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Stability properties for partial Volterra integrodifferential equations

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Stability properties of the solutions of a Volterra's population equation including infinite delay and diffusion terms are studied via a linearized stability argument, which leads to investigating on operational characteristic equation. For a specific class of delay kernels time periodic solutions are shown to appear as the delay is increased.


  1. [1]

    M. C. Crandall -P. H. Rabinowitz,The Hopf bifurcation theorem in infinite dimensions, Arch. Rat. Mech. Anal.,67 (1978), pp. 53–72.

  2. [2]

    J. M. Cushing,Integrodifferential Equations and Delay Models in Population Dyanmics, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

  3. [3]

    P. de Mottoni -A. Tesei,Asymptotic stability results for a system of quasilinear parabolic equations, Applic. Anal.,8 (1979), pp. 7–21.

  4. [4]

    A. Friedman -M. Shinbrot,Volterra integral equations in Banach space, Trans. Amer. Math. Soc.,126 (1967), pp. 131–179.

  5. [5]

    S. I. Grossman -R. K. Miller,Perturbation theory for Volterra integrodifferential systems, J. Diff. Eqns.,8 (1970), pp. 457–474.

  6. [6]

    T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, Berlin - Heidelberg - New York, 1966.

  7. [7]

    K. Kirchgässner -J. Scheurle,Bifurcation, inDynamical Systems: An International Symposium, vol. 1, L. Cesari, J. Hale and J. P. LaSalle Eds., Academic Press, New York - London, 1967.

  8. [8]

    R. K. Miller,Asymptotic stability and perturbation for linear Volterra integrodifferential systems, inDelay and Functional Differential Equations and Their Applications, K. Schmitt Ed., Academic Press, New York - London, 1972.

  9. [9]

    R. K. Miller,Volterra integral equations in a Banach space, Funkc. Ekv.,18 (1975), pp. 163–194.

  10. [10]

    A. Schiaffino,On a diffusion Volterra equation, Nonl. Anal.: TMA,3 (1979), pp. 595–600.

  11. [11]

    I. Stakgold -L. E. Patne,Nonlinear problems in nuclear reactor analysis, inNonlinear Problems in the Physical Sciences and Biology, I. Stakgod, D. D. Joseph and D. H. Sattinger Eds., Springer-Verlag, Berlin - Heidelberg - New York, 1973.

  12. [12]

    A.Tesei,Asymptotic stability results for a reaction-diffusion system, Talk delivered at the Oberwolfach Meeting «Mathematische Modelle der Biologie», 3–8 june 1978.

  13. [13]

    C. C. Travis -G. F. Webb,Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc.,200 (1974), pp. 395–418.

  14. [14]

    W. Walter,Differential and Integral Inequalities, Springer-Verlag, Berlin - Heidelberg - New York, 1970.

  15. [15]

    A. Wörz-Busekros,Global stability in ecological systems with continuous time delay, SIAM J. Appl. Math.,35 (1978), pp. 123–124.

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Tesei, A. Stability properties for partial Volterra integrodifferential equations. Annali di Matematica pura ed applicata 126, 103–115 (1980). https://doi.org/10.1007/BF01762503

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  • Periodic Solution
  • Characteristic Equation
  • Specific Class
  • Linearize Stability
  • Stability Property