Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model
This work provides a new class of geometric integrators for stochastic Hamiltonian systems, relevant for researchers in stochastic dynamics and mathematical biology.
The paper develops a theoretical framework for stochastic non-canonical Hamiltonian systems and proposes stochastic K-symplectic integrators that preserve the geometric structure and positivity of solutions, applied to the Lotka-Volterra model with external noise. Numerical examples demonstrate the superiority of these integrators over non-K-symplectic ones.
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 $L^1(Ω)$ 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.