Skip to main content
SpringerLink
Log in

Population Growth and Nash Equilibria Under Viability Constraints in the Commons

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Population growth modifies the optimal equilibrium between a stationary population and its resource, producing instead a line of equilibria, characterized by fluctuating population size, resource quantity, harvest per head, and birth rates. The Pontryagin procedure allows the analytical expression of the Nash equilibria for two populations sharing a common resource and capable of growth. An alternative procedure, which avoids solving differential equations and inherently includes state constraints, involves building the capture-viability kernel of an auxiliary system. For two populations, all Nash equilibria under state constraints are obtained as the intersection of the boundaries of two capture-viability kernels. The two methods, Pontryagin and viability, yield concordant results. Viability is more flexible and avoids solving differential equations for each initial condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. A set-valued map \(F: z \rightarrow K\) is a Marchaud map if the graph and the domain of F are closed and not empty; the values F(z) are convex; and \(\exists c \; \forall z \in \mathrm{Dom}(F), \Vert F(z) \Vert := \max _{y\in F(z)}\Vert y \Vert \le c(\Vert z \Vert +1)\).

References

  1. Hardin, G.: The tragedy of the commons. Science 162(3859), 1243–1248 (1968)

    Article  Google Scholar 

  2. Hardin, G.: The tragedy of the unmanaged commons. Trends Ecol. Evol. 9(5), 199 (1994)

    Article  Google Scholar 

  3. Hardin, G.: Tragedy of the commons. In: Henderson, D.R. (ed.) Concise Encyclopaedia of Economics, 2nd edn. Library of Economics and Liberty, Indianapolis (2008)

    Google Scholar 

  4. Clark, C.W., Munro, G.R.: The economics of fishing and modern capital theory: a simplified approach. J. Environ. Econ. Manag. 2, 92–106 (1975)

    Article  Google Scholar 

  5. Gurtin, M., Murphy, L.: On the optimal harvesting of age-structured populations: some simple models. Math. Biosci. 55, 115–136 (1981a)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gurtin, M., Murphy, L.: On the optimal harvesting of persistent age-structured populations. J. Math. Biol. 13, 131–148 (1981b)

    Article  MathSciNet  MATH  Google Scholar 

  7. Murphy, L., Smith, S.: Optimal harvesting of an age-structured populations. J. Math. Biol. 29, 77–90 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Medhin, N.: Optimal harvesting in age-structured populations. J. Optim. Theory Appl. 74, 413423 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Anita, S.: Optimal harvesting for a nonlinear age-dependent population dynamics. J. Math. Anal. Appl. 226, 6–22 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Anita, S., Iannelli, M., Kim, M.-Y., Park, E.-J.: Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 58, 1648–1666 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Zhao, C., Zhao, P., Wang, W.-S.: Optimal harvesting for nonlinear age-dependent population dynamics. Math. Comput. Model. 43, 310–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Luo, Z., Du, M.-Y.: Optimal harvesting for an age-dependent \(n\)-dimensional. Appl. Math. Mech. 29, 683–695 (2008)

    Article  MATH  Google Scholar 

  13. Hritonenko, N., Yatsenko, Y.: Structure of optimal age-dependent harvesting in the Lotka–McKendrik population model. Math. Biosci. 208(1), 48–62 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hritonenko, N., Yatsenko, Y., Goetz, R.U., Xabadia, A.: A bang–bang regime in optimal harvesting of size-structured populations. Nonlinear Anal. Theory Methods Appl. 71, e2331–e2336 (2009)

    Article  Google Scholar 

  15. Bonneuil N., Fursa, E.: Population growth in the commons: the case of the pre-revolutionary Don Cossacks (forthcoming)

  16. Bonneuil, N.: Maximum under continuous–discrete-time dynamic with target and viability constraints. Optimization 61(8), 901–913 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fulton, M.: A graphical analysis of the Cournot–Nash and Stackelberg models. J. Econ. Educ. 28, 48–57 (1997)

    Article  Google Scholar 

  18. Sarkar, J., Gupta, B., Pal, D.: A geoemtric solution of a Cournot oligopoly with nonidentical firms. J. Econ. Educ. 29, 118–126 (1998)

    Article  Google Scholar 

  19. d’Agata, A.: Geometry of Cournot–Nash equilibrium with application to commons and anticommons. J. Econ. Educ. 41(2), 169–176 (2010)

    Article  Google Scholar 

  20. Bonneuil, N.: Computing the viability kernel in large state dimension. J. Math. Anal. Appl. 323(2), 1444–1454 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Boucekkine, R., Hritonenko, N., Yatsenko, Y.: Sustainable growth under physical constraints: optimal R&D, investment and replacement policies. J. Optim. Theory Appl. 163, 301–331 (2014)

    Article  Google Scholar 

  22. Boucekkine, R., Martinez, B., Ruiz-Tamarit, R.: Optimal sustainable policies under pollution ceiling: the demographic side. Math. Model. Nat. Phenom. 9(4), 38–64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamien, M.I., Schwartz, D.: Dynamic Optimization. Elsevier, N.L. Amsterdam (1991)

    MATH  Google Scholar 

  24. Aubin, J.-P.: Viability Theory. Birkhaüser, Boston (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut national d’études démographiques, 133, bld Davout, 75980, Paris cedex 20, France

    Noël Bonneuil

  2. École des hautes études en sciences sociales, Paris, France

    Noël Bonneuil

Authors
  1. Noël Bonneuil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noël Bonneuil.

Additional information

Communicated by Lionel Thibault.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonneuil, N. Population Growth and Nash Equilibria Under Viability Constraints in the Commons. J Optim Theory Appl 176, 478–491 (2018). https://doi.org/10.1007/s10957-017-1135-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1135-y

Keywords

Mathematics Subject Classification

Search

Navigation