In this paper, we study the local dynamics and bifurcations analysis of a two-dimensional discrete-time host-parasitoid model in the closed first quadrant R_+^2. We study the local dynamics along their topological types about equilibria: O(0,0),A(1,0) and B(l,r(1-l)) of the discrete-time model by utilizing method of Linearization. We also investigate necessary and sufficient parametric condition(s) under which the unique positive equilibrium B(l,r(1-l)) of the discrete-time model is locally asymptotically stable, repeller, saddle and non-hyperbolic. It is proved that about boundary equilibrium A(1,0) discrete-time model undergoes a period-doubling bifurcation. It is also proved that discrete-time model undergoes a Neimark-Sacker bifurcation when parameter varies in a small neighborhood of the unique positive equilibrium point B(l,r(1-l)) and meanwhile stable invariant close curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Some numerical simulations are presented to verify theoretical results.

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