Dynamic analysis of a delayed differential equation for <i>Tropidothorax elegans</i> pests

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.