PARRISH: MARINE TROPHIC INTERACTIONS BY DYNAMIC SIMULATION 



where f ,i. I, w„,^. are numerical parameters. For 

 prey species modeled as described here, expressing 

 P in terms of prey numbers rather than biomass 

 seems to have system stability advantages. It is 

 through Equation (7) that this species interacts 

 with the next lower species in the trophic chain 

 (web). 



The instantaneous rate of production of 

 reproductive material and growth at the current 

 body weight and prey abundance can be deter- 

 mined by use of the above terms in Equation (1). 



Using ^ = ¥r from Equation (7) in Equation (3) 



^max 



gives the current value of a to be used in Equation 

 (2) to give the current value of Q. When food sup- 

 ply is adequate; i.e., when kC - Q>S^^^ , "fecun- 

 dity" is 



and, from Equation (1), positive growth is 

 dW 



dt 



= G = kC-Q-S, 



(8) 



When food supply is so low that 0<kC - Q<S^ 

 growth is zero and fecundity is 



S = kC-Q. 



(9) 



In more extreme food shortage, when kC - Q<0, 

 fecundity is zero and growth is 



dW 

 dt 



= G = kC-Q. 



(10) 



(Note that in the last case, growth is negative; i.e., 

 dystrophy occurs). Equations (8) and (10) are in 

 differential form, representing rates of change of 

 body weight. Numerical integration of these 

 equations gives "continuous" values of body 

 weight over the entire time span of the simula- 

 tions. Figure 1 shows the relationships between 

 the component equations which describe a single 

 species. 



(B) Population Dynamics 



Numerical changes in the population of any 

 species are the net result of gains through 

 reproduction (recruitment) and losses through the 

 various sources of mortality. Therefore, the 

 dynamics of any species population can be sum- 

 marized in the expression 



dN 

 dt 



— R + Z/naT + -^PRED 



-t- D 



STARV • 



(11) 



The rate of change of population is the algebraic 

 sum of four terms: reproductive recruitment, R, 

 natural mortality, J^nat > mortality due to preda- 

 tion, -Dpj^gp, and starvation mortality, ^g^ARv (^^^ 

 sign of the reproductive term is positive; all the 

 other terms have negative signs). Equation (11) is 

 used in essentially this form for the representative 

 individual model. For the age class model, the last 

 three terms appear for all age classes. Instead of 

 including the first term, the appropriate number 

 of recruits is simply introduced as a pulse into the 

 youngest age class at the appropriate times in the 

 simulation. 



The recruitment rate, R, is expressed as a func- 

 tion of the rate of egg production, E, by the 

 Beverton and Holt (1957:49) reproductive function 



R = 



a + b^ 

 E 



(12) 



where a and h are numerical parameters. A simple 

 relationship such as Equation (12) is appropriate 

 for the present model where the response of a 

 system of essentially adult populations to purely 

 trophic variables is of interest. The egg production 

 rate, E, is the cumulative spawn of the entire ma- 

 ture population, A^^^; i.e., 



E = N^S. 



(13) 



For the age class model, this involves summing 

 over all mature age classes and over the entire 

 year. All real species have some reproductive time 

 lag or "generation time." In all except the simplest 

 animals, this lag is significant and can have im- 

 portant influence on the dynamics of the popula- 

 tion. Such lags of any desired length are in- 

 troduced in the simulations by properly coding the 

 programs so that the E produced in 1 yr is stored 

 and used in Equation (12) to compute the R for the 

 appropriate later year. 



Except for fishing mortality, natural mortality 

 is the only kind expressed in most fishery models. 

 The position taken is that all mortality not due to 

 fishing is "natural" and may be measurable in an 

 unexploited stock or by eliminating the fishing 

 mortality from statistics on an exploited stock by 

 some analytical technique. Thus defined, natural 

 mortality is almost invariably represented in 



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