A Holling type Ⅳ predator-prey system with a rational nonlinear harvesting rate and gestation delay of prey species is studied, which is formulated by delayed differential-algebra equations. Its dynamical behaviors are investigated in terms of differential-algebra system theory, bifurcation theory, and center manifold theorem. By choosing the gestation delay as a bifurcation parameter, we first show that Hopf bifurcations can occur as the delay increases through a sequence of threshold values. Second, we derive an explicit algorithm for determining the stability and direction of the Hopf bifurcations. Last, some numerical simulations are performed to illustrate the analytical results, and their biological significances are explained.