Stochastic Dynamics of Coral and Macroalgae: Analyzing Extinction and Resilience in Coral Reef Ecosystems

Understanding the interactions between coral and macroalgae and the influence of environmental factors is critical for the conservation and restoration of coral reef ecosystems. This study introduces a stochastic model that systematically investigates the combined effects of external coral recruitment, macroalgae grazing pressure, and environmental stochasticity on coral-macroalgae dynamics. The analysis begins with deterministic dynamics, followed by an evaluation of long-term stochastic behavior with and without external coral recruitment. A critical stochastic threshold with external coral recruitment, $\lambda$ , is identified, which characterizes stochastic persistence, extinction, and ergodicity within the system. Simulation results indicate tipping points associated with variations in coral and macroalgae biomass. The analysis reveals that increased external coral recruitment and grazing of macroalgae facilitate macroalgae extinction, effectively reversing blooms. Furthermore, changes in noise intensity (either reduced noise for coral or increased noise for macroalgae) accelerate macroalgae extinction and drive a shift in coral biomass from low to high levels. These dynamics underscore the reversibility of macroalgal blooms and the opposite effects of different noise types on ecosystem behavior. Additionally, coral reef resilience is significantly influenced by initial biomass conditions, with high macroalgae biomass combined with low coral biomass markedly diminishing resilience and complicating recovery, while higher coral biomass enhances the tolerable range for system recovery. The results yield theoretical insights and offer practical strategies for coral reef conservation and restoration.