Russu, Paolo (2009): Hopf bifurcation in a environmental defensive expenditures model with time delay. Published in: Chaos, Solitons and Fractals , Vol. 42, (2009): pp. 3147-3159.
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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.
|Item Type:||MPRA Paper|
|Original Title:||Hopf bifurcation in a environmental defensive expenditures model with time delay|
|Keywords:||delay differential equation; stability; Hopf bifurcation; defensive expenditures;|
|Subjects:||C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium|
|Depositing User:||paolo russu|
|Date Deposited:||31. Oct 2011 12:38|
|Last Modified:||16. Feb 2013 07:03|
Becker R. Intergenerational equity: The capital-environment trade-off. J Environ Econ Manage 1982;9:165–85.
Cazzavillan G, Musu I. Transitional dynamics and uniqueness of the balanced-growth path in a simple model of endogenous growth with an environmental asset. FEEMWorking Paper 65.2001; 2001.
Eagles PFJ, McCool SF, Haynes CD. Sustainable tourism in protected areas. United Nations Environment Programme, World Tourism Organization and IUCN, The World Conservation Union; 2002.
Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.
Liao X, Ran J. Hopf bifurcation in love dynamical models with nonlinear couples and time delays. Chaos, Solitons & Fractals 2007;321:853–65.
Ruan SG, Wei JJ. On the zero of transcendental function with applications to stability of delay differential equations with two delay delays. Dynam Cont Dis 2003;10:863–974.
Russu P. Tourism, recreation and optimal environmental defensive expenditures. XXX Convegno AMASES, Trieste; 2006.
Sun C, Han M, Lin Y, Chen Y. Global qualitative analysis for a predator-prey system with delay. Chaos, Solitons & Fractals 2007;32:1582–96.