Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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STABILITY AND BIFURCATION ANALYSIS OF A LOTKA-VOLTERRA MODEL WITH TIME DELAYS

  • Xu, Changjin (Faculty of Science, Hunan Institute of Engineering) ;
  • Liao, Maoxin (School of Mathematical Sciences and Computing Technology, Central South University)
  • Received : 2010.07.06
  • Accepted : 2010.07.19
  • Published : 2011.01.30
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Abstract

In this paper, a Lotka-Volterra model with time delays is considered. A set of sufficient conditions for the existence of Hopf bifurcation are obtained via analyzing the associated characteristic transcendental equation. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form method and center manifold theory. Finally, the main results are illustrated by some numerical simulations.

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References

  1. S.G. Ruan, J.J. Wei, On the zero of some transcendential functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A 10 (2003), 863-874
  2. S.W. Zhang, L.S. Chen, Global attractivity for nonautonomous predator-prey system with time delay, J. Wuhan Univ. (Nat. Sci. Ed.) 49 (2003), 284-288. (In chinese)
  3. R. Xu, M.A.J. Chaplain, F.A. Davidson, A Lotka-V0lterra type food chain model with stage structure and time delays, J. Math. Anal. Appl. 315 (2006), 90-105 https://doi.org/10.1016/j.jmaa.2005.09.090
  4. Y.M. Chen, Z. Zhou, Stable periodic of a discrete periodic Lotka-Volterra competition system, J. Math. Anal. Appl. 277 (2003), 358-366. https://doi.org/10.1016/S0022-247X(02)00611-X
  5. J.A. Cui, L.S. Chen, Permanence and extinction in Logistic and systems with diffusion, J. Math. Anal. Appl. 258 (2001), 512-535. https://doi.org/10.1006/jmaa.2000.7385
  6. F. Cao, L.S. Chen, Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model, system Sci. & Math.Sci. 11 (1998), 107-111
  7. Y.L. Song, Y.H. Peng, J.J. Wei, Bifurcation for a predator-prey system with two delays, J. Math. Anal. Appl. 337 (2008), 466-479. https://doi.org/10.1016/j.jmaa.2007.04.001
  8. B. Hassard, D. Kazarino and Y. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
  9. X.P. Yan, W. T. Li, Bifurcation and global periodic solutions in a delayed facultative mutualism system, Physica D 227 (2007), 51-69. https://doi.org/10.1016/j.physd.2006.12.007
  10. X.P. Yan, C.R. Zhang, Hopf bifurcation in a delayed Lokta-Volterra predator-prey system, Nonlinear Anal.: Real World Appl. 9 (2008), 114-127. https://doi.org/10.1016/j.nonrwa.2006.09.007
  11. C.B. Yu, J.J. Wei, X.F. Zou, Bifurcation analysis in an age-structured model of a single species living in two idential patches, Appl. Math. Modelling 34 (2009), 1068-1077.
  12. J. Hale, S. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
  13. I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Academic Press, New York, 1991.
  14. A.A. Berryman, The origins and evolutions of predator-prey theory, Ecology 73 (1992), 1530-1535. https://doi.org/10.2307/1940005
  15. Z. Lu, Y. Takeuchi, Permanence and global attractivity for competitive Lokta-Volterra system with delay, J. Nonlinear Anal. TMA 22 (1994), 847-856. https://doi.org/10.1016/0362-546X(94)90053-1
  16. D. Mukherjee, A.B. Roy, Uniform persistence and global attractivity theorem for generalized prey-predator system with time delay, Nonlinear Anal. 38 (2003), 449-458.