IDEAS home Printed from
   My bibliography  Save this paper

Hopf bifurcation in a environmental defensive expenditures model with time delay


  • Russu, Paolo


In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, tau0. It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.

Suggested Citation

  • Russu, Paolo, 2009. "Hopf bifurcation in a environmental defensive expenditures model with time delay," MPRA Paper 34397, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:34397

    Download full text from publisher

    File URL:
    File Function: original version
    Download Restriction: no

    References listed on IDEAS

    1. Becker, Robert A., 1982. "Intergenerational equity: The capital-environment trade-off," Journal of Environmental Economics and Management, Elsevier, vol. 9(2), pages 165-185, June.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Russu, Paolo, 2011. "Controlling complex dynamics in a protected-area discrete-time model," MPRA Paper 36598, University Library of Munich, Germany.
    2. Russu, Paolo, 2012. "On the Optimality of Limit Cycles in Nature Based-Tourism," MPRA Paper 36599, University Library of Munich, Germany.

    More about this item


    delay differential equation; stability; Hopf bifurcation; defensive expenditures;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:34397. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Joachim Winter). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.