FISHERY BULLETIN: VOL. 84, NO. 3 



propriate for other population parameters such as 

 anchovy larvae survival but are ignored here. The 

 modeling emphasizes those features directly impact- 

 ing the adult sardine population because it is the only 

 population for which detailed data are available for 

 making comparisons. 



Initial Conditions: 

 State of the Ecosystem 



The sardine ecosystem will assumed to be in an 

 approximate equilibrium state prior to 1932, ignor- 

 ing random population fluctuations. The sardine 

 population appears to be consistently near virgin 

 levels for the few years that data are available 

 before 1932 (Fig. 3), and I speculate that the other 

 populations are at reasonably consistent levels as 

 well. There is some justification for overall stabil- 

 ity at the sardine-anchovy-competitor trophic level 

 and the predator trophic level, if not for individual 

 fish species or population groups (Sette 1969; Steele 

 1979). 



Estimates of population biomasses prior to the 

 1932-52 collapse period were summarized by Atkin- 

 son (1980) from data given by Murphy (1966) and 

 Riffenburgh (1969). The biomasses presented below 

 correspond to the assumed equilibrium state at the 

 start of a fishing year. A fishing year is defined to 

 begin in the summer after the main spring spawn- 

 ing season of the sardine and anchovy. 



The initial state in 1932 is also defined by this 

 biomass vector, P. 



Parameter Estimation for the 

 Sardine Ecosystem Model 



First, I point out that the above model represen- 

 tation is not intended to be a comprehensive descrip- 

 tion of the sardine ecosystem or to have general ap- 

 plication for predicting future population dynamics, 

 at least not as developed here. However, it is pro- 

 posed as a reasonable representation to demonstrate 

 the similarity between simulated results and ob- 

 served system dynamics during the 1932-52 time 

 frame provided appropriate parameter estimates 



can be determined. The value of the difference for- 

 mation in dealing with the parameter uncertainty 

 issue will be illustrated in the discussion below of 

 parameter estimation procedures. 



Two model parameters in the equations of Table 

 1 were estimated directly from available data in the 

 literature (Murphy 1967; MacCall 1979; Clark and 

 Phillips 1932; Huppert et al. 1980): adult sardine sur- 

 vival, S zo = 1.40 (excludes fishing mortality ef- 

 fects), and adult anchovy survival, S 5 = 1.20. The 

 driver terms in the model, d 3 (£) and E^t), were 

 also estimated from available data during the 

 simulation period. These terms could not, of course, 

 be definitized without the benefit of present hind- 

 sight. In a predictive situation, such terms would 

 generally have a large degree of uncertainty, 

 because projected fishing pressure is highly 

 speculative while larvae survival has a strong 

 stochastic component. Here, however, the available 

 data will be used to the extent possible to resolve 

 model terms. 



Estimates of sardine fishing parameter, 6 3 (t), 

 were derived from Murphy's (1966) data and are 

 shown plotted in Figure 5. The simplified model used 

 in the simulations ignores detailed yearly variations 

 and focuses on the major trends. A linear increase 

 is assumed during the period from a rate of about 

 0.1 in 1932 to a rate >0.4 in 1936. The fishing rate 

 is assumed to remain constant for the remainder of 

 the simulation period. 



The assumed model for the sardine larvae survival 

 term, E^t), is presented in Figure 6 along with 

 Sette's (1969) data from which it was derived. These 

 data represent numbers of fish at age class two ver- 

 sus the year spawned. The survival rate model 

 assumes that these observed fluctuations in the data 

 primarily reflect random survival effects during the 

 first year of life. E x {t) was obtained by normaliz- 

 ing Sette's data with respect to the spawning 

 population biomass and defining a relative scale such 

 that the integrated value over the 20-yr period from 

 1932 to 1952 was equal to one. 



The remaining model parameters, which repre- 

 sent the great majority of those in the equations of 

 Table 1, could not be directly estimated to any 

 degree of accuracy from available literature data. 

 Instead, these estimates were derived from the 

 special nonlinear programming analysis of mine 

 (1980, in press) mentioned previously. I treated 

 these ecosystem model parameters as variables with 

 upper and lower bounds reflecting their uncertain- 

 ty ranges. The bounds established by me for the sar- 

 dine ecosystem parameters were typically an order 

 of magnitude. Implicit parameter constraints were 



544 



