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Published in:
06-09-2021
Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model
Published in: BIT Numerical Mathematics | Issue 2/2022
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Abstract
In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model driven by multiplicative noises, which plays an important role in ecosystem. The model is shown to possess both a unique positive solution and a stochastic symplectic geometric structure, and hence can be interpreted as a stochastic Hamiltonian system. To inherit the intrinsic biological characteristic of the original system, a class of stochastic Runge–Kutta methods is presented, which is proved to preserve positivity of the numerical solution and possess the discrete stochastic symplectic geometric structure as well. Uniform boundedness of both the exact solution and the numerical one are obtained, which are crucial to derive the conditions for convergence order one in the \(\mathbb {L}^1(\varOmega )\)-norm. Numerical examples illustrate the stability and structure-preserving property of the proposed methods over long time.