Analysis of some experimental time series by Gause: application of simple mathematical models

For the approximation of some of well-known time series of Paramecia aurelia and Paramecia caudatum (under the separated cultivation of both species) population size changing in time, some well-known models were used. For all considering models values of parameters were estimated with least square method (with global fitting) in two different ways: with and without additional limits for parameter’s values. In the case without additional limits for model’s parameters deviations between theoretical (model) trajectories and experimental time series were tested for Normality (Kolmogorov–Smirnov test, and Shapiro–Wilk test) with zero average, and for existence/absence of serial correlation (Durbin–Watson criteria). The best results were observed for Gompertz’ and Verhulst’ models. Under the assumption that parameter K (maximum value of population size) is greater than all elements of initial sample the best results were observed for Gompertz’ model. In the last case the canonical technique for analysis of set of deviations can be applied in restricted form and needs in further development. In such a situation we cannot test the set of deviations on Normality with zero average (for big samples) because after a certain time moment all experimental points will be at one side of theoretical curve; at this situation we have to have a serial correlation in the sequence of deviations, etc.