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Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion

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Nonlinear Analysis and its Applications to Differential Equations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 43))

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Abstract

Consider the Lotka-Volterra predator-prey system

$$ \begin{gathered} \dot u(t) = u(t)[r_1 - au(t) - a_1 v(t - \sigma )], \hfill \\ \dot v(t) = v(t)[ - r_2 + b_1 u(t - \tau ) - bv(t)], \hfill \\ \end{gathered} $$
(1.1)

where are positive constants and a, abare non-negative constants.

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References

  1. E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey systemJ. Math. Anal. Appl.204 (1996), 840–853.

    Article  MathSciNet  MATH  Google Scholar 

  2. S.—N. Chow and J.K. HaleMethods of Bifurcation TheorySpringer-Verlag, New York, 1982.

    Book  MATH  Google Scholar 

  3. T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delaysTransactions of the A.M.S.352 (2000), 2217–2238.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Faria, Bifurcations aspects for some delayed population models with diffusionFields Institute Communications21 (1999)

    Google Scholar 

  5. T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, to appear inJ. Math. Anal. Appl.

    Google Scholar 

  6. T. Faria and L.T. Magalhäes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcationJ. Differential Equations 122(1995), 181–200.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. K. Hale and S. M. Verduyn-LunelIntroduction to Functional Differential EquationsSpringer-Verlag, New-York, 1993.

    MATH  Google Scholar 

  8. X.-Z. He, Stability and delays in a predator-prey systemJ. Math. Anal. Appl. 198(1996), 335–370.

    Article  Google Scholar 

  9. Y. KuangDelay Differential Equations with Applications in Population DynamicsAcademic Press, Boston, 1993.

    MATH  Google Scholar 

  10. X. Lin, J.W.-H. So and J. Wu, Centre manifolds for partial differential equations with delaysProc. Roy. Soc. Edinburgh122A (1992), 237–254.

    Article  MathSciNet  Google Scholar 

  11. P. Táboas, Periodic solutions of a planar delay equationProc. Roy. Soc. Edinburgh116A (1990), 85–101.

    Article  Google Scholar 

  12. J. WuTheory and Applications of Partial Functional Differential EquationsSpringer—Verlag, New York, 1996.

    Book  MATH  Google Scholar 

  13. T. Zhao, Y. Kuang and H. L. Smith, Global existence of periodic solutions in a class of delayed Gauss-type predator-prey systemsNonlinear Anal. 28(1997), 1373–1394.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciências, and Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Lisboa, 1749-016, Portugal

    Teresa Faria

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  1. Teresa Faria
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Editor information

Editors and Affiliations

  1. CMAF, University of Lisbon, Lisbon, 1649-003, Portugal

    M. R. Grossinho , M. Ramos , C. Rebelo  & L. Sanchez , ,  & 

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Faria, T. (2001). Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion. In: Grossinho, M.R., Ramos, M., Rebelo, C., Sanchez, L. (eds) Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 43. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0191-5_17

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  • DOI: https://doi.org/10.1007/978-1-4612-0191-5_17

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6654-9

  • Online ISBN: 978-1-4612-0191-5

  • eBook Packages: Springer Book Archive

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