Analysis of Turing patterns and amplitude equations in general forms under a reaction–diffusion rumor propagation system with Allee effect and time delay

In this paper, we divide the population into three groups: susceptible individuals ( S ), infectious individuals ( I ) and removed individuals ( R ), and propose a rumor propagation dynamic model with Allee effect and cross-diffusion. Next, we have analyzed a general form of cross-diffusion model with time delay, and drawn a general conclusion of linear stability analysis of Turing bifurcation. However, Turing bifurcation analysis cannot give the specific shapes of the patterns under certain conditions. With the help of the “Multiple Scale Analysis” method, we derive the expression of the amplitude equation for the general form of weakly nonlinear models . Finally, we apply the above theorems to the analysis of our previously proposed model, and derive the appearance condition of the Turing bifurcation and the expression of the amplitude equation respectively. Through the numerical simulations, we have verified the correctness of the above theoretical analysis.