Stability and Neimark-Sacker bifurcation analysis for a discrete single genetic negative feedback autoregulatory system with delay

In this paper, we deal with a discrete single genetic negative feedback autoregulatory system with delay by using Euler method. Choosing the delay $\tau $ as the bifurcation parameter and analyzing the associated characteristic equation corresponding to the unique positive fixed point, it is found that the stability of the positive equilibrium and Neimark-Sacker bifurcation may occur when $\tau $ crosses some critical values. Then the explicit formula which determines the stability, direction, and other properties of bifurcating periodic solution is derived by using the center manifold theorem and normal form theory. Finally, in order to illustrate our theoretical analysis, numerical simulations are also included in the end.