ATKINSON: FISH POPULATION DYNAMICS 



c 



SARDINE 

 LARVAE 



(© ANCHOVY 

 LARVAE 



(3) SARDINE 

 ADULT 



(D ANCHOVY 

 ADULT 



(6) COMPETITOR 

 GROUP 



© PREDATOR 

 GROUP 



(ADDITIONAL 

 FOOD SUPPLY) 



) 



Figure 4.— Schematic showing interactions between sardine ecosystem groups as modeled by Riffenburg (1969). Competitive 

 relationships are indicated by the connecting lines with dual arrowheads, while predator-prey relationships are defined by 

 arrows pointing to the predator. 



sented in Table 1. These equations reflect the 

 general form of Equation (9) for a single spatial com- 

 partment. Parameters are defined for all processes 

 other than transport, including competition, pred- 

 ator-prey, survival/growth, births, and fecundity. 

 These parameters are assumed to be independent 

 of the year during the 1932-52 simulation period, 

 except for 1) the sardine fishing rate, 6 3 (t), and 2) 

 a sardine larvae survival factor, E^t). The latter 

 are related to the parameters presented earlier by 



6 3 (t) = 1 - S f3 (t) 



E,(t) = S.iD/S, 



where S f3 is defined in Equation (5) and S : is the 

 average (reference) sardine larvae survival rate dur- 

 ing 1932-52. The time-varying fishing rate and lar- 

 vae survival factor represent the "drivers" perturb- 

 ing the ecosystem during the sardine collapse period. 

 Time-varying representations may also be ap- 



Table 1— Difference equations describing biomass dynamics of the sardine ecosystem 

 populations. Note that age sub-groups are indexed as separate populations to simplify 

 the nomenclature. Also, all populations in exponentials are assumed to be at time f. 



Population 1 - 

 P,(t + 1) 



Population 2 - 

 P 2 (t + 1) 



Population 3  

 P 3 (t +1) 



Population 4 



P*(t + 1] 

 Population 5 



P 5 (t +1) 



Population 6 



P 6 {t + 1) 



Population 7 - 



P 7 (t + 1) 



sardine larvae 



= f 3 S 3 .o[1 - <M0]exp(-a 33 P 3 - a 35 P 5 - 03 6 P 6 )exp(-/3 37 P 7 )P 3 (f) 



sardine yearling 



= E : (t)S, exp(-a u P, - a 14 P 4 )exp(-/3 l3 P 3 - /5 15 P 5 - p, 6 P 6 )P,(t) 



sardine adult 



= S 2 exp(-a 22 P 2 - a 25 P 5 )exp(-/3 26 P 6 )P 2 (f) 



+ S 30 [1 - d 3 (f)]exp(-a 33 P 3 - a 35 P 5 - a 36 P 6 )exp(-/3 37 P 7 )P 3 (0 

 anchovy larvae 



= f 5 S 5 exp(-a 53 P 3 - a 55 P 5 - a 5 6 P 6)exp(-/3 57 P 7 )P 5 (f) 

 anchovy adult 

 = S 4 exp(-a 41 P, - a 44 P 4 )exp(-/? 43 P 3 - /3 45 P 5 - /? 46 P 6 )P 4 (f) 



+ S 5 exp(-o 53 P 3 - a 55 P 5 - a 56 P 6 )exp(-/3 57 P 7 )P 5 (f) 

 competitor group 



= S 6 exp(-a 63 P 3 - a 65 P 5 - a 66 P 6 )exp(-/? 67 P 7 )P 6 (0 

 predator group 

 = S 7 exp(-a 77 P 7 j[1 - ff 7 exp(-y 73 P 3 - y 75 P 5 - y 76 P 6 )]P 7 (f) 



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