452 J. L. Tiwari et al. 



simulation procedure involved in the analysis of such models can help us 

 circumvent the process of tuning and calibration. 



Since all ecosystem models are data-based and since the fluctuations 

 of various rate processes are based on experimental observations, a more 

 realistic and appropriate approach to the analysis of these specific 

 ecological systems would be the one in which naturally occurring and 

 experimentally observed fluctuations are taken into consideration. This 

 will enable us to utilize the maximum amount of information from the 

 experimental data and to assign a range of values to the parameters in 

 terms of probability distributions. With Monte Carlo simulation 

 techniques a probabilistic description of these quantities can easily be 

 incorporated into the model and the results of the simulation will give the 

 mean and variance of each of the variables of interest. This would be of 

 practical value in any natural resource management scheme, for allowance 

 can be made for the expected fluctuation from the average value. A 

 probabilistic description of the system is also more realistic and desirable 

 because of our ignorance of the exact values and because of uncertainties 

 associated with these complex natural systems. 



To formalize these concepts and to facilitate our discussion, we begin 

 from our general equation (5) describing the dynamics of the system under 

 consideration. 



X=/[X(/),U(0,W(0,/] , X(ro) = Xo (5) 



The introduction of random elements in this equation makes X a vector of 

 stochastic differential equations and the solution, X, is now a vector of 

 stochastic processes. There are three levels — initial conditions, input 

 variables, and parameters — at which stochastic elements can be intro- 

 duced. Results from a number of independent Monte Carlo reaUzations 

 obtained through computer simulation procedures can then be used to 

 estimate the mean and associated variance of the system variables or 

 rate processes. 



A simplified version of the plankton subsystem was analyzed by 

 Tiwari and Hobbie (1976a) using random differential equation models. 

 Initial values of the state variables, input variables (light and 

 temperature), and parameters were defined as random variables having 

 univariate Gaussian distributions. The mean and variances for these 

 variables were estimated from the data. For each simulation run, 

 numerical values of these variables were sampled from the prespecified 

 distribution and this process was repeated nine times. From these 

 realizations, mean and standard deviations were estimated for each of the 

 state variables (for details see Tiwari and Hobbie 1976a). 



The results of a simulation in which randomness was incorporated at 

 all three levels are summarized in Figures 10-9 and 10-10. These two 

 graphs show the temporal behavior of average biomasses together with the 



