Dynamics mechanism of time-delay reaction–diffusion rumor-propagation model with saturation control

In this paper, we propose a time-delay reaction–diffusion rumor-propagation model with saturation control and carry out the dynamic mechanism analysis. Firstly, we derive the Turing-instability condition for the model without delay, and then treat the time delay as the bifurcation parameter. Secondly, we analyze the influence of parameter variation on system stability and the results show that when the time delay exceeds the threshold uncontrollably, the system will lose stability and generate Hopf bifurcation. However, the system will remain stable under control. In addition, the influence of saturation control on system stability is also discussed. Furthermore, applying the normal form theory and the center manifold reduction of partial functional differential equations, the explicit formulas for determining the bifurcation direction and the stability of bifurcation periodic solutions are acquired. Finally, the theoretical results are verified by numerical simulation.