Neimark-Sacker bifurcation analysis on a numerical discretization of Gause-type predator-prey model with delay

This paper presents Neimark-Sacker bifurcation analysis for a kind of discrete Gause-type predator-prey system with time delay, which is obtained by using Euler discretization method. For biological reasons, we are only interested in positive solutions of system, thus some parameter conditions are given for the existence of a unique positive fixed point. Then by choosing delay as bifurcation parameter, and analyzing the associated characteristic equation, we obtain stability result of the positive fixed point. It is also demonstrated that Neimark-Sacker bifurcation occurs when the delay crosses some critical values. An explicit formula which determines the stability, direction and other properties of bifurcating periodic solution is derived by using the center manifold and normal form theory. Finally, a numerical example is given to support the analytic results.