Dynamics Analysis Of Spruce Budworm Reaction-diffusion Model With Delay

The spruce budworm is one of the most destructive insect that lives in the spruce and fir forests of United States and Eastern Canada,and the periodical outbreak of spruce budworm can lead to enormous loss of forest industry.In North America,the study of disaster prevention and control of spruce budworm is costly every year.Therefore,establishing the appropriate mathematical model and understanding the detailed dynamics are very important to develop a strategy for disaster prevention and control of spruce budworm and have very important theoretical and practical significance.This dissertation mainly investigates the stability and bifurcation,which includes global Hopf bifurcation and Turing-Hopf bifurcation,of several kinds spruce budworm model with time delay and diffusion.The extended existence of periodic solution is obtained by global Hopf bifurcation results,which can be used to explain the internal causes that spruce budworm population always periodically outbreaks.The existence of stable spatial inhomogeneous periodic solution is obtained by Turing-Hopf bifurcation results,witch can be used to explain the internal causes that the periodical outbreaks of spruce budworm population has spatial inhomogeneity.The main work is as follows:(?)For spruce budworm single population model with diffusion and age structure,the effects of mature time delay on system dynamics are investigated in one space dimension and homogeneous Neumann boundary conditions.The existence of nonnegative steady state and the stability(including global stability)of zero steady state are obtained.The stability and existence of local Hopf bifurcation near positive steady state are investigated by analyzing the distribution of eigenvalues.The direction of Hopf bifurcation and stability of bifurcating periodic solution are determined by the center manifold reduction and normal form method for partial functional differential equations.And the existence of global Hopf bifurcation is established by using the global Hopf bifurcation theory of partial functional differential equations.Finally,some numerical simulations are carried out to illustrate the analytical results and the reason why spruce budworm population always periodically outbreaks is found,as the bird population is a constant.(?)For spruce budworm and bird populations model with diffusion and age structure,the effects of mature time delay on system dynamics are investigated in one space dimension and homogeneous Neumann boundary conditions.The existence of nonnegative steady state and the stability(including global stability)of all boundary steady states are obtained.The stability and existence of local Hopf bifurcation near positive steady state are investigated by analyzing the distribution of eigenvalues.The direction of Hopf bifurcation and stability of bifurcating periodic solution are determined by the center manifold reduction and normal form method for partial functional differential equations.And the existence of global Hopf bifurcation is established by using the global Hopf bifurcation theory of partial functional differential equations.Finally,some numerical simulations are carried out to illustrate the analytical results and the reason why spruce budworm population always periodically outbreaks is found,as there exists an interactional effect between spruce budworm and bird populations.(?)In one space dimension and homogeneous Neumann boundary conditions,the existence of Turing-Hopf bifurcation of general reaction-diffusion equations with two variables is investigated by analyzing the distribution of eigenvalues.By using the above research results for a class of modified Leslie-Gower diffusive model,the stability and existence of Turing-Hopf bifurcation of positive steady state is obtained.The normal form of the Leslie-Gower diffusive model is determined by the center manifold reduction and normal form method for reaction-diffusion equations.Finally,by taking a special case of Leslie-Gower diffusive model(i.e.,spruce budworm and bird populations diffusive model)as a numerical example,the results of theoretical analysis are shown through the numerical simulation.The dynamics classification of topological equivalence in the neighbourhood of Turing-Hopf bifurcation point in two-parameter plane and existence of stable spatially inhomogeneous periodic solution are obtain.Which can be used to explain the phenomenon that the periodical outbreaks of spruce budworm population has spatial inhomogeneity.