How environmental stochasticity can destroy the persistence of macroalgae in a coral reefs ecosystem

In this study, we mainly investigate how environmental stochasticity can destroy the persistence of macroalgae in a coral reefs ecosystem by analyzing the noise-induced tipping behavior. Firstly, detailed mathematical analysis for all feasible system parameters shows that the deterministic system has rich dynamics, including two types of bifurcations and two types of bistabilities. This also reveals that the dynamic behavior of coral reefs ecosystem could be highly sensitive to the system parameters and initial values. For the stochastic system, two kinds of noise-induced tipping behaviors are numerically found: Transition from coral-free state to macroalgae-free state; transition from coexistence state to macroalgae-free state. We then mainly analyze the impacts of noise intensity on the probability and time that coral reef ecosystem tips between different states, evaluate the extinction risk of macroalgae for different initial values, and eventually assign extinction warning levels to these values. Our analysis reveals that as a fragile marine ecosystem, the evolution trend of the coral reefs depends not only on the system parameters and initial values, but also on the intensity of the stochasticity experienced by the system.