Stability And Hopf Bifurcation Analysis In A Pest Control Model With Two Delays

Since the 20th century,biological mathematics has achieved great development.It is found that differential equations with two delays or multi-delays can describe various kinds of natural phenomena more accurately.So there is a new trend that differential equations with two delays or multi-delays are extensively used to characterize the evolution of the system in field such as physics,economics,ecology,population biology.In this paper,we study the pest control model with two delays and theoretically analyze some dynamic properties of the model with the theory,expecting that our work will play a guiding role in agricultural pest control.The pest inoculation period delay parameter?₁ and the predator inoculation period delay parameter?₂ are considered in the mode.Thus,our model can better reflect the law between the three populations?tea plant,pests and predators?.Firstly,we calculate the equilibrium point of the system,translate the system to the origin and linearize the system at the origin so as to obtain the characteristic equation corresponding to the linearized system.Then we discuss the distribution of roots of the characteristic equation by using eigenvalue analysis method.Since the characteristic equation is a transcendental equation with two delay parameters,six cases are discussed according to two parameters.Consequently,we obtain the condition for local stability of the equilibrium and Hopf bifurcation.Secondly,we transform the delay differential equation into abstract functional differential equation.By using the center manifold theory and Poincare canonical form method,we discuss the stability of the periodic solutions and the direction of Hopf bifurcation.A numerical simulations are given to verify the accuracy of the theoretical results obtained above.Finally,we make a brief summary of this article and point out some questions which are worthy of further investigation.