Three Types Of Ecological Models Of Hopf Bifurcation, Stability And Persistence

In the periodicity, oscillation and the permanence of the solution of differential equation reveal the long-term behavior of the dynamical system, and have wide application in the ecology. To protect the ecological balance and save endangered ecological population has very important actual significance. Biological regular and phenomenon in the biological were described by using differential equations, which drew the attention of a lot of experts and scholars. And many new topics that have strong practical background were forms.This paper will discuss the periodicity, oscillation and the permanence for three classes of ecological models, including, the Hopf bifurcation periodic solution for a Class General Ecological Model, the dynamics effect of the cannibalism rate for a class predator-prey structured population model and the persistent property of a N-population feedback control system with delaysIn chapter 2, Stability of the positive equilibrium and the existence and uniqueness of Hopf bifurcation for a class general ecological model with delays are investigated. At first, the necessary and sufficient conditions of unconditional local stability for the model are obtained by using the unconditional stability of linear system and the theory of characteristic value. Furthermore, regarding to the theory of the function orthogonality, the existence and uniqueness of Hopf bifurcation for the model is given. Finally, some examples show the feasibility of the conditions and main results about theorems and Solution curve fitting Maps are presented by using Mat lab.In chapter 3, it mainly discusses the dynamics effect on the cannibalism rate to a class predator-prey structured population model, and draw out a critical value of the Hopf bifurcation for the cannibalistic attack rate. Linear stability analysis is also provided through the theory of characteristic value and Lyapunov Direct Method. The effects of parameters on the stability are discussed. Furthermore, cannibalism is proofed to have a stabilizing effect on the population, whereas the prey growth effect is on the contrary. Finally, the stability effect of the cannibalism rate is emphasized by Matlab numerical simulations and shows realization.In chapter 4, the persistent property of a class N-population feedback control system with delays andω-periodic functions are investigated. The conditions for the persistent property theory of feedback control system are given by the property ofω-periodic functions and using the comparison theorem of differential equation.