Stability Analysis and Hopf Bifurcation Research for DNA Strand Displacement with Time delay

In this paper, a nonlinear system with time delay mode for DNA strand displacement based on toehold exchange firstly is proposed. The boundedness of all the state variables of the DNA strand displacement system is proved. After that, local stability analysis is performed around the positive equilibrium point to obtain a necessary and sufficient condition for local stability, which depend on the coefficients of the characteristic equation corresponding to linearization of the system. In addition, the existence of the Hopf bifurcation is analyzed, which shows that the time delay parameter will guarantee that the system is stable within a certain range. However, Hopf bifurcation and chaos occur when the time delay parameter increases continuously and even exceeds the critical value. At last, numerical simulations have verified the correctness and validity of the DNA strand displacement studies.