Modeling 411 



some range, the values attained by a state variable are the state space of 

 that variable. The state of the total system at time t can be defined as the 

 values of the state variables. This state can be represented as a vector in n- 

 dimensional phase space which will indicate the movement of the system. 

 The time sequence of these vectors will form trajectories which show the 

 long term behavior of the system. 



The state of the system at any time t can be determined by solving the 

 equations describing the interactions. Various types of equations can be 

 used to describe these interactions, depending upon the nature of the 

 problem and the objectives of the study. Thus, differential equations may 

 be used for a continuous system and difference equations for a discrete 

 system. For a certain class of problems, differential-difference equations 

 might be more appropriate. To include the space effects in the model (for 

 "distributed" systems), the system can be formulated by partial 

 differential equations. In certain situations integral equations might be 

 more appropriate and useful. 



Once the system is defined and described by a set of mathematical 

 equations we can proceed to analyze its dynamic behavior. The equations 

 describing the biological systems are generally of the non-linear type and 

 are analytically intractable; often they can only be analyzed by computer 

 simulation techniques. This is essentially a numerical approximation of 

 the results for a particular set of parameter values and initial conditions. It 

 is not a general solution to the differential equations describing the 

 dynamics of the system. 



Once the model is successfully duplicating the observed behavior of 

 the system under consideration, the effects of various perturbations can be 

 investigated. We can ask the questions: what is the effect of changing an 

 interaction or a set of interactions between the state variables, of changing 

 parameter values, of adding or deleting some state variables and 

 interactions, etc. 



The basic questions and problems which emerge in the analysis of all 

 systems (Zadeh and Desoer 1963) are essentially the specification of the 

 objects and their attributes to be considered in the study. Furthermore, the 

 interactions between the attributes of each object and the interactions 

 between the attributes of different objects must be characterized in 

 appropriate mathematical forms. The relationships between system inputs 

 and outputs, and also a system of relationships between the attributes of 

 the system as a whole, should be defined. 



The complexity of the internal structure of the system is dependent 

 upon the complexity of the patterns of interactions between the variables 

 of the system. The "connectance" — the probability that any pair of 

 variables will interact — of a large and complex ecosystem may be an 

 important factor affecting the stability of the system (Gardener and Ashby 

 1970, Somorjai and Goswami 1972, May 1971, 1972, 1973, Siljak 1974, 

 Maynard Smith 1974). These analytical and computer simulation results 

 show that in large systems there is a very sharp transition from stable to 



