Modeling 413 



which may be written in vector form as 



X = d\/dt = 



dxi/dt 

 dxi/dt 



(2) 



_ dx„/dt _ 



These rates will be assumed to be functions of the instantaneous 

 values of the states, the time t, known inputs «, (/), / = 1, 2, . . ., n, and 

 random inputs w, (/) / = 1, 2, . . ., J, such that we have a system of 

 differential equations 



x,=/, [xi(/),X2(/), . . .,x„it), w,(0, "2(0, • • ■,Un{t), 

 WiCO, vvzCO, ••-, wj(0] 



with initial conditions that may also be random: 



x,{t)=x, 



/=1,2, . . .,n. 



(3) 



(4) 



These can be expressed conveniently in the form of a vector 

 differential equation and initial condition vector 



X = /[X(0,U(0,W(O,/], X(ro) = X, 



(5) 



The behavior of the system can be represented by a set of trajectories 

 in ^-dimensional space. The state of the system at any instant of time is 

 determined by the values of the variables, jci,X2, . . .,x„. The variables will 

 usually be the biomasses of the species and the concentrations of dissolved 

 organic carbon and nutrients constituting the system. The state of the 

 system can then be represented as a point in /j-dimensional phase space. 

 To each point in this space we can attach a vector indicating the movement 

 of the system. These vectors can be joined to form trajectories which show 

 the long-term dynamic behavior of the system. 



For convenience, the mathematical model of the total aquatic system 

 has been divided into two submodels, benthic (Tiwari et al. 1978) and 

 planktonic. The state variables of the models are types of organisms 

 (algae, bacteria, Daphnia, chironomids), nutrients, detritus and dissolved 

 organic carbon (see Tables 10-1 to 10-8). These variables are the objects of 

 the system and the processes associated with each of the variables are the 

 attributes. The dynamic behavior of the system is characterized and 

 constrained by the set of interactions between these objects and attributes. 



