Influence of Stochastization on One-Step Models

It is assumed that the introduction of probability in mathematical model makes it more adequate. There are practically no methods of the agreed (depending on structure of the system) introduction of probability in deterministic models. Authors have improved the method of constructing stochastic models for the class of one-step processes and illustrated it by models of population dynamics. Population dynamics was chosen for study because its deterministic models are sufficiently well explored that allows to compare the obtained results with the results already known. We have examined the impact of the introduction of stochastics in the deterministic model, on the example of population dynamics system of type “predator–prey”. Previously obtained stochastic differential equations are studied by the methods of the qualitative theory of differential equations. Stationary state and first integral of the system are obtained. To demonstrate the results the numerical simulations on the basis of Runge–Kutta method for stochastic differential equations are performed. The first integral of deterministic system (phase volume) in the stochastic case does not remain unchanged, but increases, which ultimately leads to the death of one or both populations. One of the disadvantages of the classical system of type “predator–prey” is preservation of the amplitude of populations oscillations. In the stochastic model the process terminates with the death of one or both populations, which from the authors’ point of view makes the model more adequate.

stochastic differential equations, “predator–prey” model, master equation, Fokker–Planck equation