The Hopf Bifurcation Analysis Of A Prey-predator System With Time Delay And Sex-structure For The Prey

In this paper, the impact of time delay on the dynamical behavior of predator-preysystems with sex-structure. We assume that prey population is divided into male andfemale prey and predator only catch the mature prey in Chapters two and three. Weapply the same methods to study the same contents.Firstly, by linearizing the system at the positive equilibrium, analyzing the associatedcharacteristic equation, and using the theory of Hopf bifurcation and the theory of stability,the author obtain the stability of positive equilibrium and the sufcient condition of theexistence of Hopf bifurcation.Secondly, the impact of the time delay on the stability of the delay diferential equation(DDE) model is investigated, where time delay is regarded as a bifurcation parameter. Itis found that a Hopf bifurcation can occur as the time delay passes some critical values.Using the normal form theory and center manifold argument, the explicit formulae whichdetermine the stability, direction and other properties of bifurcating periodic solutions arederived.Finally, Numerical simulations are performed to support theoretical results and somecomplex dynamic behaviors are observed.