Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator

Abstract In this paper, a delayed predator–prey system with stage structure for predator is proposed and studied. It is found that the time delay is harmless for permanence of the stage-structured system. If αβ αβ  > 1, we show that the unique positive equilibrium is locally asymptotically stable when time delay τ ∗ is sufficiently small, while loss of stability by a Hopf bifurcation can occur as the delay increases.

[1]  Shengqiang Liu,et al.  Extinction and permanence in nonautonomous competitive system with stage structure , 2002 .

[2]  Wan-Tong Li,et al.  Positive periodic solutions of a class of delay differential system with feedback control , 2004, Appl. Math. Comput..

[3]  Wan-Tong Li,et al.  Existence and global stability of positive periodic solutions of a predator-prey system with delays , 2003, Appl. Math. Comput..

[4]  Robert M. May,et al.  Time‐Delay Versus Stability in Population Models with Two and Three Trophic Levels , 1973 .

[5]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[6]  R. M. Nisbet,et al.  THE SYSTEMATIC FORMULATION OF TRACTABLE SINGLE-SPECIES POPULATION MODELS , 1983 .

[7]  Wan-Tong Li,et al.  Periodic solutions of a periodic Lotka-Volterra system with delays , 2004, Appl. Math. Comput..

[8]  M. Zhien.  Stability of predation models with time delay , 1986 .

[9]  Fordyce A. Davidson,et al.  Global stability of a stage-structured predator-prey model with prey dispersal , 2005, Appl. Math. Comput..

[10]  Alan Hastings,et al.  Density Dependence and Age Structure: Nonlinear Dynamics and Population Behavior , 1997, The American Naturalist.

[11]  William Gurney,et al.  The systematic formulation of tractable single-species models incorporating age structure , 1983 .

[12]  Jianhong Wu,et al.  Persistence and global asymptotic stability of single species dispersal models with stage structure , 1991 .

[13]  Guilie Luo,et al.  Extinction and permanence in competitive stage structured system with time-delays , 2002 .

[14]  H. I. Freedman,et al.  Nonoccurence of Stability Switching in Systems with Discrete Delays , 1988, Canadian Mathematical Bulletin.

[15]  Fordyce A. Davidson,et al.  Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay , 2004, Appl. Math. Comput..

[16]  W. Gurney,et al.  Stability switches in distributed delay models , 1985 .

[17]  H. I. Freedman,et al.  A time-delay model of single-species growth with stage structure. , 1990, Mathematical biosciences.

[18]  Wendi Wang,et al.  A predator-prey system with stage-structure for predator , 1997 .

[19]  H. I. Freedman,et al.  The trade-off between mutual interference and time lags in predator-prey systems , 1983 .

[20]  Lansun Chen,et al.  The stage-structured predator-prey model and optimal harvesting policy. , 2000, Mathematical biosciences.

[21]  H. I. Freedman,et al.  Analysis of a model representing stage-structured population growth with state-dependent time delay , 1992 .

[22]  H. I. Freedman,et al.  Global stability of Volterra models with time delay , 1991 .

[23]  Yanni Xiao,et al.  Dynamical behavior for a stage-structured SIR infectious disease model , 2002 .