Hopf Branch Of The Three-group System With Time Delay

Delay phenomena are existing extensively in the real world and engineering. Manymodels in ecology can be formulated as system of differential equations with time delays.Most of existing models have only one time delay, while relative to the natural environment,the number of delay models are more complex, but better reflect the objective world, andthus it has more practical.The main purpose of this paper is to study three species predator-prey models withdelays, the stability and Hopf bifurcation of positive equilibrium are studied by choosing thedelay s as a bifurcation parameter, which shows the positive equilibrium is from stable tounstable to stable as delay increase,and Hopf bifurcation occurs.Further,an explicit formulafor determining the stability and the direction of periodic solution bifurcating from positiveequilibrium is derived by the normal form theory and center manifold argument.Finally,numerical simulations are carried to illustrate the main results.Paper is divided into the following sections:As an introduction, in chapter 1,the back-ground and some theoretical tools of thesubject are given.In chapter 2,a three species system with delays is studied. The stability of the positiveequilibrium and the existence of Hopf bifurcation are investigated. The direction and stabilityof the Hopf bifurcation are investigated and some numerical simulations are carried out .In chapter 3,we consider a kind of food chain systems with time delays ,derive thesu?cient conditions for permanence.The stability of the non-negative equilibria and theexistence of Hopf bifurcation are investigated. numerical simulations are finally performedfor justifying the theoretical results.Chapter 4 which on the basis of the model in Chapter 3 discuss the situation whenτ₂ =0, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium whenτ=τ₀. The estimation of the length of delay to preserve stability has also been calculated.Then the direction and stability of the bifurcated periodic solutions are determined, using the normal form theory and the center manifold reduction.