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<p>Stochastic predator-prey systems incorporating the foraging arena scheme are characterized by highly nonlinear, non-globally Lipschitz drift coefficients and strict requirements for non-negative solutions. These features pose significant computational challenges, since standard numerical techniques, such as the classical Euler-Maruyama method, may fail to preserve positivity and can diverge due to the super-linear growth of the coefficients. In this work, a novel explicit time-stepping method, the Logarithmic Truncated Euler-Maruyama (Log-TEM) scheme, is constructed to strongly approximate this complex system. By synergistically integrating a Lamperti-type logarithmic transformation with an explicit truncation strategy, the proposed framework guarantees the unconditional positivity of the numerical solution, thereby ensuring biological plausibility. We provide a rigorous theoretical analysis to establish the exponential integrability of the numerical solution and prove a strong convergence rate. Furthermore, beyond finite-time convergence, we investigate the preservation of long-time dynamical properties. It is demonstrated that the Log-TEM scheme faithfully reproduces the extinction dynamics and theoretical extinction criteria of the exact solution. Finally, numerical simulations are presented to validate these theoretical findings and illustrate the effectiveness of the proposed scheme.</p>

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An explicit logarithmic truncated numerical scheme for stochastic predator-prey systems: dynamics preservation and convergence

  • 期刊论文

  • 发表日期：

    2026年6月16日

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  • Han Ma,
  • Yan Ma,
  • Wei You

摘要

Stochastic predator-prey systems incorporating the foraging arena scheme are characterized by highly nonlinear, non-globally Lipschitz drift coefficients and strict requirements for non-negative solutions. These features pose significant computational challenges, since standard numerical techniques, such as the classical Euler-Maruyama method, may fail to preserve positivity and can diverge due to the super-linear growth of the coefficients. In this work, a novel explicit time-stepping method, the Logarithmic Truncated Euler-Maruyama (Log-TEM) scheme, is constructed to strongly approximate this complex system. By synergistically integrating a Lamperti-type logarithmic transformation with an explicit truncation strategy, the proposed framework guarantees the unconditional positivity of the numerical solution, thereby ensuring biological plausibility. We provide a rigorous theoretical analysis to establish the exponential integrability of the numerical solution and prove a strong convergence rate. Furthermore, beyond finite-time convergence, we investigate the preservation of long-time dynamical properties. It is demonstrated that the Log-TEM scheme faithfully reproduces the extinction dynamics and theoretical extinction criteria of the exact solution. Finally, numerical simulations are presented to validate these theoretical findings and illustrate the effectiveness of the proposed scheme.