BIFURCATION ANALYSIS OF A DDE MODEL OF THE CORAL REEF

‎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.