Coral reef ecosystems owe their ecological success - and vulnerability to climate change - to the symbiotic metabolism of corals and Symbiodinium spp. The urgency to understand and predict the stability and breakdown of these symbioses (i.e., coral 'bleaching') demands the development and application of theoretical tools. Here, we develop a dynamic bioenergetic model of coral-Symbiodinium symbioses that demonstrates realistic steady-state patterns in coral growth and symbiont abundance across gradients of light, nutrients, and feeding. Furthermore, by including a mechanistic treatment of photo-oxidative stress, the model displays dynamics of bleaching and recovery that can be explained as transitions between alternate stable states. These dynamics reveal that "healthy" and "bleached" states correspond broadly to nitrogen- and carbon-limitation in the system, with transitions between them occurring as integrated responses to multiple environmental factors. Indeed, a suite of complex emergent behaviors reproduced by the model (e.g., bleaching is exacerbated by nutrients and attenuated by feeding) suggests it captures many important attributes of the system; meanwhile, its modular framework and open source R code are designed to facilitate further problem-specific development. We see significant potential for this modeling framework to generate testable hypotheses and predict integrated, mechanistic responses of corals to environmental change, with important implications for understanding the performance and maintenance of symbiotic systems.