The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy

In this paper, by using theories and methods of ecology and ordinary differential equation, the dynamics complexity of a prey–predator system with Beddington-type functional response and impulsive control strategy is established. Conditions for the system to be extinct are given by using the Floquet theory of impulsive equation and small amplitude perturbation skills. Furthermore, by using the method of numerical simulation with the international software Maple, the influence of the impulsive perturbations on the inherent oscillation is investigated, which shows rich dynamics, such as quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises, etc. The numerical results indicate that computer simulation is a useful method for studying the complex dynamic systems.

[1]  S. Rinaldi,et al.  Food chains in the chemostat: Relationships between mean yield and complex dynamics , 1998 .

[2]  Vikas Rai,et al.  Crisis-limited chaotic dynamics in ecological systems , 2001 .

[3]  Weiming Wang,et al.  Mechanization for solving SPP by reducing order method , 2005, Appl. Math. Comput..

[4]  Sanyi Tang,et al.  Chaos in functional response host–parasitoid ecosystem models , 2002 .

[5]  Rong Yuan,et al.  Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response , 2004 .

[6]  M. Lakshmanan,et al.  Occurrence of multiple period-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems , 2003 .

[7]  Lansun Chen,et al.  Chaos in three species food chain system with impulsive perturbations , 2005 .

[8]  S. Rinaldi,et al.  Reduced order models for the prediction of the time of occurrence of extreme episodes , 2001 .

[9]  Xianning Liu,et al.  Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator ☆ , 2003 .

[10]  Zhilong Xiu,et al.  Dynamic complexity of a two-prey one-predator system with impulsive effect , 2005 .

[11]  Singular perturbation analysis of a model for a predator-prey system invaded by a parasite. , 1996, Bio Systems.

[12]  Lansun Chen,et al.  Chaos in a food chain chemostat with pulsed input and washout , 2005 .

[13]  M. A. Aziz-Alaoui,et al.  Analysis of the dynamics of a realistic ecological model , 2002 .

[14]  Chris Cosner,et al.  On the Dynamics of Predator–Prey Models with the Beddington–DeAngelis Functional Response☆ , 2001 .

[15]  M. Varriale,et al.  A study of a three species food chain , 1998 .

[16]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[17]  Zhenqing Li,et al.  The dynamics of a Beddington-type system with impulsive control strategy☆ , 2006 .

[18]  S. Gakkhar,et al.  Order and chaos in predator to prey ratio-dependent food chain , 2003 .

[19]  Lansun Chen,et al.  A Holling II functional response food chain model with impulsive perturbations , 2005 .

[20]  S Rinaldi,et al.  Remarks on food chain dynamics. , 1996, Mathematical biosciences.

[21]  Xinzhi Liu,et al.  Permanence of population growth models with impulsive effects , 1997 .

[22]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[23]  Bing Liu,et al.  Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control , 2004 .

[24]  Qui Zhipeng,et al.  The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response. , 2004, Mathematical biosciences.

[25]  Robert M. May,et al.  Necessity and chance: deterministic chaos in ecology and evolution , 1995 .

[26]  Bernd Blasius,et al.  Categories of chaos and fractal basin boundaries in forced predator–prey models , 2001 .

[27]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[28]  Tzy-Wei Hwang,et al.  Global analysis of the predator–prey system with Beddington–DeAngelis functional response , 2003 .

[29]  Tzy-Wei Hwang,et al.  Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response , 2004 .

[30]  Kevin S. McCann,et al.  Bifurcation Structure of a Three-Species Food-Chain Model , 1995 .

[31]  Stefano Lenci,et al.  Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation , 2000 .

[32]  Weiming Wang,et al.  A new algorithm for symbolic integral with application , 2005, Appl. Math. Comput..

[33]  Weiming Wang,et al.  A new algorithm for integral of trigonometric functions with mechanization , 2005, Appl. Math. Comput..

[34]  Drumi D. Bainov,et al.  Impulsive Differential Equations: Asymptotic Properties of the Solutions , 1995, Series on Advances in Mathematics for Applied Sciences.

[35]  Lansun Chen,et al.  The study of predator–prey system with defensive ability of prey and impulsive perturbations on the predator , 2005 .

[36]  Fengyan Wang,et al.  A food chain model with impulsive perturbations and Holling IV functional response , 2005 .

[37]  Hristo V. Kojouharov,et al.  Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response , 2005, Appl. Math. Comput..