Stochastic modification of the predator–prey model based on the normal approximation method

A stochastic modification of the two-dimensional dynamic predator–prey model is constructed based on the application of the normal approximation method. The construction takes into account random disturbances related to nonparametric white noise. The normal approximation method assumes the approximation of unknown distributions by a normal distribution and the transition to a deterministic system of higher dimension compared to the dimension of the original stochastic system. The transition from the classical predator–prey system was made to a stochastic system, and after transformations within the framework of the normal approximation method, a five-dimensional deterministic system of differential equations with respect to the moments of the first and second orders was obtained. The resulting system is analyzed taking into account the qualitative effects of stochastization and studied taking into account the dynamics of phase variables (two mathematical expectations, two variances and covariance). For different sets of parameters the trajectories of solutions for the resulting system of equations with respect to moments are constructed. The results can be used in solving problems of modeling nonlinear systems taking into account random disturbances, as well as problems of constructing nonlinear stochastic filters.