Predator gestation delay and nonlocal competition play key roles in controlling population density and maintaining ecosystem stability. In order to control the Dendrolimus superans that cause serious damage to forests, we propose a predator-prey reaction-diffusion equation with Holling type-II functional response function, gestation delay, and nonlocal competition. We investigated the existence conditions of the Hopf bifurcation and obtained its normal form of Hopf bifurcation by employing the multiple time scales method. We selected the appropriate parameters for numerical simulation and found that the gestation delay is helpful to maintain the stability of the population density of Dendrolimus superans.

Sclerodermus guani is a facultative natural enemy with dual functional responses of predation and parasitism, which has been widely used in the biological control of Monochamus saltuarius, the vector insect of Pine Wilt Disease (PWD). However, most existing studies treat its predation and parasitism behaviors in isolation and ignore the synergistic effect of the dual functions. Based on this, we innovatively construct a facultative predation-parasitism reaction-diffusion model with top-hat nonlocal perception, dual predation-parasitism functional responses, and parasitic time delay, and we systematically analyze the spatiotemporal dynamic behaviors of the model. Theoretical analysis shows that, in the absence of time delay, the nonlocal perception intensity $\eta$ and perception range $R$ can induce local over-predation of the host by natural enemies and population collapse, forming stable spatially nonhomogeneous steady-state solutions. When the parasitic time delay exceeds the critical value, the delay effect leads to periodic population oscillation. The synergistic effect of nonlocal perception and time delay induces Turing-Hopf bifurcation, presenting complex spatiotemporally coupled patterns. Numerical simulations based on parameters with realistic biological backgrounds verify the correctness of the theoretical analysis, and intuitively demonstrate the dynamic modes corresponding to different parameters, including local asymptotic stability, stable spatially nonhomogeneous steady-state solutions, and spatially nonhomogeneous periodic solutions. The results reveal the regulation mechanism of top-hat nonlocal perception and parasitic time delay on the spatiotemporal dynamics of the Monochamus saltuarius-Sclerodermus guani system. They not only enrich the spatiotemporal dynamics theory of facultative predation-parasitism systems, but also provide a quantitative mathematical basis for optimizing the release timing and release strategy of Sclerodermus guani in prevention and control as well as lay a theoretical foundation for the biological control of PWD.

In this paper, we establish an infectious disease model of Tropidothorax elegans to study the impact of them on plants. Our model involves the time delay for Tropidothorax elegans to hatch eggs, which is influenced by temperature. Second, we theoretically analyze the existence and the stability of the equilibrium and the normal form near the Hopf bifurcating critical point. Next, we choose three groups of parameters for numerical simulations to verify theoretical analysis of our model. Then, based on numerical simulations, we give bioanalysis which are consistent with the patterns of Tropidothorax elegans pests, such as dying off in large numbers of adults during the winter and one or two generations a year.

This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations.

In this paper, a reaction-diffusion dual carbon model associated with Dirichlet boundary condition is proposed under the influence of economic development in China. First, we enumerate and analyse some influencing factors of carbon emission and carbon absorption, and select economic development as the influence factor of carbon emission. Second, we establish a model associated with dual carbon and analyse the existence and stability of equilibrium and the existence of bifurcations. Finally, we analyse and predict for the value of parameters. Numerical simulations are presented to support our theory results. Combined with theoretical analysis and numerical simulations, we obtain that China can achieve carbon peak before 2030. If we want to achieve carbon neutral before 2060, it requires efforts from all of parts of society. Therefore, we put forward some practical suggestions to achieve carbon neutrality and carbon peak on schedule in China for the next few decades.

The excessive emission of greenhouse gases leads to abnormal climate change. Under this background, China puts forward the dual carbon target. In this paper, we use the analytic hierarchy process to determine the important influencing factor of carbon emissions. Next, we establish a delayed differential equation model of carbon emission-absorption under the influence of China's urbanization. We analyze the existence and stability of the positive equilibrium. Finally, we determine the ranges of parameters and study the impact of urbanization on China's dual carbon target through numerical simulations. The numerical simulation also shows that the system may have globally asymptotically stable equilibrium. Through the simulation results, we conclude whether the dual carbon target of China can be achieved by the scheduled time and give some suggestions that could be taken to achieve this target. The projected results provide some guidance for policy adjustments and also have practical significance in protecting the ecological environment.

As the COVID-19 continues threatening public health worldwide, when to vaccinate the booster shots becomes the hot topic. In this paper, based on the characteristics of COVID-19 and its vaccine, an $SAIR$ model associated with temporary immunity is proposed to study the effect on epidemic situation. Second, we theoretically analyze the existence and stability of equilibrium and the system undergoes Hopf bifurcation when delay passes through some critical values. Third, we study the dynamic properties of Hopf bifurcation and derive the normal form of Hopf bifurcation to determine the stability and direction of bifurcating periodic solutions. After that, numerical simulations are carried out to demonstrate the application of the theoretical results. Particularly, in order to ensure the validity, statistical analysis of data is conducted to determine the values for model parameters. Next, we study the impact of the infection rates on booster vaccination time to simulate the mutants, and the results are consistent with the facts. Finally, we predict the mean time of completing a round of vaccination worldwide with the help fitting and put forward some suggestions by comparing with the critical time of booster vaccination.