Complex Dynamic Behaviors of an Impulsively Controlled Predator-prey System with Watt-type Functional Response
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we consider a discrete predator-prey system with Watt-type functional response and impulsive controls. First, we find sufficient conditions for stability of a prey-free positive periodic solution of the system by using the Floquet theory and then prove the boundedness of the system. In addition, a condition for the permanence of the system is also obtained. Finally, we illustrate some numerical examples to substantiate our theoretical results, and display bifurcation diagrams and trajectories of some solutions of the system via numerical simulations, which show that impulsive controls can give rise to various kinds of dynamic behaviors.
Abstract
In this paper, we consider a discrete predator-prey system with Watt-type functional response and impulsive controls. First, we find sufficient conditions for stability of a prey-free positive periodic solution of the system by using the Floquet theory and then prove the boundedness of the system. In addition, a condition for the permanence of the system is also obtained. Finally, we illustrate some numerical examples to substantiate our theoretical results, and display bifurcation diagrams and trajectories of some solutions of the system via numerical simulations, which show that impulsive controls can give rise to various kinds of dynamic behaviors.
참고문헌 (38)
- J. F. M. Al-Omari, "Stability and Optimal Harvesting in Lotka-Volterra Competition Model for Two-species with Stage Structure", Kyungpook Mathematical Journal, 47(1), pp.31~56 (2007), 경북대학교 자연과학대학 수학과
- J. F. Andrews, "A mathematical model for the continuous culture of macroorganisms untilizing inhibitory substrates", Biotechnol. Bioeng., 10, pp.707~723 (1968)
- R. Arditi, "Coupling in predator-prey dynamics:Ratio-dependence", J. Theor. Biol., 139, pp.311~326 (1989)
- HUNKI BAEK, "DYNAMICS OF AN IMPULSIVE FOOD CHAIN SYSTEM WITH A LOTKA-VOLTERRA FUNCTIONAL RESPONSE", Journal of the Korean Society for Industrial and Applied Mathematics, 12(3), pp.139~151 (2008), 한국산업응용수학회
- H. Baek, "A food chain system with Holling-type IV functional response and impulsive perturbations", Computers and Mathematics with Applications, 60, pp.1152~1163 (2010)
- 백헌기, "On the Dynamical Behavior of a Two-Prey One-Predator System with Two-Type Functional Responses", Kyungpook Mathematical Journal, 53(4), pp.647~660 (2013), 경북대학교 자연과학대학 수학과[DOI]
- Hunki Baek, "Stability for a Holling Type IV Food Chain System With Impulsive Perturbations", Kyungpook Mathematical Journal, 48(3), pp.515~527 (2008), 경북대학교 자연과학대학 수학과
- Hunki Baek, "Extinction and Permanence of a Holling I Type Impulsive Predator-prey Model", Kyungpook Mathematical Journal, 49(4), pp.763~770 (2009), 경북대학교 자연과학대학 수학과
- Hunki Baek, "Permanence of a Three-species Food Chain System with Impulsive Perturbations", Kyungpook Mathematical Journal, 48(3), pp.503~514 (2008), 경북대학교 자연과학대학 수학과
- D. D. Bainov, "Impulsive Differential Equations:Periodic Solutions and Applications, vol. 66, of Pitman Monographs and Surveys in Pure and Applied Mathematics" (1993), Longman Science & Technical
- D. D. Bainov, "Oscillatory and asymptotic properties of solutions of nonlinear impulsive differential equations of third order with retarded argument", Kyungpook Math. J., 39(1), pp.111~118 (1999)
- D. D. Bainov, "Nonoscillatory solutions of a class of impulsive differential equations of n -th order with retarded argument", Kyungpook Math. J., 39(1), pp.33~46 (1999)
- D. D. Bainov, "Global stability of the solutions of impulsive functional differential equations", Kyungpook Math. J., 39(2), pp.239~249 (1999)
