Coral is an animal that plays an important role in the marine ecosystem. Consequently, a number of researchers have studied questions of conservation of the red coral population by introducing a data-based and high-dimensional discrete model. Numerical simulations of this model have given some first insights into the effects of changes in the mortality rate, as well as the effects of overfishing. We analyze their discrete time model for red coral populations to shed light on the long-term dynamics of the population. The model exhibits both fixed points and a Hopf bifurcation, as a function of the basic reproductive number. We demonstrate that after the Hopf bifurcation, neighborhoods of the fixed points converge to closed curves, which in turn approach extinction in certain parameter regimes. Furthermore, the numerical results have been verified through computer assisted proofs as a first step toward rigorous mathematical results.
