FISHERY BULLETIN: VOL, 73, NO. 4 



the simulations is the extremely limited range of 

 diet of the model species; their factual namesakes 

 have rather catholic tastes. 



For these hypothetical species, a set of values 

 for an arbitrary standard equilibrium condition 

 was established as follows. Populations for all 

 species were arbitrarily set at values that seemed 

 reasonable relative to each other and in respect to 

 the various body weights and reproductive rates. 

 Using the various parameters selected, for each 

 species, the r value corresponding to the standard 

 equilibrium state was then computed. From this r 

 and the equilibrium population of the prey, the 

 predation parameters of Equation (7) were com- 

 puted. Using this procedure for each fish species, 

 working up the trophic chain, a complete set of 

 equilibrium values for all species became avail- 

 able. A compatible trophic web was thus created 

 arbitrarily, having at least static stability; i.e., 

 (IN /(It and dW/dt were zero for all species. Table 2 

 provides the values of parameters and of basic 

 variables at standard equilibrium state for the 

 model species used in these simulations. Where a 

 consistent set of laboratory and field data on 

 species in a real trophic web were available to be 

 used in the model for predictive purposes, some of 

 these procedures would be unnecessary. 



Like all simulations, those run with the model 

 require that initial conditions be specified. 

 Typically in these runs, the initial conditions were 

 those of the standard equilibrium state with the 

 exception of some single variable value which was 

 displaced so as to perturb the system. For example, 

 a simulation run started with all variables at 

 equilibrium except B^ might be analogous to the 

 natural occurrence of sudden catastrophic mor- 

 tality in a prey species. Initial conditions are dis- 

 cussed further under RESULTS. In each case, the 

 simulation was allowed to run for an arbitrary 

 length of time, or until automatically terminated 

 when some variable reached a prescribed limiting 

 value. Usually runs were continued until a stable 

 state (the original standard equilibrium or other- 

 wise) was approached, or until a distinct mono- 

 tonic trend with a predictable outcome was de- 

 tected. 



All the simulations were programmed using the 

 IBM^ System/360 CSMP (Continuous System 

 Modeling Program) (International Business 



''Reference to trade names does not imply endorsement by the 

 National Marine Fisheries Service, NOAA. 



Machines Corporation 1969, 1971) and run on an 

 IBM 360/50 Data Processing System. 



RESULTS 



A limitation of the approach taken here, as with 

 any simulation model for numerical solution, is 

 that mathematically exact and general solutions 

 are not obtainable. A full solution of the system 

 represents a very complex multidimensional re- 

 sponse surface. In the very simplest case of one 

 modeled fish species and a food base species, there 

 are three basic dependent variables whose in- 

 tegrated values appear in the solution; viz., N^-^^^ , 

 H^Fish , and B^.lm. "representative individual" 

 model with n fish species and the food base, there 

 are 2n + I basic dependent variables, and in a 

 similar model with x explicit age classes per 

 species, there are 2nx + 1 basic dependent varia- 

 bles. 



A system with the complexity and nonlineari- 

 ties of this type of model is capable of behaving 

 quite differently in different regions of the state 

 space. Since it is impossible to explore the entire 

 response surface thoroughly, measures must be 

 taken to limit simulation effort to regions of 

 interest. Eventually a detailed and systematic 

 exploration of regions of known interest using es- 

 tablished optimization techniques (e.g., Box et al. 

 1953; Box 1954; Box and Hunter 1957) may be 

 useful with the model. 



For the present, the scope of simulation effort 

 has been limited by selecting parameter values 

 that seem reasonable and compatible for each of a 

 small group of rather common fish species and by 

 building out from a system already investigated to 

 a larger system of which the original is a subset. In 

 a number of cases where moderate changes to 

 values of parameters or even to the form of com- 

 ponent functions have been made, system 

 dynamics have been somewhat altered or the sys- 

 tem has even moved toward a new stable state. 

 Usually, however, in a system with any regula- 

 tory capacity (stability) at all, the change has not 

 been drastic. Rather large perturbations in initial 

 values of the basic dependent trophic variables of 

 such a system have not usually displaced the sys- 

 tem to a distant stable region or resulted in 

 breaking the trophic web (eliminating one or more 

 species). This behavior of most of the systems 

 simulated gives evidence that there is at least one 

 region of some useful size in the total state 

 space-i.e., the region in which the arbitrary 



704 



