# American Institute of Mathematical Sciences

March 2019, 24(3): 1079-1093. doi: 10.3934/dcdsb.2019007

## Some remarks on an environmental defensive expenditures model

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain 2 Departament of Engineering, University Niccolò Cusano, Via Don Carlo Gnocchi, 3 00166, Roma, Italy 3 Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121, Ancona (AN), Italy

Received  April 2017 Revised  October 2017 Published  January 2019

Fund Project: This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2015-63723-P and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492

In this paper, we consider the environmental defensive expenditures model with delay proposed by Russu in [16] and obtain different results about stability of equilibria in the case of absence of delay. Moreover we provide a more detailed analysis of the stability for equilibria and Hopf bifurcation in the case with delay. Finally, we discuss possible modifications of the model in order to make it more accurate and realistic.

Citation: Tomás Caraballo, Renato Colucci, Luca Guerrini. Some remarks on an environmental defensive expenditures model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1079-1093. doi: 10.3934/dcdsb.2019007
##### References:
 [1] A. Antoci, S. Borghesi and P. Russu, Environmental defensive expenditures, expectations and growth, Population and Environment, 27 (2005), 227-244. doi: 10.1007/s11111-006-0019-0. [2] R. Becker, Intergenerational equity: The capital-environment trade-off, J. Environ. Econ. Manage., 9 (1982), 165-185. doi: 10.1016/0095-0696(82)90020-1. [3] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [4] T. Caraballo, X. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199. doi: 10.1137/14099930X. [5] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [6] G. Cazzavillan and I. Musu, Transitional dynamics and uniqueness of the balanced-growth path in a simple model of endogenous growth with an environmental asset, FEEM Working Paper, 2001 (2001), Paper No 65, 12pp. doi: 10.2139/ssrn.286694. [7] M. Dell'Era and M. Sodini, Closed form solution for dynamic of sustainable tourism., University Library of Munich, Germany, 2009. [8] P. F. J. Eagles, S. F. McCool and C. D. Haynes, Sustainable Tourism in Protected Areas. United Nations Environment Programme, World Tourism Organization and IUCN, The World Conservation Union, 2002. [9] M. Ferrara, Green economy, sustainable growth theory and demographic dynamics: a modern theoretical approach, Proceedings of the 3rd International Conference on Applied Mathematics, Simulation, Modelling, Circuits, Systems and Signals, pages, 11-12, 2009. World Scientific and Engineering Academy and Society (WSEAS). [10] M. Ferrara and L. Guerrini, Economic development and sustainability in a two-sector model with variable population growth rate, Journal of Mathematical Sciences: Advances and Applications, 1 (2008), 323-339. [11] J. K. Hale and S. V. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] B. Hassard, D. Kazarino and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press, 1981. [13] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [14] V. Kolmanovskii and L. Shaikhet, General method of Lyapunov functionals construction for stability investigation of stochastic difference equations, Dynamical Systems and Applications, World Sci. Ser. Appl. Anal., World Sci. Publ., River Edge, NJ, 4 (1995), 397-439. doi: 10.1142/9789812796417_0026. [15] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [16] P. Russu, Hopf bifurcation in a environmental defensive expenditures model with time delay, Chaos, Solitons and Fractals, 42 (2009), 3147-3159. doi: 10.1016/j.chaos.2009.04.021. [17] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Mathematical Studies, No. 33. Van Nostrand Reinhold Co., London, 1971.

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##### References:
 [1] A. Antoci, S. Borghesi and P. Russu, Environmental defensive expenditures, expectations and growth, Population and Environment, 27 (2005), 227-244. doi: 10.1007/s11111-006-0019-0. [2] R. Becker, Intergenerational equity: The capital-environment trade-off, J. Environ. Econ. Manage., 9 (1982), 165-185. doi: 10.1016/0095-0696(82)90020-1. [3] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [4] T. Caraballo, X. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199. doi: 10.1137/14099930X. [5] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [6] G. Cazzavillan and I. Musu, Transitional dynamics and uniqueness of the balanced-growth path in a simple model of endogenous growth with an environmental asset, FEEM Working Paper, 2001 (2001), Paper No 65, 12pp. doi: 10.2139/ssrn.286694. [7] M. Dell'Era and M. Sodini, Closed form solution for dynamic of sustainable tourism., University Library of Munich, Germany, 2009. [8] P. F. J. Eagles, S. F. McCool and C. D. Haynes, Sustainable Tourism in Protected Areas. United Nations Environment Programme, World Tourism Organization and IUCN, The World Conservation Union, 2002. [9] M. Ferrara, Green economy, sustainable growth theory and demographic dynamics: a modern theoretical approach, Proceedings of the 3rd International Conference on Applied Mathematics, Simulation, Modelling, Circuits, Systems and Signals, pages, 11-12, 2009. World Scientific and Engineering Academy and Society (WSEAS). [10] M. Ferrara and L. Guerrini, Economic development and sustainability in a two-sector model with variable population growth rate, Journal of Mathematical Sciences: Advances and Applications, 1 (2008), 323-339. [11] J. K. Hale and S. V. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] B. Hassard, D. Kazarino and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press, 1981. [13] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [14] V. Kolmanovskii and L. Shaikhet, General method of Lyapunov functionals construction for stability investigation of stochastic difference equations, Dynamical Systems and Applications, World Sci. Ser. Appl. Anal., World Sci. Publ., River Edge, NJ, 4 (1995), 397-439. doi: 10.1142/9789812796417_0026. [15] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [16] P. Russu, Hopf bifurcation in a environmental defensive expenditures model with time delay, Chaos, Solitons and Fractals, 42 (2009), 3147-3159. doi: 10.1016/j.chaos.2009.04.021. [17] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Mathematical Studies, No. 33. Van Nostrand Reinhold Co., London, 1971.
Figure of experiment 1: the fixed point is not stable, the solution diverges
Figures for Experiment 2: stable limit cycle.
Stability of the fixed point for $r<\delta$.
The solution of the system with delay and for $\tau = 8.6$ and $\tau = 9.8$ respectively.
The solution of the system with delay and for $\tau = 2.1, 2.3,117,121$.
There exists a positive stable fixed point and the second component of the solution takes negative values.
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