Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.

[1]  Rui Xu,et al.  Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays , 2006 .

[2]  Lansun Chen,et al.  A delayed epidemic model with stage-structure and pulses for pest management strategy , 2008 .

[3]  Dongmei Xiao,et al.  Complex dynamic behaviors of a discrete-time predator–prey system , 2007 .

[4]  Dejun Tan,et al.  Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion ☆ , 2006 .

[5]  K. Gopalsamy,et al.  Time lags and global stability in two-species competition , 1980 .

[6]  Zhicheng Wang,et al.  The existence of periodic solutions for some models with delay , 2002 .

[7]  C. Çelik,et al.  The stability and Hopf bifurcation for a predator–prey system with time delay , 2008 .

[8]  Zhidong Teng,et al.  Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy , 2006 .

[9]  Xiaoxing Chen,et al.  Periodicity in a nonlinear discrete predator–prey system with state dependent delays , 2007 .

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

[11]  Chunjin Wei,et al.  Eco-epidemiology model with age structure and prey-dependent consumption for pest management , 2009 .

[12]  Lansun Chen,et al.  Staged-structured Lotka-Volterra predator-prey models for pest management , 2008, Appl. Math. Comput..

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

[14]  Maoan Han,et al.  Global qualitative analysis for a predator–prey system with delay , 2007 .

[15]  C. Çelik,et al.  Dynamical behavior of a ratio dependent predator-prey system with distributed delay , 2011 .

[16]  Yasuhiro Takeuchi,et al.  Stability analysis on a predator-prey system with distributed delays , 1998 .

[17]  Jianjun Jiao,et al.  A delayed stage-structured Holling II predator–prey model with mutual interference and impulsive perturbations on predator , 2009 .

[18]  C. Çelik,et al.  Hopf bifurcation of a ratio-dependent predator–prey system with time delay , 2009 .

[19]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[20]  S. Ruan.  Absolute stability, conditional stability and bifurcation in Kolmogrov-type predator-prey systems with discrete delays , 2001 .

[21]  Fengyan Wang,et al.  Chaos in a Lotka–Volterra predator–prey system with periodically impulsive ratio-harvesting the prey and time delays , 2007 .

[22]  Rong Yuan,et al.  Stability and bifurcation in a harvested one-predator–two-prey model with delays , 2006 .

[23]  Xiang-Ping Yan,et al.  Hopf bifurcation in a delayed Lokta–Volterra predator–prey system , 2008 .

[24]  Jianjun Jiao,et al.  The dynamics of an age structured predator–prey model with disturbing pulse and time delays ☆ , 2008 .

[25]  Shawgy Hussein,et al.  Stability and Hopf bifurcation for a delay competition diffusion system , 2002 .

[26]  Xiaofeng Liao,et al.  Hopf bifurcation and chaos analysis of Chen’s system with distributed delays , 2005 .

[27]  Guirong Jiang,et al.  Impulsive state feedback control of a predator-prey model , 2007 .

[28]  Yongli Song,et al.  Stability and bifurcation analysis on a Logistic model with discrete and distributed delays , 2006, Appl. Math. Comput..

[29]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[30]  Zhidong Teng,et al.  Persistence in nonautonomous predator-prey systems with infinite delays , 2006 .

[31]  Anthony W. Leung,et al.  Periodic solutions for a prey-predator differential delay equation , 1977 .

[32]  Xiang-Ping Yan,et al.  Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system , 2006 .

[33]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .