Dynamics of a Discrete Lotka–Volterra Information Diffusion Model
Abstract
To explore the process of online social network information interaction, in this paper, we analyze the dynamics of a discrete Lotka–Volterra information diffusion model. Using the center manifold theorem, the conditions for transcritical bifurcation and flip bifurcation are obtained. With the help of approximation by a flow and Picard iteration, we explore the qualitative structures and stability of degenerate fixed point of the model with eigenvalues ±1. What’s interesting is that our results reveal a new and complex qualitative structure for fixed point, which are different from the previous reports and called degenerate saddle point. Additionally, the qualitative structures provide a new idea for investigation the stability of degenerate fixed point. Meanwhile, near the maximum user density, the dynamic results of degenerate fixed point indicate that if the intervention rate is greater than the inverse of the maximum user density, then the higher user density decreases, the lower user density increases when intrinsic growth rates are small (between 0 and 2). However, when the intrinsic growth rate is greater than 2, the high user density will continue to increase until it approaches the maximum user density indefinitely, while the small user density will approach 0, which provide us with new insights into information diffusion. Finally, we show the results of the model by numerical simulations, and the characteristics of information diffusion near the degenerate fixed point are predicted by theoretical analysis.
