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Impulsive Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on the Unification of Discrete and Continuous Systems
Department of Mathematics, Ankara University, Ankara, Turkey
Department of Mathematics, ?ankaya University, Ankara, Turkey
Department of Mathematics, Ondokuz Mays University, Samsun, Turkey
- 1 Department of Mathematics, Ankara University, Ankara, Turkey
- 2 Department of Mathematics, ?ankaya University, Ankara, Turkey
- 3 Department of Mathematics, Ondokuz Mays University, Samsun, Turkey
Applied Mathematics·Volume 06 (2015)·Pages 1649–1664·Published 5 August 2015·DOI10.4236/am.2015.69147
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Abstract
In this study, the impulsive predator-prey dynamic systems on time scales calculus are studied. When the system has periodic solution is investigated, and three different conditions have been found, which are necessary for the periodic solution of the predator-prey dynamic systems with Beddington-DeAngelis type functional response. For this study the main tools are time scales calculus and coincidence degree theory. Also the findings are beneficial for continuous case, discrete case and the unification of both these cases. Additionally, unification of continuous and discrete case is a good example for the modeling of the life cycle of insects.
KeywordsTime Scales CalculusPredator-Prey Dynamic SystemsPeriodic SolutionsCoincidence Degree TheoryBeddington-DeAngelis Type Functional Response
- Wang, W., Shen, J. and Nieto, J. (2007) Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses. Discrete Dynamics in Nature and Society, 2007, Article ID: 81756, 15 p.
- Beddington, J.R. (1975) Mutual Interference between Parasites or Predators and Its Effect on Searching Efficiency. Journal of Animal Ecology, 44, 331-340. http://dx.doi.org/10.2307/3866
- DeAngelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975) A Model for Trophic Interaction. Ecology, 56, 881-892. http://dx.doi.org/10.2307/1936298
- Fan, M. and Agarwal, S. (2002) Periodic Solutions for a Class of Discrete Time Competition Systems. Nonlinear Studies, 9, 249-261.
- Fan, M. and Wang, K. (2000) Global Periodic Solutions of a Generalized n-Species GilpinAyala Competition Model. Computers & Mathematics with Applications, 40, 1141-1151. http://dx.doi.org/10.1016/S0898-1221(00)00228-5
- Fan, M. and Wang, K. (2001) Periodicity in a Delayed Ratio-Dependent Predatorprey System. Journal of Mathematical Analysis and Applications, 262, 179-190. http://dx.doi.org/10.1006/jmaa.2001.7555
- Fan, M. and Wang, Q. (2004) Periodic Solutions of a Class of Nonautonomous Discrete Time Semi-Ratio-Dependent Predatorprey Systems. Discrete and Continuous Dynamical Systems, Series B, 4, 563-574. http://dx.doi.org/10.3934/dcdsb.2004.4.563
- Fang, Q., Li, X. and Cao, M. (2012) Dynamics of a Discrete Predator-Prey System with Beddington-DeAngelis Function Response. Applied Mathematics, 3, 389-394. http://dx.doi.org/10.4236/am.2012.34060
- Huo, H.F. (2005) Periodic Solutions for a Semi-Ratio-Dependent Predatorprey System with Functional Responses. Applied Mathematics Letters, 18, 313-320. http://dx.doi.org/10.1016/j.aml.2004.07.021
- Li, Y.K. (1999) Periodic Solutions of a Periodic Delay Predatorprey System. Proceedings of the American Mathematical Society, 127, 1331-1335. http://dx.doi.org/10.1090/S0002-9939-99-05210-7
- Wang, Q., Fan, M. and Wang, K. (2003) Dynamics of a Class of Nonautonomous Semi-Ratio-Dependent Predator-Prey Systems with Functional Responses. Journal of Mathematical Analysis and Applications, 278, 443-471. http://dx.doi.org/10.1016/S0022-247X(02)00718-7
- Xu, R., Chaplain, M.A.J. and Davidson, F.A. (2005) Periodic Solutions for a Predato-Prey Model with Holling-Type Functional Response and Time Delays. Applied Mathematics and Computation, 161, 637-654. http://dx.doi.org/10.1016/j.amc.2003.12.054