FISHERY BULLETIN: VOL. 84, NO. 3 



The problem scenario for my NLP formulation is 

 that of predicting the dynamic response of eco- 

 system populations to a given perturbation. The 

 response is characterized over some period of in- 

 terest by the objective function which, depending 

 on the particular problem, can be equated to average 

 population numbers, final population levels, worst- 

 year fishery catch, or some other dynamic feature. 

 The ecological parameters in the dynamics model 

 become the variables with bounds corresponding to 

 the estimated parameter uncertainty range. 



Implicit parameter constraints are added to the 

 formulation based on available population history 

 data, ecosystem stability observations, or any known 

 or postulated relationships between parameters. The 

 historical population data are substituted directly 

 into the difference equations, or other assumed 

 dynamics equations. In effect, such constraints force 

 the response modes of the dynamics model to include 

 past population observations, albeit ones that oc- 

 curred under different (known) conditions than 

 those of interest in the future. Stability observations 

 also infer conditions on the dynamics equations and, 

 hence, model parameters. However, there are prac- 

 tical issues in formulating such conditions. Lyapunov 

 stability analysis techniques (Brogan 1974), while 

 applicable to nonlinear system analysis, are not 

 readily defined for the complex difference equations. 



Efficient NLP computational procedures have 

 been applied by me (1980) to solve the special eco- 

 system formulation described above. A search takes 

 place through bounded parameter space for extreme 

 (minimum and maximum) objective function values 

 while maintaining the equality of the implicit con- 

 straints, i.e., the search proceeds on the "constraint 

 surface" in parameter space. The key to an effec- 

 tive problem solution is the computational require- 

 ments of the dynamics model which is used in both 

 constraint formulation and for evaluating the objec- 

 tive function at each search step. While the NLP 

 approach does not give definitive estimates of in- 

 dividual model parameters, it strongly delimits their 

 range of values via the interrelationships established 

 by the implicit constraints (Atkinson 1980). 



ECOSYSTEM SIMULATIONS USING 

 THE DIFFERENTIAL EQUATION MODEL 



The discrete-time multispecies dynamics model 

 given by Equation (9) has been implemented as a 

 FORTRAN computer program and used to perform 

 a variety of simulations of theoretical and applied 

 fisheries scenarios (Atkinson 1980). A case of some 

 practical interest, the collapse of the sardine popula- 



tion within the California Current region, will be 

 described and used to demonstrate the potential 

 model utility. 



General Description of 



the Sardine Population Collapse 



off California 



The waters of the California Current flow south- 

 ward along the west coast of North America cover- 

 ing the general region are illustrated in Figure 2. 

 While the California Current supports a diverse 

 group of fish, the sardine fishery was by far the most 

 important in the early years of this century until the 

 dramatic collapse of the sardine population in 

 1930-60. A large increase in fishing effort took place 

 during this time and apparently caused, or at least 

 was associated with the sardine population collapse. 

 The estimated history of the sardine population from 

 1930 to 1960 as derived by Murphy (1966) is shown 

 in Figure 3. 



Two sets of anchovy population estimates for the 

 1930-60 time frame are also presented in Figure 3. 

 Although these data are confused by significant gaps 

 and strong fluctuations from year to year, there does 

 appear to be a significant population increase from 

 levels in the 1940's and early 1950's to that near 

 the end of the 1950's. Since the anchovy is the chief 

 competitor of the sardine with similar food require- 

 ments and overlapping habitat boundaries, the 

 general indication is that the anchovy replaced the 

 sardine within the trophic structure (Murphy 1966; 

 Gulland 1971). Murphy's (1966) 3-yr averaged data 

 provides the clearest evidence of this increasing 

 trend. Smith's (1972) yearly estimates show that the 

 anchovy population actually declined from 1940-41 

 to 1950 (the next year in which data was available), 

 before a sharp rise occurred. The significant varia- 

 tions evident in both anchovy and sardine data are 

 probably caused by random environmental in- 

 fluences on recruitment success (Lasker 1978; Par- 

 rish et al. 1981; Methot 1983). 



Soutar and Isaacs (1974) presented some interest- 

 ing longer term data on the sardine and anchovy 

 (plus other pelagic fish) as derived from sedimen- 

 tary scale depositions in anaerobic basins off South- 

 ern California and Baja California. The deposition 

 rate, which is averaged by 5-yr periods, provides a 

 relative picture of the population variations over the 

 last 150 yr (up to 1970). The data for the 1930-60 

 time frame indicate similar trends to that above, i.e., 

 decreasing sardine levels and increasing anchovy 

 levels. However, significant sardine and anchovy 



540 



