Pattern formation in a reaction–diffusion rumor propagation system with Allee effect and time delay

This paper analyzes the diffusion behavior of the suspicious and the infected cabins in cyberspace by establishing a rumor propagation reaction–diffusion model with Allee effect and time delay. The Turing instability conditions of the system are emphatically studied. After considering the effect of time delay on the rumor propagation system, we have studied the correlation between the stability of the system under the influence of small time delay and the homogeneous system near the equilibrium point; the critical condition of the delay-induced spatial instability is given as well. Then, we prove the existence of Hopf bifurcation induced by time delay in some cases. Further considering the possibility of diffusion coefficient changing with time, the critical parameter curves of stability and instability of approximate systems are given by means of Floquet theory, and the necessary conditions of Turing instability of periodic coefficient are studied. In the numerical simulations, we find that the variation in diffusion coefficient and time delay will change the pattern type, and the periodical diffusion behavior will affect the arrangement of the crowd gathering area in the pattern.