SAILA and PARRISH: EXPLOITATION EFFECTS 



P in the graph is what is usually called net pro- 

 duction (an energy rate). Net production P, 

 is equal to gross production, i.e., assimilated 

 food, kR, minus respiration, Q. Summing inputs 

 at the graph vertex P2, the value of P2 is: 



P2 = k2Ri2 — Q2 . (8) 



Respiration, Q, is expressed as the product 

 of biomass and a "respiration coefficient," q: 



Q = qB, 



where B is the biomass (standing energy crop) 

 of a species. 



M is natural mortality, considered propor- 

 tional to biomass; 



Ml = niiBi, M2 = m2B2. (9) 



Constancy of these coefficients is assumed. 



F is fishing mortality which is used if an ex- 

 ploited population is considered. Death due to 

 any other specific cause can be separately con- 

 sidered in a similar manner. 



U is energy in the undigested (unassimilated) 

 portion of food eaten, and k is the "digestion 

 coefficient." The predator assimilates a frac- 

 tion k of the ration R eaten, and the remaining 

 energy, U = (1 — k)R, is lost. 



Upon first inspection of the graph, it may be 

 disconcerting to see vertices representing quan- 

 tities such as biomass (energy) in the graph 

 with vertices representing quantities such as 

 production (energy per unit time). The con- 

 fusion is resolved by realizing that the graph 

 is not a pure flow network. It merely shows 

 some assumed relationships, and at each vertex 

 the same rules apply. For example, at vertex Bi 

 in Figure 2(a) : 



Bi = ( + l)Pi + (—l)Ri2 



+ (1 — m,)5,, (10) 



or. 



Pi — Bi + Bi — miBi = R12. (11) 



Net production — Natural mortality = Re- 

 mainder eaten by predators. 



Thus interpreted, the graph represents the re- 

 lationships correctly. 



An important feature of this formulation is 

 the attempt to approximate the density depen- 



dence of feeding rate. Formulations for species 

 interactions such as the classic equations of 

 Lotka and Volterra express rates of change of 

 the number or biomass of a species as products 

 of coefficients and numbers or biomass of the 

 interacting species. This approach has involved 

 the assumption that feeding rate is independent 

 of the abundance of prey and it is an oversim- 

 plification which results in an inherently un- 

 stable system. The experimental work of Ivlev 

 (1961) provided a density-dependent feeding re- 

 lation: 



R = H {1 



-VP 



), 



(12) 



R 



where: H = the "maximum ration" of the 

 predator, the most it would ever 

 eat (or the maximum rate at 

 which it would feed) no matter 

 how much food were available; 

 the "actual ration" of the pred- 

 ator, the amount actually eaten 

 (or the rate at which it feeds) 

 under an actual condition of 

 food availability; 

 the density or biomass of the 

 prey population; 

 a coefficient. 



and 



P 

 V 



A linear approximation of this relationship, fol- 

 lowing Menshutkin (1969), can be used in the 

 graph model. A parameter 812 is defined as: 



812 = D12 — H12 , (13) 



where: D12 — the amount of the prey biomass 

 accessible to the predator D12 = 

 ^12^1. A constant of propor- 

 tionality to prey biomass is as- 

 sumed. 

 H12 = as defined above. Since H12 is 

 obtained as a fraction of pred- 

 ator biomass {H12 = ^12^2), the 

 assumption is introduced that 

 all predator individuals feed at 

 the same rate. 



The "actual ration" of the predator is then de- 

 fined as: 



R12 =ai2//i2 + 612812 , (14) 



where ai2 and 612 are fractional coefficients. 



387 



