Abstract
We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.
Research Article | Open Access
Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses
Received03 Jan 2007
Accepted12 Jul 2007
Published30 Sept 2007
References
P. A. Abrams and L. R. Ginzburg, “The nature of predation: prey dependent, ratio dependent or neither?,” Trends in Ecology & Evolution, vol. 15, no. 8, pp. 337–341, 2000.
View at: Publisher Site | Google ScholarS. Ahmad and A. C. Lazer, “Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 40, no. 1–8, pp. 37–49, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings of the American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. A. Berryman, “The orgins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992.
View at: Publisher Site | Google ScholarF. Chen, “On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 33–49, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. Fan and K. Wang, “Periodicity in a delayed ratio-dependent predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 262, no. 1, pp. 179–190, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetT.-W. Hwang, “Global analysis of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 395–401, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. López-Gómez, R. Ortega, and A. Tineo, “The periodic predator-prey Lotka-Volterra model,” Advances in Differential Equations, vol. 1, no. 3, pp. 403–423, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNetL. A. Real, “The kinetics of functional response,” The American Naturalist, vol. 111, no. 978, pp. 289–300, 1977.
View at: Publisher Site | Google ScholarH. R. Thieme, “Uniform persistence and permanence for nonautonomous semiflows in population biology,” Mathematical Biosciences, vol. 166, no. 2, pp. 173–201, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetW. Wang and J. H. Sun, “On the predator-prey system with Holling- functional response,” Acta Mathematica Sinica, vol. 23, no. 1, pp. 1–6, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetX.-Q. Zhao, Dynamical Systems in Population Biology, vol. 16 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, NY, USA, 2003.
View at: Zentralblatt MATH | MathSciNetH. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
View at: Zentralblatt MATH | MathSciNetL.-L. Wang and W.-T. Li, “Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 341–357, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. W. Jia, “Persistence and periodic solutions for a nonautonomous predator-prey system with type III functional response,” Journal of Biomathematics, vol. 16, no. 1, pp. 59–62, 2001 (Chinese).
View at: Google Scholar | Zentralblatt MATH | MathSciNetC. Liu, G. Chen, and C. Li, “Integrability and linearizability of the Lotka-Volterra systems,” Journal of Differential Equations, vol. 198, no. 2, pp. 301–320, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. J. Schreiber, “Coexistence for species sharing a predator,” Journal of Differential Equations, vol. 196, no. 1, pp. 209–225, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993.
View at: Zentralblatt MATH | MathSciNetV. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
View at: Zentralblatt MATH | MathSciNetJ. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593–610, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 288–303, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
View at: Zentralblatt MATH | MathSciNetS. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, vol. 394 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
View at: Zentralblatt MATH | MathSciNetM. Choisy, J.-F. Guégan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects,” Physica D, vol. 223, no. 1, pp. 26–35, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. D'Onofrio, “On pulse vaccination strategy in the SIR epidemic model with vertical transmission,” Applied Mathematics Letters, vol. 18, no. 7, pp. 729–732, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetE. Funasaki and M. Kot, “Invasion and chaos in a periodically pulsed mass-action chemostat,” Theoretical Population Biology, vol. 44, no. 2, pp. 203–224, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATHS. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006.
View at: Publisher Site | Google ScholarS. Gao, Z. Teng, J. J. Nieto, and A. Torres, “Analysis of an SIR epidemic model with pulse vaccination and distributed time delay,” Journal of Biomedicine and Biotechnology, vol. 2007, Article ID 64870, 2007.
View at: Google ScholarS. Zhang, D. Tan, and L. Chen, “The periodic -species Gilpin-Ayala competition system with impulsive effect,” Chaos, Solitons & Fractals, vol. 26, no. 2, pp. 507–517, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetR. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.
View at: Zentralblatt MATH | MathSciNetX.-Q. Zhao, “Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications,” The Canadian Applied Mathematics Quarterly, vol. 3, no. 4, pp. 473–495, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2007 Weibing Wang et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.