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.
翻译:系统转换的随机后勤模型是生态系统中的一个重要模式。虽然这一模型的分析解决方案是积极的,但目前的数值方法无法在近似值中保留这种边界。 因此,提出适当的数字方法来解决这一模型,以维护模型解决方案的正反和动态行为非常重要。 在本文中,我们为这一模型展示了一种保护短短的尤勒-马鲁亚山模型的假设性,这种模型具有明确和易于执行的优势。在没有额外的限制条件的情况下,研究数字算法的高度趋同,并获得了1/2顺序趋同率。在这种模型的特殊情况下,不改变第一顺序的强烈趋同率。此外,还实现了长期动态特性的近似,包括随机性持久性、灭绝性和分布的稳定性。提供了一些模拟和实例,以证实理论结果并展示方法的有效性。