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Fishery Bulletin 92(2). 1994 



LU 



3 



Q_ 

 O 



1957). In modeling fish populations, one 

 could just as easily use the exponential yield 

 model of Fox ( 1970) or a model of more flex- 

 ible shape, such as that of Pella and 

 Tomlinson (1969) or its alternative formu- 

 lation by Fletcher ( 1982). (Fletcher's formu- 

 lation lacks the estimated exponent that has 

 been found to complicate estimation [Ricker 

 1975, p. 326].) Unfortunately, those formu- 

 lations can not supply an analytical formu- 

 lation similar to Equations 6 and 8, which 

 means that numerical integration would 

 have to be used, as in the GENPROD com- 

 puter program of Pella and Tomlinson 

 (1969). Another alternative would be to use 

 a discrete-time model, rather than the con- 

 tinuous-time model presented here. Such 

 models are simpler mathematically, but 

 usually entail assumptions that the growth, 

 recruitment, and catching seasons are brief. 

 The logistic model was used here because 

 it is a simple case, not because using other 

 models would be impractical or inferior. 



For what types of stocks are the models 

 presented here appropriate? Research is 

 lacking to answer this question definitively, 

 but general comments are possible. One 

 group of fishes for which production mod- 

 els seem to work well is the tropical tunas. 

 These species are characterized by rela- 

 tively fast growth, relatively constant re- 

 cruitment, and reduced annual seasonality 

 in the life processes. Density dependence in 

 growth has been demonstrated in a related 

 species, Scomber japonicus (Prager and 

 MacCall, 1988); such plasticity in growth 

 would allow the compensation inherent in 

 a production model to be expressed in a way 

 beyond recruitment variability. For modeling fish 

 stocks with more seasonality in growth, reproduc- 

 tion, and harvest, a discrete-time production model 

 might prove superior to the continuous-time model 

 presented here. 



In many fish stocks, recruitment is extremely vari- 

 able. Ordinary production models may not work well 

 when applied to stocks with large recruitment fluc- 

 tuations that are unrelated to population size, espe- 

 cially when the catch-effort series is short. If recruit- 

 ment fluctuations can be linked to external factors 

 (such as variation in rainfall or sea-surface tempera- 

 ture), a production model incorporating these factors 

 might work well (Freon, 1986). It would be simple to 

 modify the logistic model to incorporate an environ- 

 mental factor, perhaps as an influence on r on an 

 annual basis. 



120 



100 



- 20000 



1 5000 P 



5 



25000 



1400 



1200 



1000 



800 - 



600 



400 - 



200 



Figure 1 



Data used to fit production model examples loosely based on 

 swordfish, Xiph ias gladius , in the North Atlantic Ocean. (A) Stan- 

 dardized effort rate (•) and total yield (o). (B) CPUE trajectory 

 (•) computed from data in (A), and index of abundance (°) used to 

 tune the second example. The index, which was used for illustrative 

 purposes only, is not a good measure of total-stock abundance. 



Other extensions 



Many other extensions to the production model have 

 been published. An incomplete list includes these: 

 Fox (1975, 1977) presented a logistic production 

 model with mixing of two stocks; Deriso (1980) and 

 Hilborn (1990) demonstrated different methods of 

 fitting production models to age-structured popula- 

 tions (but see also Ludwigand Walters, 1985); Freon 

 (1986) introduced environmental variables into a 

 production model that used the equilibrium assump- 

 tion; Laloe (1989) and Die et al. ( 1990) incorporated 

 fished area into production models; Polovina ( 1989) 

 demonstrated a system of production models in which 

 some parameters are common among models; and 

 Hoenig and Warren (1990) demonstrated Bayesian 



