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2011 | 21 | 1 | 97-107
Tytuł artykułu

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.
Słowa kluczowe
Rocznik
Tom
21
Numer
1
Strony
97-107
Opis fizyczny
Daty
wydano
2011
otrzymano
2010-07-16
poprawiono
2010-10-08
Twórcy
autor
  • (i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China
  • (ii) Faculty of Science, Hunan Institute of Engineering, Fuxing Rd 88, Xiangtan 411004, PR China
autor
  • (i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China
  • (ii) Faculty of Science, Hunan Institute of Engineering, Fuxing Rd 88, Xiangtan 411004, PR China
  • School of Mathematics and Physics, Nanhua University, Changsheng Rd 26, Hengyang 421001, PR China
autor
  • (i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China
  • (ii) Faculty of Science, Hunan Institute of Engineering, Fuxing Rd 88, Xiangtan 411004, PR China
  • Department of Mathematics, Zhangjiajie College of Jishou University, Renming Rd 120, Zhangjiajie 427000, PR China
Bibliografia
  • Bhattacharyya, R. and Mukhopadhyay, B. (2006). Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, Ecological Complexity 3(2): 160-169.
  • Faria, T. (2001). Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications 254(2): 433-463.
  • Gao, S.J., Chen, L.S. and Teng, Z.D. (2008). Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Applied Mathematics and Computation 202(2): 721-729.
  • Hale, J. (1977). Theory of Functional Differential Equations, Springer-Verlag, Berlin.
  • Hassard, B., Kazarino, D. and Wan, Y. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
  • Kar, T. and Pahari, U. (2007). Modelling and analysis of a preypredator system stage-structure and harvesting, Nonlinear Analysis: Real World Applications 8(2): 601-609.
  • Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer, Dordrecht.
  • Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA.
  • Kuang, Y. and Takeuchi, Y. (1994). Predator-prey dynamics in models of prey dispersal in two-patch environments, Mathematical Biosciences 120(1): 77-98.
  • Li, K. and Wei, J. (2009). Stability and Hopf bifurcation analysis of a prey-predator system with two delays, Chaos, Solitons & Fractals 42(5): 2603-2613.
  • May, R.M. (1973). Time delay versus stability in population models with two and three trophic levels, Ecology 4(2): 315-325.
  • Prajneshu Holgate, P. (1987). A prey-predator model with switching effect, Journal of Theoretical Biology 125(1): 61-66.
  • Ruan, S. and Wei, J. (2003). On the zero of some transcendential functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series A 10(1): 863-874.
  • Song, Y.L. and Wei, J. (2005). Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Journal of Mathematical Analysis and Applications 301(1): 1-21.
  • Teramoto, E.I., Kawasaki, K. and Shigesada, N. (1979). Switching effects of predaption on competitive prey species, Journal of Theoretical Biology 79(3): 303-315.
  • Xu, R., Chaplain, M.A.J. and Davidson F.A. (2004). Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments, Nonlinear Analysis: Real World Applications 5(1): 183-206.
  • Xu, R. and Ma, Z.E. (2008). Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure, Chaos, Solitons & Fractals 38(3): 669-684.
  • Yan, X.P. and Li, W.T. (2006). Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Applied Mathematics and Computation 177(1): 427-445.
  • Yan, X.P. and Zhang, C.H. (2008). Hopf bifurcation in a delayed Lokta-Volterra predator-prey system, Nonlinear Analysis: Real World Applications 9(1): 114-127.
  • Zhou, X.Y., Shi, X.Y. and Song, X.Y. (2008). Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, Applied Mathematics and Computation 196(1): 129-136.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv21i1p97bwm
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