In this paper, the dynamics of a delayed phytoplankton-zooplankton model is considered. Taking the delay due to the gestation of zooplankton as param- eter, we describe the local Hopf bifurcation by center manifold th...In this paper, the dynamics of a delayed phytoplankton-zooplankton model is considered. Taking the delay due to the gestation of zooplankton as param- eter, we describe the local Hopf bifurcation by center manifold theorem and normal form, then we discuss the global existence of periodic solution. At last, some simulations are given to support our result.展开更多
This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifur...This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.展开更多
基金supported by the National Natural Science Foundations of China(No.11431008)NSF of Shanghai grant(No.15ZR1423700)
文摘In this paper, the dynamics of a delayed phytoplankton-zooplankton model is considered. Taking the delay due to the gestation of zooplankton as param- eter, we describe the local Hopf bifurcation by center manifold theorem and normal form, then we discuss the global existence of periodic solution. At last, some simulations are given to support our result.
基金Project supported by the National Natural Science Foundation of China (Nos. 10771215 and10771094)
文摘This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.
