Deterministic and stochastic analysis of a coral reef ecosystem with grazed macroalgae

Based on the work of Mumby et al. [Thresholds and the resilience of Caribbean coral reefs, Nature 450 (2007) 98–101], this study is devoted to investigate the deterministic and stochastic features of a coral reef ecosystem in which macroalgae and coral compete to occupy algae turfs, and the macroalgae are grazed by parrotfish. By taking the grazing rate as the focused parameter, we completely analyze the global dynamics of the deterministic coral reef ecosystem, including bistable phenomenon, linear equilibria and degenerate attractor. It is found that for different grazing rates, the coral reef system may exhibit rich dynamics which are closely dependent on the initial values. Furthermore, we derive a stochastic coral reef model in the form of continuous-time Markov chain (CTMC), and estimate the extinction probabilities of macroalgae and coral by using the branching process theory. Analytical results reveal that the macroalgae or coral species will go to extinction in a positive probability, even if it can survive in the deterministic environment.