Synthesis and Analysis of Multidimensional Mathematical Models of Population Dynamics

Abstract

The designing of multidimensional models of population dynamics taking into account the relations of competition and mutualism is described. The model examples in three-dimensional and four-dimensional cases are considered, qualitative and numerical investigation of deterministic models is carried out. The deterministic description of each model is given by the system of ordinary nonlinear differential equations. The transition to the corresponding multidimensional nondeterministic models defined by differential inclusions, fuzzy and stochastic differential equations is made, and stability analysis is performed. Synthesis of the stochastic models “competitor-competitor-mutualist” and “competitor-mutualist-competitor-mutualist” is carried out. The structure of multidimensional stochastic models with competition and mutualism is described, Fokker-Planck equations are written, the rules of the transition to stochastic multidimensional differential equations in the Langevin form are formulated. The comparative analysis of deterministic and stochastic models is carried out. The numerical experiment for the studied models has been carried out with the help of the developed software package for the numerical solution of the differential equations systems by Runge-Kutta stochastic methods. Algorithms for generating trajectories of the Wiener process and multipoint distributions and numerical algorithms of Runge-Kutta stochastic method are used. The numerical experiment in a number of cases showed a significant proximity of stochastic and deterministic dynamic models trajectories. The conditions under which the introduction of stochastics influences poorly the stability of the system and it is possible to consider its deterministic approach for the system studying.