- Volume 16 Issue 3
DOI QR Code
DYNAMICS OF A ONE-PREY AND TWO-PREDATOR SYSTEM WITH TWO HOLLING TYPE FUNCTIONAL RESPONSES AND IMPULSIVE CONTROLS
- Baek, Hunki (DEPARTMENT OF MATHEMATICS EDUCATION, CATHOLIC UNIVERSITY OF DAEGU)
- Received : 2011.08.13
- Accepted : 2012.09.15
- Published : 2012.09.25
In this paper, we investigate the dynamic behaviors of a one-prey and two-predator system with Holling-type II functional response and defensive ability by introducing a proportion that is periodic impulsive harvesting for all species and a constant periodic releasing, or immigrating, for predators at different fixed time. We establish conditions for the local stability and global asymptotic stability of prey-free periodic solutions by using Floquet theory for the impulsive equation, small amplitude perturbation skills. Also, we prove that the system is uniformly bounded and is permanent under some conditions via comparison techniques. By displaying bifurcation diagrams, we show that the system has complex dynamical aspects.
Supported by : Catholic University of Daegu
- J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10(1968), 707-723. https://doi.org/10.1002/bit.260100602
- H. Baek, Dynamic complexites of a three - species Beddington-DeAngelis system with impulsive control strategy, Acta Appl. Math., 110(1)(2010), 23-38. https://doi.org/10.1007/s10440-008-9378-0
- H. Baek, Qualitative analysis of Beddingtion-DeAngelis type impulsive predator-prey models, Nonlinear Analysis : Real World Applications, 11(3)(2010), 1312-1322. https://doi.org/10.1016/j.nonrwa.2009.02.021
- D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations:asymptotic properties of the solutions, Singapore:World Scientific, 1993.
- B. A. Croft, Arthropod biological control agents and pesicides. Wiley, New York (1990).
- J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math. 32(1977), 82-95. https://doi.org/10.1137/0132006
- A. Donofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biosci, 179(2002), 57-72. https://doi.org/10.1016/S0025-5564(02)00095-0
- P. Georgescu and G. Morosanu, Impulsive perturbations of a three-trophic prey-dependent food chain system, Mathematical and Computer Modeling, 48(2008), 975-997. https://doi.org/10.1016/j.mcm.2007.12.006
- M. P. Hassell, The dynamics of competition and predation. p.68. Arnod, London (1976).
- C. S. Holling, The functional response of predator to prey density and its role in mimicy and population regulatio. Mem. Entomol. Soc. Can., 45(1965), 1-60.
- V. Lakshmikantham, D. Bainov, P.Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, 1989.
- A. Lakmeche and O. Arino, Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynamics of Continuous, Discrete and Impulsive Systems, 7(2000), 265-287.
- B. Liu, Y. Zhang and L. Chen, Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy, Int. J. of Bifur. and Chaos, 15(2)(2005), 517-531. https://doi.org/10.1142/S0218127405012338
- B. Liu, Z. Teng and L. Chen, Analsis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. of Comp. and Appl. Math., 193(1)(2006), 347-362 https://doi.org/10.1016/j.cam.2005.06.023
- X. Liu and L. Chen, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons and Fractals, 16(2003), 311-320. https://doi.org/10.1016/S0960-0779(02)00408-3
- J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment, Bull. Math. Biol., 58(1996), 425-447. https://doi.org/10.1007/BF02460591
- M. G. Roberts and R. R. Kao, The dynamics of an infectious disease in a population with birth purses, Math. Biosci., 149(1998), 23-36. https://doi.org/10.1016/S0025-5564(97)10016-5
- S. Ruan, D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math, 61(2001), 1445-1472.
- W. Sokol and J.A. Howell , Kineties of phenol oxidation by ashed cell, Biotechnol. Bioeng., 23(1980), 2039-2049.
- B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60(1998), 1-26. https://doi.org/10.1006/bulm.1997.0010
- S.Y. Tang and L. Chen, Density-dependent birth rate, birth pulse and their population dynamic consequences, J. Math. Biol., 44(2002), 185-199. https://doi.org/10.1007/s002850100121
- S. Tang, Y. Xiao, L. Chen and R.A. Cheke, Integrated pest management models and their dynamical behaviour, Bulletin of Math. Biol., 67(2005), 115-135. https://doi.org/10.1016/j.bulm.2004.06.005
- W. Wang, H. Wang and Z. Li, The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy, Chaos, Solitons and Fractals, 32(2007), 1772-1785. https://doi.org/10.1016/j.chaos.2005.12.025
- W. Wang, H. Wang and Z. Li, Chaotic behavior of a three-species Beddington-type system with impulsive perturbations, Chaos Solitons and Fractals, 37(2008), 438-443. https://doi.org/10.1016/j.chaos.2006.09.013
- R. D. Yang and A. E. Humphrey, Dynamics and steady state studies of phenol biodegeneration in pure and mixed cultures, Biotechnol. Bioeng, 17(1975), 1211-1235. https://doi.org/10.1002/bit.260170809
- S. Zhang and L. Chen, Chaos in three species food chain system with impulsive perturbations, Chaos Solitons and Fractals, 24(2005), 73-83. https://doi.org/10.1016/j.chaos.2004.07.014
- S. Zhang and L. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos Solitons and Fractals, 24(2005), 1269-1278. https://doi.org/10.1016/j.chaos.2004.09.051
- S. Zhang, F.Wang and L. Chen, A food chain model with impulsive perturbations and Holling IV functional response, Chaos, Solitons and Fractals, 26(2005), 855-866. https://doi.org/10.1016/j.chaos.2005.01.053
- S. Zhang, L. Dong and L. Chen, The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, Solitons and Fractals, 23(2005), 631-643. https://doi.org/10.1016/j.chaos.2004.05.044