery char- 

 acteristics. We quantify selection differentials 

 on growth parameters and body size. If growth 

 traits are heritable, those life-history and fish- 

 ery characteristics with the largest selection 

 differentials are most likely to generate an evo- 

 lutionary response. Armed with such knowl- 

 edge, fishery managers can weigh potential evolutionary 

 effects when choosing a fishing strategy. 



Draw uniform random number to determine cohort of 

 individual; probabilities based on stable age structure 



Draw bivanate normal random numbers to determine 

 values of growth parameters L , K 



Draw uniform random number to determine spawning 

 time step 



Unfished population 



Fished population 



Draw uniform random number to determine 

 mortality 



Alive? 



Alive? 



t = spawning 

 time step 7 



: = spawning 

 time step? 



Draw uniform random 



number to determine if 



spawning occurred 



Draw uniform random 

 number to determine 

 if spawning occurred 



Store growth 

 parameters 



Store growth 

 parameters 



Figure 1 



Flow diagram of the individual-based model. 250,000 individu- 

 als were initialized and then duplicated; one copy entered an 

 unfished population, the other entered a fished population. Both 

 populations were simulated for a single year with monthly time 

 steps. Selection differentials on the growth parameters were 

 computed as the difference between mean trait values of the 

 unfished and fished parents. 



Materials and methods 



To compute selection differentials caused by size-selective 

 fishing we used an individual-based model (Fig. 1). To 

 initialize the model, 250,000 individual phenotypes were 

 generated. Each was assigned a set of life-history param- 

 eters and then duplicated. One copy entered an unfished 

 population that experienced only natural mortality; the 



other copy entered a fished population that experienced 

 both natural and fishing mortality. Growth, survival, and 

 reproductive success of individuals were simulated with 

 monthly time steps for a single year. At the end of the 

 simulation, selection differentials on growth parameters 

 were computed as the percent change between the mean 

 values of spawners in the two populations. 



Model structure 



The model comprised three basic life-history functions: 

 growth, survival, and reproduction. For each individual. 



