Stability and Neimark-Sacker bifurcation analyses of a discrete-time predator-prey system with Leslie type

In this thesis, the dynamical behaviour of a discrete-time predator-prey model of Leslie type is presented. This model is obtained from continuous-time predator-prey model by using Euler method. The model has two populations which are prey and predator living in the same environment and interacting with each other. In this model the change of populations, modeled by approximation of nonlinear dynamical systems, with respect to time is governed by difference equations. First, the existence of the positive equilibrium point of the discrete system is shown and the conditions for the stability are found. Then, the conditions of existence for Flip bifurcation and Neimark-Sacker bifurcation arising from this positive equilibrium point are determined. More specifically, these bifurcations are driven by using the center manifold theorem and the normal form theory by choosing the integral step size as a bifurcation parameter. Finally, some numerical simulations are presented to support and extend the theoretical results.