Coral reef systems can undergo rapid transitions from coral-dominated to macroalgae-dominated states following disturbances, and models indicate that these may sometimes represent shifts between alternative stable states. While several mechanisms may lead to alternate stable states on coral reefs, only a few have been investigated theoretically. We explore a model that illustrates that reduced vulnerability of macroalgae to herbivory as macroalgae grow and mature could be an important mechanism: when macroalgae are palatable to herbivores as juveniles, but resistant as adults, coral-dominated and algae-dominated states are bistable across a wide range of parameter space. We compare two approaches to global sensitivity analysis to rank the relative importance of the model parameters in determining the presence and magnitude of alternative stable states, and find that the two most influential parameters are the death rate of coral and the rate of maturation of algae out of the vulnerable stage. The Random Forest approach for global sensitivity analysis, recently adopted by ecologists, provides a more efficient method for ranking the relative importance of parameters than a variance-based approach that has been used frequently by computer scientists and engineers. Our results suggest that managing reefs to reduce chronic stressors that cause coral mortality and/or enhance the growth rates of algae can help prevent reefs from becoming locked in a macroalgae-dominated state.