The dynamics of a prey-dependent consumption model concerning impulsive control strategy

A mathematical model for dynamics of a prey-dependent consumption model concerning impulsive control strategy is proposed and analyzed. We show that there exists a globally stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further, the conditions for the permanence of system are given. We show the existence of nontrivial periodic solution if the pest-eradication periodic solution loses its stability. When the unique positive periodic solution lose its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamics, which implies that the impulsive control model we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation and chaos.

[1]  Alberto d’Onofrio,et al.  Pulse vaccination strategy in the sir epidemic model: Global asymptotic stable eradication in presence of vaccine failures , 2002 .

[2]  R R Kao,et al.  The dynamics of an infectious disease in a population with birth pulses. , 1998, Mathematical biosciences.

[3]  H. Barclay.  Models for Pest Control Using Predator Release, Habitat Management and Pesticide Release in Combination , 1982 .

[4]  V. Volterra.  Variations and Fluctuations of the Number of Individuals in Animal Species living together , 1928 .

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

[6]  Lansun Chen,et al.  Density-dependent birth rate, birth pulses and their population dynamic consequences , 2002, Journal of mathematical biology.

[7]  Stephen D. Wratten,et al.  Measures of Success in Biological Control , 2000 .

[8]  Sanyi Tang,et al.  The effect of seasonal harvesting on stage-structured population models , 2004, Journal of mathematical biology.

[9]  Lansun Chen,et al.  The periodic competing Lotka-Volterra model with impulsive effect. , 2004, Mathematical medicine and biology : a journal of the IMA.

[10]  B. Shulgin,et al.  Pulse vaccination strategy in the SIR epidemic model , 1998, Bulletin of mathematical biology.

[11]  F. Parker,et al.  Management of Pest Populations by Manipulating Densities of Both Hosts and Parasites Through Periodic Releases , 1971 .

[12]  Lansun Chen,et al.  Impulsive vaccination of sir epidemic models with nonlinear incidence rates , 2004 .

[13]  A. J. Lotka.  UNDAMPED OSCILLATIONS DERIVED FROM THE LAW OF MASS ACTION. , 1920 .

[14]  Zvia Agur,et al.  Theoretical examination of the pulse vaccination policy in the SIR epidemic model , 2000 .

[15]  H. I. Freedman.  Graphical stability, enrichment, and pest control by a natural enemy , 1976 .

[16]  Xuebin Chi,et al.  The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission , 2002 .