In this paper, a rumor-spreading model with Holling type III functional response was established. The existence of the equilibrium points was discussed. According to the Routh–Hurwitz criteria, the locally asymptotic stability of the equilibrium points was analyzed. The global stability of the equilibrium points was proven based on Lasalle’s invariance principle and generalized Bendixson–Dulac theorem. Numerical simulations were carried out to illustrate the impact of different parameters on the spread of rumors. When the stifling rate $\lambda$ increases, or the predation capacity $\beta$ or the system coming rate $k$ decreases, the number of rumor-spreaders is reduced to extinction. The results provide theory, method and decision support for effectively controlling the spread of rumors.

This article considers the competition between the propagation of traditional product information and innovative green product information, and it proposes a hybrid model with advertisement and promotion strategies. On this basis, an innovation green product information propagation model is developed through the optimization of the advertisement strategies of the adopters of innovative green product information and the promotion strategies of the adopters of traditional product information, according to Pontryagin’s maximum principle to seek the optimal strategies for maximizing the influence of innovative green product information, and using numerical calculations to simulate the propagation state of product information. The results show that advertisement strategies play a decisive role in the propagation of innovative green product information in the market. If the promotion strategies are also considered, the propagation effect of innovative green product information will be more effective.