Stability And Hopf Bifurcation For Ecological System With Delay

Ecological system can be affected by various environmental factors in its evolution process and delays of one type or another exist widely in the evolution of ecological system. Study on the dynamics of ecological system with delay have attracted great attention of many scholars in the field of system biology. It not only has great theoretical significance but also important practical significance to investigate the dynamics of ecological system with delay. It can also reveal the actual motion law of ecological system. Stability and Hopf bifurcation of ecological system with delay are investigated by means of stability theory and bifurcation theory of dynamical system, center manifold theorem and normal form theory. The major work of this thesis is summarized as follows.(1) Stability and Hopf bifurcation of a delayed epidemic model with constant recruit-ment rate and saturation recovery is investigated. It has been found that there exist a stability switch and the Hopf bifurcation occurs as the delay increases by re-garding the latent period of the disease as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. Further, explicit formula determining direction of the Hopf bifurcation and stability of the bifurcat-ing periodic solutions is derived by using center manifold theorem and normal form theory. Finally, some numerical simulations are presented to verify the obtained theoretical results.(2) Stability and Hopf bifurcation of a three-species food chain system with two de-lays is investigated. The locally asymptotic stability of the system and existence of the Hopf bifurcation are discussed by analyzing the distribution of roots of the associated characteristic equation of the system when the two delays are equal and unequal, respectively. Direction of the Hopf bifurcation and stability of the bifur-cating periodic solutions are further investigated by using center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate the obtained theoretical results.(3) Stability and Hopf bifurcation of a delayed predator-prey system with stage-structure for both the predator and the prey is studied. Sufficient conditions for the locally asymptotic stability of the system and existence of the Hopf bifurcation are obtained by regarding different combination of the two delays as the bifurcation parameter. Further, explicit formula determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions is given by using center manifold theorem and normal form theory. Finally, some numerical simulations are presented to support the obtained theoretical results.(4) Stability and Hopf bifurcation of a model for the transmission of computer virus in network with two delays and quarantine strategy is investigated. The locally asymptotic stability of the system and existence of the Hopf bifurcation are discussed by regarding different combination of the delay due to the period that the anti-virus software uses to clean viruses and the temporary immunity period delay of the computers in which the viruses have been cleared as the bifurcation parameter. And properties of the Hopf bifurcation have been also studied. Finally, some numerical simulations are given to justify the theoretical results.