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Hopf bifurcation in a environmental defensive expenditures model with time delay

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  • Russu, Paolo

Abstract

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, τ0. 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.

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  • Russu, Paolo, 2009. "Hopf bifurcation in a environmental defensive expenditures model with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3147-3159.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:3147-3159
    DOI: 10.1016/j.chaos.2009.04.021
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    References listed on IDEAS

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    1. Liao, Xiaofeng & Ran, Jiouhong, 2007. "Hopf bifurcation in love dynamical models with nonlinear couples and time delays," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 853-865.
    2. Sun, Chengjun & Han, Maoan & Lin, Yiping & Chen, Yuanyuan, 2007. "Global qualitative analysis for a predator–prey system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1582-1596.
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    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. Kaslik, Eva & Neamţu, Mihaela, 2020. "Dynamics of a tourism sustainability model with distributed delay," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    3. Russu, Paolo, 2012. "On the Optimality of Limit Cycles in Nature Based-Tourism," MPRA Paper 36599, University Library of Munich, Germany.

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