- M. A. Basudan, "On population growth model with density dependence", Kyungpook Math. J., 41(1), pp.127~136 (2001)
- J. R. Beddington, "Mutual interference between parasites or predator and its effect on searching efficiency", J. Animal Ecol., 44, pp.331~340 (1975)
- J. Cost, "Comparing predator-prey models qualitatively and quantitatively with ecological time-seires dat", PhD-Thesis, pp.1998
- M. B. Dimitrova, "Criteria for oscillation of impulsive differential equations of first order with deviating argument", Kyungpook Math. J., 40(1), pp.29~37 (2000)
- M. Fan, "Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response", J. of Math. Anal. and Appl., 295, pp.15~39 (2004)
- S. Gakkhar, "Chaos in seasonally perturbed ratio-dependent preypredator system", Chaos, Solitons and Fractals, 15, pp.107~118 (2003)
- C. S. Holling, "The functional response of predator to prey density and its role in mimicry and population regulations", Mem. Ent. Sec. Can., 45, pp.1~60 (1965)
- S. -B. Hsu, "Global stability for a class of predator-prey systems", SIAM J. Appl. Math., 55(3), pp.763~783 (1995)
- V Lakshmikantham, "Theory of Impulsive Differential Equations" (1989), World Scientific Publisher
- B. Liu, "Dynamic complexities in a Lotka-Volterra predatorprey model concerning impulsive control strategy", Int. J. of Bifur. and Chaos, 15(2), pp.517~531 (2005)
- Yicheng Liu, "Second Order Impulsive Neutral Functional Differential Inclusions", Kyungpook Mathematical Journal, 48(1), pp.1~14 (2008), 경북대학교 자연과학대학 수학과
- S. Ruan, "Golbal analysis in a predator-prey sytem with non-monotonic functional response", SIAM J. Appl. Math., 61(4), pp.1445~1472 (2001)
- E, Saez, "Dynamics of a predator-prey model", SIAM J. Appl. Math., 59(5), pp.1867~1878 (1999)
- G. T.Skalski, "Funtional responses with predator interference: viable alternatives to the Holling type II mode", Ecology, 82, pp.3083~3092 (2001)
- W. Sokol, "Kineties of phenol oxidation by ashed cell", Biotechnol. Bioeng., 23, pp.2039~2049 (1980)
- Gani Trendafilov Stamov, "Almost Periodic Processes in Ecological Systems with Impulsive Perturbations", Kyungpook Mathematical Journal, 49(2), pp.299~312 (2009), 경북대학교 자연과학대학 수학과
- Gani Tr.Stamov, "Second Method of Lyapunov for Existence of Almost Periodic Solutions for Impulsive Integro-Differential Equations", Kyungpook Mathematical Journal, 43(2), pp.221~231 (2003), 경북대학교 자연과학대학 수학과
- W. Wang, "The dynamic complexity of a three-species Beddingtontype food chain with impulsive control strategy", Chaos, Solitons and Fractals, 32, pp.1772~1785 (2007)
- X. Wang, "Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy", Chaos, Solitons and Fractals, 37(3), pp.706~718 (2008)
- Xiao Wang, "Global Attractivity and Oscillations in a Nonlinear Impulsive Parabolic Equation with Delay", Kyungpook Mathematical Journal, 48(4), pp.593~611 (2008), 경북대학교 자연과학대학 수학과
- K. E. F Watt, "A mathematical model for the effect of densities of attacked and attacking specises on the number atacked", Can. Entomol., 91, pp.129~144 (1959)
- Rui Xu, "Periodic Solutions of a Delayed Population Model with One Predator and Two Preys", Kyungpook Mathematical Journal, 44(4), pp.519~535 (2004), 경북대학교 자연과학대학 수학과
- Fengqin Zhang, "Anti-periodic Boundary Value Problem for Impulsive Differential Equations with Delay", Kyungpook Mathematical Journal, 48(4), pp.553~558 (2008), 경북대학교 자연과학대학 수학과
- S. Zhang, "The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator", Chaos, Solitons and Fractals, 23, pp.631~643 (2005)
- S. Zhang, "Chaos in periodically forced Holling type IV predatorprey system with impulsive perturbations", Chaos, Solitons and Fractals, 27, pp.980~990 (2006)
