Analysis of the stochastic model “prey-migration area-predator-superpredator”

Current research areas of dynamic migration and population models include the analysis of trajectory dynamics and solving parametric optimization problems using computer methods. In this paper we consider the population model “prey-migration area-predator-superpredator”, which is given by a system of four differential equations. The model takes into account trophic interactions, intraspecific and interspecific competition, as well as migration of the prey to the refuge. Using differential evolution parameters are found that ensure the coexistence of populations of prey, predator and superpredator, respectively, in the main habitat and the existence of a population of prey in a refuge. The transition to stochastic variants of the model based on additive noise, multiplicative noise and the method of constructing self-consistent models is performed. To describe the structure of the stochastic model the Fokker-Planck equations are used and a transition to a system of equations in the Langevin form is performed. Numerical solution of stochastic systems of differential equations is implemented by the Euler-Maruyama method. Computer experiments are conducted using a Python software package, and trajectories for deterministic and stochastic cases are constructed. A comparative analysis of deterministic model and corresponding stochastic models is carried out. The results can be used in solving problems of mathematical modeling of biological, ecological, physical, chemical and demographic processes.