Stochastic dynamics of coral reef system with stage-structure for crown-of-thorns starfish

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.