412 J. L. Tiwariet al. 



unstable behavior as the connectance or the strength of interaction in the 

 system exceeds a critical value. Thus both the complexity of the 

 ecosystems and the magnitude of the interactions between its components 

 must be taken into consideration and analyzed, for they will affect the 

 behavior of the system. 



The first step in the modeling process was for the biologists to identify 

 and define the appropriate variables and the interactions between these 

 variables to be included in the models. Experiments were designed to 

 quantify the relevant interactions and necessary data were collected to 

 measure and estimate variables, parameters, and constants. The models 

 evolved through a series of workshop meetings where interactions were 

 identified and their mathematical forms discussed and evaluated. These 

 ideas were incorporated into the model and computer simulations were 

 made to evaluate these concepts and results. The mathematical model of 

 the aquatic system presented here represents our present "best" 

 understanding of the structure and function of these systems. It is 

 appropriate to mention here that the model does not include every existing 

 species and interaction. Only those variables and processes are included 

 which were measured experimentally and whose parameters could be 

 estimated from the existing data. The exceptions to this were processes 

 that we knew were very important yet could not measure directly. In part 

 because of these unknowns, we do not present our models as finished 

 products, but instead believe they are mere steps towards eventual 

 understanding of aquatic ecosystems. They should provide a basis for 

 further experimentation and more refined mathematical analyses of such 

 systems. 



General Formulation and Notational Convention 



Let the state of the system under consideration be defined at any 

 instant of time / by parameters x,, /+1, 2, . . ., n, which we shall denote for 

 convenience by the vector X: 



X = 



X2 



(1) 



Xn 



The rate of change of the system with respect to time will be expressed 

 by the derivation relation 



x = dA:,/d/ / = 1, 2, . . ., « 



