The outbreak of crown-of-thorns starfish (CoTS) is one of the most critical biological disturbances in coral reef systems and contributes greatly to the degradation of coral. Although many studies have focused on coral-CoTS interactions, there is no effective method for suppressing CoTS outbreaks in the long term, in part because of incomplete knowledge of their life cycle. In this paper, a stochastic system is formulated to characterize the interactions among coral, immature CoTS, and mature CoTS, where environmental noise is introduced into the growth of coral. The existence and uniqueness of a global positive solution with any given initial value are investigated, sufficient conditions are established for the extinction of CoTS, and the existence and uniqueness of a stable stationary distribution are explored by applying the Markov semigroup theory. In particular, by solving the Fokker–Planck equation, the approximate expression of the probability density function of the distribution around its quasi positive equilibrium is expounded under certain parametric restrictions. By employing Milstein’s higher-order method, numerical simulations are presented to validate the theoretical results. This study reveals that coral are more vulnerable and are more likely to go extinct in the presence of strong noise, while weak noise is more conducive to the existence of a stable stationary distribution, which implies the long-term persistence of coral and CoTS.

中文翻译：

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棘冠海星阶段结构珊瑚礁系统的随机动力学


棘冠海星（CoTS）的爆发是珊瑚礁系统中最严重的生物干扰之一，极大地促进了珊瑚的退化。尽管许多研究都集中在珊瑚与 CoTS 的相互作用上，但从长远来看，没有有效的方法来抑制 CoTS 的爆发，部分原因是对其生命周期的了解不完整。在本文中，制定了一个随机系统来表征珊瑚、未成熟 CoTS 和成熟 CoTS 之间的相互作用，其中环境噪声被引入珊瑚的生长中。研究了给定初值的全局正解的存在性和唯一性，建立了CoTS灭绝的充分条件，并应用马尔可夫半群理论探讨了稳定平稳分布的存在性和唯一性。特别是，通过求解福克-普朗克方程，阐述了在一定参数限制下，围绕其准正平衡分布的概率密度函数的近似表达式。采用Milstein的高阶方法，进行数值模拟来验证理论结果。这项研究表明，在强噪声存在下，珊瑚更脆弱，更有可能灭绝，而弱噪声更有利于稳定静止分布的存在，这意味着珊瑚和CoTS的长期存在。