Abstract
We investigate the dynamics of a class of multi-species competition predator interaction models with Beddington-DeAngelis functional response. Sufficient conditions for existence of a positive periodic solution are given and sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the Lyapunov method. In addition, some numerical simulation shows that our models can occur in many forms of complexities including periodic oscillation and strange chaotic strange attractor.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331–340 (1975)
Kuang, Y., Baretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998)
Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theoret. Pop. Biol. 56, 65–75 (1999)
Abrams, P.A., Ginzburg, L.R.: The nature of predation: Prey-predator. Ratio-dependent or neither. Trends Ecol. Evol. 15, 337–341 (2000)
Fan, M., Kuang, Y.: Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 295, 15–39 (2004)
Rui, X.: Global etability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response. J. Math. 67, 1683–1693 (2012)
Xiaohu, W., Shuyong, L., Xu, D.: Globally exponential stability of periodic solutions for impulsive neutral-type neura networks with delays. J. Math. 64, 65–75 (2011)
Lantang, M., Gliu, X.: Positive periodic soution for ratio-dependent n-specices discrete time system. J. Math. 56, 577–589 (2011)
Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)
Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)
Zhang, J., Gui, Z.J.: Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays. Journal of Computational and Applied Mathematics 224, 602–613 (2009)
Yan, Y., Wang, K.H., Gui, Z.J.: Periodic Solution of Impulsive Predator-Prey Models with the Beddington-DeAngelis Functional Response. In: 5th International Congress on Mathematical Biology, pp. 86–91. World Academic Press (2011)
Anokhin, A., Berezansky, L., Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. Math. Anal. Appl. 193, 923–941 (1995)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific Series on Nonlinear Sciences. Ser. A, Singapore (1995)
Lin, Z., Liu, J., Pedersen, M.: Periodicity and blowup in a two-species cooperating model. Nonlinar Analysis. Real World Applications 12, 479–486 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yan, Y., Wang, K., Gui, Z. (2012). Dynamics and Simulations of Multi-species Competition-Predator System with Impulsive. In: Liu, B., Ma, M., Chang, J. (eds) Information Computing and Applications. ICICA 2012. Lecture Notes in Computer Science, vol 7473. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34062-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-34062-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34061-1
Online ISBN: 978-3-642-34062-8
eBook Packages: Computer ScienceComputer Science (R0)