FISHERY BULLETIN: VOL. 84, NO. 3 



4000 



z 



2 3000 



z 

 o 



_j 2000 h 



a. 

 O 



a. 



1000 - 



1920 



30 



40 

 YEAR 



50 



1960 



Figure 3.— Estimated adult populations of sardine and anchovy 

 during the 1930-60 sardine collapse period. The solid line cor- 

 responds to yearly sardine estimates by Murphy (1966). The dashed 

 line with triangles corresponds to 3-yr average anchovy esti- 

 mates also by Murphy; the initial point is a 2-yr estimate with a 

 data gap until 1951. The circles correspond to yearly anchovy 

 estimates by Smith (1972); a data gap exists between 1941 and 

 1950. 



variations are also evident in earlier times before 

 fishing pressure became a significant factor in the 

 ecosystem. For example, the sardine history showed 

 extremely low levels in 1865-80 comparable to the 

 levels after 1940. The earlier anchovy record, while 

 also having periods of relatively high and low 

 sedimentation rate, appears to have been at con- 

 sistently higher levels before 1930-60, even higher 

 than the recent increase of the late 1950's. Soutar 

 and Isaacs (1974) stated that relatively unproduc- 

 tive conditions have apparently existed for the past 

 30 yr or so and have generally affected fish popula- 

 tions of the California Current. 



Model Formulation 



subsystem defined by Riffenburgh (1969) and shown 

 in Figure 4. While not a comprehensive description 

 of this ecosystem, I use this representation to 

 demonstrate the application of the difference model 

 in a reasonably complex fishery situation. The sar- 

 dine ecosystem will be simulated during the period 

 from 1932 to 1952 spanning the years of the major 

 sardine collapse. 



The sardine population is divided into three age 

 groups: larval-year stages, yearlings, and adults. 

 The larval year is the most vulnerable period of the 

 sardines' development during which it goes through 

 many fundamental changes. The yearlings are the 

 in-between stage to the sexually mature adult 

 members of the population, which are defined to be 

 2-yr-olds and above. Early stages of the sardine feed 

 on phytoplankton while the adults feed primarily on 

 zooplankton (Huppert et al. 1980). The adults are 

 also predators of their own larval stages and those 

 of the anchovy as indicated in Figure 4. 



The anchovy population is divided into two groups, 

 larvae and adult, which have similar intergroup rela- 

 tionships and feeding habits to the corresponding 

 sardine groups. Competitor and predator groups to 

 the sardine and anchovy are defined as lumped 

 assemblages, both encompassing a broad range of 

 diverse fish species; the competitor group also con- 

 tains many invertebrates. The pelagic fish com- 

 petitors (e.g., jack mackerel) are assumed to behave 

 similarly to the sardine and anchovy except that 

 some of the larger members feed on the sardine 

 yearling stage (Riffenburgh 1969). The predators 

 (e.g., hake and baracuda) feed on the adults of the 

 sardine-anchovy-competitor trophic level and also 

 have other prey that have been decoupled from the 

 modeled subsystem. Phytoplankton and zooplankton 

 groups are modeled implicitly as carrying capacity 

 terms. 



Additional model assumptions are that 1) spatial 

 features are not critical (i.e., one spatial compart- 

 ment is used), and 2) seasonal effects can be ignored 

 (i.e., a yearly time step is defined). These two 

 assumptions are probably not justifiable in the time 

 period after 1950 or so, because of the shift of 

 dominance from the northern sardine subpopulation 

 to the southern one. Important differences in such 

 factors as natural survival rates, maturation charac- 

 teristics, and fishing effort exist for these subpopula- 

 tions (Murphy 1966). 



The waters of the California Current region, with 

 their chemical and biological constituents, can be 

 viewed as an ecological system (Sette 1969). The 

 present model focuses on the sardine and anchovy 



Discrete-Time Difference Equations 



The difference model representing the seven inter- 

 acting populations of the sardine ecosystem is pre- 



542 



