Stochastic generalized Kolmogorov systems with small diffusion: II. Explicit approximations for periodic solutions in distribution

TL;DR

This paper introduces explicit approximation methods for periodic solutions in distribution of stochastic Kolmogorov systems with small diffusion, enhancing computational efficiency and relaxing previous constraints.

Contribution

It presents two new approximation methods, PNOA and PLNA, that simplify calculations and relax conditions for positive definiteness in stochastic Kolmogorov systems.

Findings

Methods accurately approximate periodic solutions in distribution.

Relaxed conditions for positive definiteness of covariance matrices.

Numerical experiments validate theoretical results.

Abstract

This paper is Part II of a two-part series on coexistence states study in stochastic generalized Kolmogorov systems under small diffusion. Part I provided a complete characterization for approximating invariant probability measures and density functions, while here, we focus on explicit approximations for periodic solutions in distribution. Two easily implementable methods are introduced: periodic normal approximation (PNOA) and periodic log-normal approximation (PLNA). These methods offer unified algorithms to calculate the mean and covariance matrix, and verify positive definiteness, without additional constraints like non-degenerate diffusion. Furthermore, we explore essential properties of the covariance matrix, particularly its connection under periodic and non-periodic drift coefficients. Our new approximation methods significantly relax the minimal criteria for positive definiteness of the solution of the discrete-type Lyapunov equation. Some numerical experiments are provided to support our theoretical results.