Hopf bifurcation in a environmental defensive expenditures model with time delay
AbstractIn 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.
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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 34397.
Date of creation: 2009
Date of revision:
delay differential equation; stability; Hopf bifurcation; defensive expenditures;
Find related papers by JEL classification:
- C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
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- 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.
- Russu, Paolo, 2011. "Controlling complex dynamics in a protected-area discrete-time model," MPRA Paper 36598, University Library of Munich, Germany.
- 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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