有限和无限范围内状态转换的逻辑斯蒂模型的显式数值逼近
Explicit numerical approximation for logistic models with regime switching in finite and infinite horizons
摘要：
具有状态转换的随机Logistic模型是生态系统中的一个重要模型。虽然该模型的分析解是正的，但现有的数值方法在近似中无法保持这种边界。因此，提出合适的数值方法来求解该模型，保持模型解的正性和动力学行为是非常重要的。在本文中，我们提出了一个保持正性截断Euler-Maruyama格式，它具有显式和易于实现的优点。在没有附加约束条件的情况下，研究了数值算法的强收敛性，得到了1/2阶收敛速度。在不切换的特殊情况下，得到了该模型的一阶强收敛速度。进一步，实现了系统长时间动力学性质的近似，包括系统的随机持久性、连续性和分布稳定性。仿真和算例验证了理论分析的正确性和方法的有效性。
英文摘要：
The stochastic logistic model with regime switching is an important model in the ecosystem. While analytic solution to this model is positive, current numerical methods are unable to preserve such boundaries in the approximation. So, proposing appropriate numerical method for solving this model which preserves positivity and dynamical behaviors of the model's solution is very important. In this paper, we present a positivity preserving truncated Euler-Maruyama scheme for this model, which taking advantages of being explicit and easily implementable. Without additional restriction conditions, strong convergence of the numerical algorithm is studied, and 1/2 order convergence rate is obtained. In the particular case of this model without switching the first order strong convergence rate is obtained. Furthermore, the approximation of long-time dynamical properties is realized, including the stochastic permanence, extinctive and stability in distribution. Some simulations and examples are provided to confirm the theoretical results and demonstrate the validity of the approach.
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