PARRISH: MARINE TROPHIC INTERACTIONS BY DYNAMIC SIMULATION 



fishing, and would include death due to disease, 

 senility, accident, environmental stress, etc. Field 

 data for this limited class of mortality are rather 

 scarce. For the hypothetical species in these 

 simulations, approximate conventional M values 

 were taken from Beverton and Holt's (1959) ta- 

 bles, and for species under predation, these were 

 at times modified. In the age class models, 

 different values of M are used for different age 

 classes. Beverton and Holt (1959) discussed the 

 variation of mortality with age more fully. 



The expression for mortality due to predation, 

 -^PRED' comes directly from the ration of the 

 predator species, Cp , modeled as described in the 

 preceding discussion of energy balance. Thus, the 

 rate of change of prey population due to predation, 



-^PRED' IS 



D 



PRED 



^ _^pCpNp 



w 



(15) 



prey 



where Np is the number of predators, each with 

 ration, Cp, W^^^yis the weight of a prey individual, 

 and the summation is over all predator species 

 which consume the particular prey. Equations (15) 

 and (7) provide the coupling between each model 

 species and the other species with which it 

 interacts in the trophic web. 



Despite the scarcity of knowledge on starvation 

 in fish, it would seem that a complete model for a 

 system controlled by trophic variables should 

 include some reasonable attempt at a formulation 

 of this source of mortality. An expression was 

 developed that can approximate the general form 

 of the survival versus time curves from Ivlev's 

 (1961b:266) starvation experiments with fish. This 

 expression states that under pressure of starva- 

 tion alone, the surviving number, A'^, of an initial 

 population, Nq, after time, t, will be 



1 



A^ = NQ—{m + n - me-''). 



(16) 



The m and n are numerical parameters, and the 

 parameter, .s, comes from the boundary condition 

 at 100% mortality, after the critical time, t^, , to 

 extinction has been found from the integrated 

 form of the energy balance equation under star- 

 vation conditions (see Appendix). The form of the 

 function of Equation (16) is plotted in Figure 2. 

 Use of Equation (1) in computing t,. and s provides 

 an appropriate curve for any ration. The model 

 uses the differential form of Equation (16), 



Q2 0.4 



TIME STARVING, YRS. 



0.6 



Figure 2. -Survival curve of a population undergoing only star- 

 vation mortality at zero ration. 



m 



^STARV = -^oTT^e'', 



(17) 



as the fourth term in Equation (11). 



When exploitation by man is included in the 

 system, the only modification is the addition of 

 another term to Equation (11). In accordance with 

 conventional fishery theory and the concept of 

 chance encounter between fish and fishing gear, 

 this term is exactly like Equation (14); i.e., fishing 

 mortality, />fish> is 



D 



FISH 



= -FA^. 



(18) 



The numerical parameter, F, is an expression of 

 the intensity of fishing effort and the 

 vulnerability of the prey to the fishing gear. 



Equations (12), (14), (15), (17)-and (18) where 

 appropriate-provide all the terms for determin- 

 ing rate of change of population from Equation 

 (11). (Figure 1 summarizes their relationships.) 

 Numerical integration of Equation (ll)-less the 

 first term for the age class model-gives "con- 

 tinuous" values of population of the species over 

 the entire time span of the simulation. Biomass of 

 an entire species population at any instant is the 

 product of instantaneous values of W and N 

 (summation of a group of such products in age 

 class models). Production over any desired period 

 is obtained by integrating the incremental growth 

 rates, G, and reproductive products, S (if desired), 

 over that period. 



701 



