  The reduction in coral reef densities, characterized by the proliferation of
macroalgae, has emerged as a global threat. In this paper, we present a
discrete-time coral reef dynamical model that incorporates macroalgae. We
explore all ecologically possible equilibrium points for the proposed model.
The conditions for the local stability of the interior equilibrium point are
analyzed, which represents the coexistence of both coral and macroalgae.
Furthermore, we investigate the model's behavior using the center manifold
theorem and bifurcation theory. Our analysis reveals that the model undergoes
codimension-one bifurcations, specifically period-doubling and Neimark-Sacker
bifurcations. To address the chaos resulting from the emergence of the
Neimark-Sacker bifurcation, we apply the OGY feedback control method and a
hybrid control methodology. Finally, we provide numerical simulations not only
to validate the obtained results but also to demonstrate the complex dynamic
behaviors that arise. These behaviors include reversal period-doubling
bifurcation, period-4, 8, and 24 bubble bifurcations, as well as chaotic
behavior.
