Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model

TL;DR

This paper introduces stochastic K-symplectic integrators for non-canonical Hamiltonian systems, specifically applied to the stochastic Lotka--Volterra model, ensuring geometric structure preservation and solution positivity.

Contribution

It develops a theoretical framework for stochastic K-symplectic structures and proposes integrators that preserve these structures and positivity in stochastic non-canonical Hamiltonian systems.

Findings

Proposed integrators preserve geometric structure.

Integrators maintain solution positivity.

Numerical examples verify theoretical properties.

Abstract

We give a theoretical framework of stochastic non-canonical Hamiltonian systems as well as their modified symplectic structure which is named stochastic K-symplectic structure. The framework can be applied to the study of the Lotka--Volterra model perturbed by external noises. In terms of internal properties of the stochastic Lotka--Volterra model, we propose different kinds of stochastic K-symplectic integrators which could inherit the positivity of the solution. The K-symplectic conditions are also obtained to ensure that the proposed schemes admit the same geometric structure as the original system. Besides, the first-order condition of the proposed schemes in $L^1(\Omega)$ sense are given based on the uniform boundedness of both the exact solution and the numerical one. Several numerical examples are illustrated to verify above properties of proposed schemes compared with non-K-symplectic ones.