Quantum mechanics

Algorithm

Demography

Bifurcation

Statistics

Public relations

Applied mathematics

Stability (learning theory)

Computer science

Turing

Rumor

Environmental health

Statistical physics

Correctness

Physics

Nonlinear system

Diffusion

Amplitude

Mathematics

Political science

Population

Allee effect

Sociology

Mathematical analysis

Reaction–diffusion system

Medicine

Programming language

Machine learning

Information Sciences

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.

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