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.