# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021179
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Environmental degradation and indeterminacy of equilibrium selection

 1 Department of Economic and Business Sciences, University of Sassari, Sassari, Italy 2 Department of Statistics, Informatics and Applications, University of Florence, Italy 3 INdAM (National Institute of High Mathematics), Group of Analysis and Probability, Rome, Italy 4 Department of Law, University of Naples, Federico II, Naples, Italy 5 Department of Finance, Faculty of Economics, Technical University of Ostrava, Ostrava, Czech Republic

* Corresponding author: mauro.sodini@unipi.it

Received  December 2020 Revised  May 2021 Early access August 2021

This paper analyzes an intertemporal optimization problem in which agents derive utility from three goods: leisure, a public environmental good and the consumption of a produced good. The global analysis of the dynamic system generated by the optimization problem shows that global indeterminacy may arise: given the initial values of the state variables, the economy may converge to different steady states, by choosing different initial values of the control variable.

Citation: Angelo Antoci, Marcello Galeotti, Mauro Sodini. Environmental degradation and indeterminacy of equilibrium selection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021179
##### References:

show all references

##### References:
Global indeterminacy scenario. Parameter values: $\overline{E} = 1.3$; $\alpha = 0.3$, $\beta = 24.84$, $\gamma = 1.3$, $\delta = 1.06,$ $\varepsilon = 0.4,$ $\rho = 0.04$. Initial values of the state variables $E(0) = 1.04025820702807;$ $K(0) = 5.99124478008392$. For $L(0) = 0.425697415099600$ the trajectory evolves on the stable manifold of $P_{2}$ converging to the saddle $P_{2}$. For a larger value of $L(0)$ (in figure $L(0) = 0.681115864159360$) trajectories converge to the sink $P_{1}$. For lower values of $L(0)$ (in figure $L(0) = 0.383127673589640$) trajectories reach the plane $K = 0$ in finite time. The dashed curve shows the branch of the unstable manifold of $P_{2}$ converging to the sink $P_{1}$.
Hopf bifurcation and global indeterminacy. The parameter values are the same as in the previous figure except for $\delta = 1.035.$ Trajectories converging to the limit cycle $\Gamma$ surrounding $P_{1}$ coexist with trajectories on the stable manifold of the saddle $P_{2}$. Initial values of the state variables for the trajectories in figure are $E(0) = 2.20617886672602,$ $K(0) = 5.35230152562421$. The red trajectory converging to the saddle $P_{2}$ starts with $L(0) = 0.605972067919661$, the one converging to $\Gamma$ with $L(0) = 0.666569274711627$.
Evolution of the attractor of the system. The parameter values are the same as in the previous figures except for $\delta$. Starting from $\delta = \delta_{1}>1.0374$ for which $P_{1} = \left( 6.225232325,0.3491884150,0.35\right)$ is a sink (coordinates of the stationary states are independent of $\delta$), if we let $\delta$ decrease, $P_{1}$ undergoes a supercritical Hopf bifurcation ($\delta\simeq1.0374$). Letting $\delta$ decrease further we observe an expansion of the attractive limit cycle surrounding $P_{1}$. The labels near the closed curves indicate the values of $\delta$ in the simulation: $\delta_{2} = 1.035,$ $\delta _{3} = 1.03,$ $\delta_{4} = 1.028,$ $\delta_{5} = 1.027$.
Case $\delta<1.$ The parameter values are the same as in the previous figures except for $\delta = 0.95$.No attractor exists$.$ The phasespace is projected on the plane $K,E$. Three trajectories are depicted starting from the same initial values of the state variables $K,$ $E$ ($K(0) =$ $11.8279$, $E(0) = 0.6949$), but with different initial values of the jumping variable $L.$ The dashed one reaches (in finite time) the plane $K = 0$ ($L(0) = L_{1} = 0.2800$); the dash-dot one reaches (in finite time) the plane $E = 0$ ($L(0) = L_{2} = 0.4550$); the solid one tends (in infinite time) to the saddle point $P_{2}$ ($L(0) = L_{3} = 0.3585$).
 [1] Giorgio Calcagnini, Edgar J. Sanchez Carrera, Giuseppe Travaglini. Real option value and poverty trap. Journal of Dynamics & Games, 2020, 7 (4) : 317-333. doi: 10.3934/jdg.2020025 [2] M. Dolfin, D. Knopoff, L. Leonida, D. Maimone Ansaldo Patti. Escaping the trap of 'blocking': A kinetic model linking economic development and political competition. Kinetic & Related Models, 2017, 10 (2) : 423-443. doi: 10.3934/krm.2017016 [3] Xiong Li, Hao Wang. A stoichiometrically derived algal growth model and its global analysis. Mathematical Biosciences & Engineering, 2010, 7 (4) : 825-836. doi: 10.3934/mbe.2010.7.825 [4] E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461 [5] Luis C. Corchón. A Malthus-Swan-Solow model of economic growth. Journal of Dynamics & Games, 2016, 3 (3) : 225-230. doi: 10.3934/jdg.2016012 [6] Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456 [7] Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination. Mathematical Biosciences & Engineering, 2017, 14 (4) : 975-999. doi: 10.3934/mbe.2017051 [8] Gülden Gün Polat, Teoman Özer. On group analysis of optimal control problems in economic growth models. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2853-2876. doi: 10.3934/dcdss.2020215 [9] Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292 [10] Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156 [11] Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011 [12] Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050 [13] Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517 [14] Luis C. Corchón. Corrigendum to "A Malthus-Swan-Solow model of economic growth". Journal of Dynamics & Games, 2018, 5 (2) : 187-187. doi: 10.3934/jdg.2018011 [15] Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347 [16] Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209 [17] Konstantina Trivisa. Global existence and asymptotic analysis of solutions to a model for the dynamic combustion of compressible fluids. Conference Publications, 2003, 2003 (Special) : 852-863. doi: 10.3934/proc.2003.2003.852 [18] Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767 [19] Fang Liu, Zhen Jin, Cai-Yun Wang. Global analysis of SIRI knowledge dissemination model with recalling rate. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3099-3114. doi: 10.3934/dcdss.2020116 [20] Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999

2020 Impact Factor: 1.327

## Tools

Article outline

Figures and Tables