This paper studies a delayed rumor propagation model with logistic growth and saturation incidence. The next generation matrix method, some inequality techniques, the Lyapunov-LaSalle invariance principle, and the Lyapunov method are used in this paper. Our results indicate that if the basic regeneration number (which is analogous to the basic reproduction number in disease transmission models) is less than 1, the rumor-free equilibrium point (which is analogous to the disease-free equilibrium point in disease transmission models) is globally stable. If the basic regeneration number is greater than 1, then the rumor is permanent, and some sufficient conditions are obtained for local and global asymptotic stability of the rumor prevailing equilibrium point (which is analogous to the endemic equilibrium point in disease transmission models). Finally, three examples with numerical simulations are presented to illustrate the obtained theoretical results.

Based on the theory of fractional differential equations, this paper studied a single-species contraception model with feedback control terms. By comprehensively applying the Mittag-Leffler function construction method, Laplace transform, and stability criteria for fractional-order systems, the existence, uniqueness, boundedness, and local stability of the model's equilibrium points were analyzed, and the key influencing factors of the system's steady-state behavior were identified. Furthermore, the mechanisms of Hopf bifurcation and Turing bifurcation were explored, the sufficient conditions for the existence of these two types of bifurcations were derived, and the theoretical framework for the system's dynamic behavior was improved. A numerical example was designed to verify the validity of the theoretical results, and the simulation results confirmed the correctness of the stability conclusions and bifurcation conditions. In the conclusion section, the numerical simulations were reviewed and correlated with the theory; future research directions were proposed in light of the limitations of the current study, providing references for subsequent explorations.