: This thesis is based on the following paper I. Kusbeyzi , O. O. Aybar and A. Hacinliyan, “Stability and bifurcation in two species Predator-prey models”, Nonlinear Analysis: Real World Applications 12(2011) 377-387.And H. Zhu , S. A. Campbell and G. S. K. Wolkowicz, “Bifurcation and analysis of a predator-prey system with nonmonotonic functional response”, SIAM J. APPL. MATH. Vol. 63, No. 2, pp. 636-682. The predator–prey problem attempts to model the relationship in the populations of different species that share the same environment where some of the species (predators) prey on the others. The prey is assumed to exhibit linear growth given by a positive parameter. Predator species consume preys with a nonlinear interaction with another set of parameters that determine the rate of competition between predators. The natural death rate of the predator is assumed to be linear and given by a negative parameter. One of the earliest implementations, the Lotka–Volterra model serves as a starting point of more advanced models in the analysis of population dynamics. Because of its unrealistic stability characteristics, stability analysis of the model and its generalizations have recently gained much attention. To understand the behavior of a nonlinear system one can analyze the existence and stability of equilibrium points. As parameters are varied changes in the number and stability of equilibrium points lead to bifurcation. The well-known generalizations of the Lotka–Volterra model include the addition of polynomial interactions, non-monotonic response functions, time delayed and diffusion effected, time delayed non-monotonic interactions. Nutku has proposed a generalization where an additional cubic rather than a quadratic interaction is involved. In the first part of this thesis, bifurcation properties of quadratic and cubic local and monotonic interactions are studied. Consider the following prey-predator system in four case. we show that for a b and c d the hopf bifurcations exists.