We analyse a two-dimensional discrete-time population model motivated by one-step sampling of delayed weak-Allee recruitment under constant harvesting. The map is interpreted independently for populations with non-overlapping census intervals. Positive equilibria reduce to an explicit scalar harvesting function. This yields a nondegenerate equilibrium fold and a numerically resolved harvest-induced basin boundary. Jury conditions give the complete local stability tests, while closed-form period-doubling boundaries and analytical transversality conditions identify stability loss of the upper equilibrium. The delay-shift structure makes the Jacobian determinant negative, excluding a fixed-point Neimark–Sacker bifurcation. A projected map is analysed separately: extinction is a locally finite-time attracting absorbing state, and over-harvest simulations quantify collapse transients without treating h_max as a global persistence theorem. Bifurcation diagrams and Lyapunov exponents diagnose bounded sensitive trajectories, and three idealised feedback laws illustrate local stabilisation subject to explicit operational limitations.
This paper investigates the stability and Hopf bifurcation of a delayed fractional-order tumor-immune interaction model involving immune effector cells, tumor cells, and interleukin-2. The time delay is introduced to describe the delayed activation and proliferation of effector cells induced by immune stimulation. For the commensurate-order case, the existence of equilibrium points is discussed, and the local stability of the positive equilibrium is analyzed by deriving the corresponding characteristic equation. Taking the delay as the bifurcation parameter, sufficient conditions are established for stability switching and the occurrence of Hopf bifurcation at the critical delay. Numerical simulations are carried out to verify the theoretical results and to illustrate the effects of the fractional order and time delay on the dynamic behavior of the system. The results show that increasing the delay can destabilize the positive equilibrium and lead to sustained oscillations, while the fractional order affects the critical bifurcation threshold. These findings provide theoretical insight into the dynamic mechanisms of tumor-immune interactions and offer a qualitative reference for further mathematical modeling of immunotherapy-related processes.
Based on the transmission mechanism of HPV, this paper establishes a dynamic model that includes sexual and non-sexual transmission. First, the threshold R_0 of the model is obtained using the next-generation matrix method. It is found that when R_0 < 1, the disease-free equilibrium is globally asymptotically stable, while when R_0 > 1, the endemic equilibrium exists. Subsequently, numerical simulations verify that vaccination strategies targeting susceptible populations are more effective than input measures for vaccinated populations. Furthermore, sensitivity analysis is conducted on threshold R_0, and the corresponding optimal control problem is considered. It is found that when implementing a single measure, sexual contact control is the most effective, followed by non-sexual contact control. In addition, vaccination measures should not be ignored.
To explore the evolutionary characteristics of COVID-19 disease and prevention and control measures after the relaxation of the epidemic, we establish a COVID-19 transmission model with breakthrough infection and immune failure. Firstly, dynamic analysis of the model is conducted, the basic reproduction number is obtained using the next-generation matrix method, and the global stability of disease-free equilibrium is proven. Secondly, we use actual case data from the fifth wave of COVID-19 in Hong Kong before and after the policy relaxation to estimate parameters. We also conduct a local sensitivity analysis of basic reproduction number and various parameters using elasticity indices. The results suggest that minimizing the infection rate in the early stages after the policy relaxation and promptly isolating individuals upon discovery of symptoms can quickly control the spread of the epidemic. Finally, optimal control research is conducted using Bang-Bang control, and the existence of optimal solution is proven. Through the simulation study of optimal control, different control strategies are compared. The results indicate that to prevent a resurgence of the disease as quickly as possible after the policy is relaxed, it is necessary to rapidly increase vaccination coverage at all doses before the policy is implemented, ensure prompt isolation and treatment of infected individuals upon developing symptoms, and simultaneously reduce the transmission rate by wearing masks and reducing outdoor activities. This will allow the spread of the disease to be quickly controlled and economic life to return to normal.
Lycium barbarum L., a key edible and medicinal crop, faces a persistent and serious threat to its production safety from Bactericera gobica Loginova. In this study, we develop a synergistic biological control model to investigate the combined effects of additional food supply to the predatory natural enemy Deraeocoris punctulatus Fallen and the developmental delay in the parasitoid Psyllaephagus arenarius Trjapitzin. With additional food supply, our equilibrium analysis gives sharp conditions for the occurrence and stability of extinction, psyllid-only, predator-absent, parasitoid-absent, and predator-prey coexistence. Theoretical analysis reveals that variations in additional food supply can cause instability in parasitoid-absent equilibrium and induce a Hopf bifurcation. Furthermore, a Hopf bifurcation also occurs when the developmental delay of the parasitoid falls below a critical threshold at the coexistence equilibrium. Numerical simulations confirm that changes in both additional food supply and developmental delay can trigger the Hopf bifurcations. Finally, PRCC sensitivity analysis identifies the critical parameters that influence the dynamic behavior of the system, and the stability of the predator directly influences the overall control efficacy.
