Ecological Effects of Predator Harvesting and Environmental Noises on Oceanic Coral Reefs

Coral reefs provide refuge for prey and are important for the preservation of an oceanic ecosystem. However, they have been experiencing severe destruction by environmental changes and human activities. In this paper, we propose and analyze a tri-trophic food chain model consisting of coral, Crown-of-thorns starfish (CoTS), and triton in deterministic and stochastic environments. We investigate the effects of harvesting in the deterministic system and environmental noises in the stochastic system, respectively. The existence of possible steady states along with their stability is rigorously discussed. From the economic perspective, we examine the existence of the bionomic equilibrium and establish the optimal harvesting policy. Subsequently, the deterministic system is extended to a stochastic system through nonlinear perturbation. The stochastic system admits a unique positive global solution initiating from the interior of the positive quadrant. The long-time behaviors of the stochastic system are explored. Numerical simulations are provided to validate and complement our theoretical results. We show that over-harvesting of triton is not beneficial to coral reefs and modest harvesting of CoTS may promote sustainable growth in coral reefs. In addition, the presence of strong noises can lead to population extinction.