Modeling 453 



9 



E 



20 



40 

 Time, days 



60 



FIGURE 10-9. Algae biomass in 

 the plankton of a pond as simu- 

 lated by a deterministic (DET) and 

 a stochastic (SM) model. The sto- 

 chastic model simulation is the 

 mean of nine runs. The standard 

 deviation (SD) is also given. (After 

 Tiwari and Hobble 1976a.) 



standard deviation of algae and Daphnia. Also included in these figures 

 are the results of the corresponding deterministic model in which all the 

 variances were set to zero. Some of the interesting properties of these 

 models are worth mentioning here. In general the results of the 

 deterministic models were different from that of the stochastic means. In 

 fact, in most of the cases the differences were so great that the 

 deterministic results were outside the 95% confidence limit. Although this 

 is not very surprising, especially because of the nonlinear terms in the 

 differential equations, it can be pointed out that one of the assumptions 

 implicit in the rationale behind a deterministic formulation is that the 

 deterministic model is a mean of a stochastic model. Some degree of 

 increased stability was also observed in the stochastic models. The 

 behavior of the deterministic model is very sensitive to the small changes 

 in the parameter values; for some parameters a small change in the third 

 decimal place can produce very different results in the resultant biomass. 

 However, in our stochastic formulation where all the parameters were 

 treated as random variables, this type of sensitivity was not observed. Thus 

 it seems that simultaneous fluctuations in a number of parameters create 

 in the system a buffering mechanism in which too great effects of some 

 parameters are counterbalanced by small effects of others. But perhaps the 

 most useful quantity, from an application point of view, that comes from 

 these models is the standard deviation associated with the average value of 

 the state variables. With the information from a standard deviation, a 

 confidence interval around the mean can be constructed. This allows us to 



20 40 



Time, day? 



FIGURE 10-10. Daphnia bio- 

 mass in the plankton of a pond 

 as simulated by a deterministic 

 and a stochastic model. The sto- 

 chastic model simulation is the 

 mean of nine runs. The stand- 

 ard deviation is also given. 

 (After Tiwari and Hobbie 

 1976a.) 



