Some remarks on an environmental defensive expenditures model

Tomás Caraballo; Renato Colucci; Luca Guerrini

doi:10.3934/dcdsb.2019007

Article Contents

2019, Volume 24, Issue 3: 1079-1093. Doi: 10.3934/dcdsb.2019007

This issue Previous Article Robustness of dynamically gradient multivalued dynamical systems Next Article I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems

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

Early access: January 2019

Published: January 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N40; Secondary: 34K18.

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Figure 1. Figure of experiment 1: the fixed point is not stable, the solution diverges

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Figure 2. Figures for Experiment 2: stable limit cycle.

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Figure 3. Stability of the fixed point for $ r<\delta $.

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Figure 4. The solution of the system with delay and for $ \tau = 8.6 $ and $ \tau = 9.8 $ respectively.

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Figure 5. The solution of the system with delay and for $ \tau = 2.1, 2.3,117,121 $.

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Figure 6. There exists a positive stable fixed point and the second component of the solution takes negative values.

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References

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