Bifurcation Analysis In A Time-Delay Model For Predator-Prey Growth With Stage-Structure

In an ecological system, predator-prey model is well studied. As there is aprocess between immature population and mature population, in which the statesof individual are di?erent. Then,in order to study the property of a species morereasonable, people often introduce a time delay to describe the stage-structurephenomena.A time-delay model for predator-prey growth with stage structure is consid-ered here. Satio and Takeuchi proposed this model and then they establishedsu?cient conditions for global attractivity of an interior equilibrium. In chapter3 of this article, We will concentrate on the existence of periodic orbit when thecondition that enables the global attractivity is destroyed. This complementarywork make our understanding more deeply on this model.The main contents of this paper are organized as follows. At first, we investi-gate the stability and Hopf bifurcations by analyzing the distribution of the rootsof associated characteristic equation. Then an explicit formulas for determiningthe stability and the direction of periodic solutions bifurcating from Hopf bifur-cations are derived, using the normal form theory and center manifold argument.Finally, some numerical simulations are carried out for supporting the analyticresults found.