Abstract
In this paper, first we discuss a local stability analysis of model was introduced by P. J. Mumby et. al. (2007), with $frac{gM^{2}}{M+T}$ as the functional response term. We conclude that the grazing intensity is the important parameter to control the existence or extinction of the coral reef. Next, we consider this model under the influence of the time delay as the bifurcation parameter. We show that for small time delay, the stability type of the equilibria will not change, however for large enough time delay, the interior equilibrium point become unstable in contrast to the ODE case. Also for some critical grazing intensity and the time delay, a Hopf bifurcation occur and a nontrivial periodic orbit will appear. Further we discuss its corresponding stability switching directions.
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