Double merging of phase space for differential equations with small stochastic supplements under Levi and Poisson approximation conditions

Abstract

The paper is devoted to the study of limit theorems of evolving evolutionary systems of "particles" in random environment. Here the term "particle" is used broadly to include moleculas in the infected individuals considered in epidemie models, species in logistie growth models, age classes of population in demographics models, etc.
The evolutionary system is complicated by the influence of impulse perturbation and non-trivial structure of the random environment. Namely, the the switching Markov process has a split phase space of states.
We propose a new approach in construction of the approximation scheme for the impulse perturbation that allows not only to see the averaged and diffusion component of the limit process, but also to preserve Poisson jumps that models catastrophic events like mass extinction, earthquakes, etc.
We discuss limit behavior of the generators of the evolutionary systems that allows not only to claim convergence of corresponding distributions, but to use the results obtained for solving the problems of stability and dissipativity of the limit processes.