The stability and Neimark-Sacker(N-S) bifurcation of a single population harvest model with time delay and piecewise constant variables are discussed. The range of the parameter for the local stability and the existence of N-S bifurcation of this model are derived, by using the theory of characteristic value,it is proved that the model undergoes N-S bifurcation when r=r0=b\[eb-2+((eb-2)2+4)1/2\]/2(eb-1). The direction and stability of N-S bifurcation are derived by using the bifurcation theory and the center manifold theorem; it is proved that the only stable(unstable) closed unchanged curve appears if d<0 (>0). Some examples and numerical simulations are presented to illustrate the correctness and realizability of theoretical results and the complex dynamical behaviors of this model.

delay differential equation; piecewise constant variable; stability;Neimark-Sacker bifurcation