Stochastic positivity-preserving symplectic splitting methods for stochastic Lotka--Volterra predator-prey model

Abstract: In this paper, we present two stochastic positive-preserving symplectic methods for the stochastic Lotka-Volterra predator-prey model driven by a multiplicative noise. To inherit the intrinsic characteristic of the original system, the stochastic Lie--Trotter splitting method and the stochastic Strang splitting method are introduced, which are proved to preserve the positivity of the numerical solution and possess the discrete stochastic symplectic conservation law as well. By deriving the uniform boundedness of the $p$-th moment of the numerical solution, we prove that the strong convergence orders of these two methods are both one in the $L^2(\Omega)$-norm. Finally, we validate the theoretical results through two and four dimensional numerical examples.