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Graphs for the Replicator Equations and the “Tragedy of the Exhaustion of the Common Resource”

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Abstract

Active users have appeared who require the solution of inverse problems on a graph with conditions of uncertainty for modeling processes in the economic sphere. Such tasks were not considered by mathematicians earlier. This article provides an analogy with mathematical models which are described by a replicator system (RS) of equations related to the stated theme. The exact and asymptotic solutions are developed in the case of “stiff” RS equations. The effects of an a priori dying clone and of the boundary layer for small values of time are found. An analogy is drawn with some objects having similar features, which really exist in the economy. The effect of the existence of shadow, invisible super-consumers is described.

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Notes

  1. We talked with skeptics but we found out that they do not drink water from the water supply, they avoid the consumption of palm oil and other factors that lead to the pollution of their blood vessels, etc.

REFERENCES

  1. Í. Haken, Information and Self-Organization. A Macroscopic Approach to Complex Systems, Theoretical, Mathematical and Computer Physics (Springer, Berlin, 2005).

    Google Scholar 

  2. I. V. Andrianov, R. G. Barantsev, and L. I. Manevich, Asymptotic Mathematics and Synergetics (Editorial URSS, Moscow, 2004) [in Russian].

    Google Scholar 

  3. M. Eigen and P. Schuster, “The hypercycle. A principle of natural self-organization. Part: Emergence of the hypercycle,” Naturwissensch. 64, 541–565 (1977).

    Article  Google Scholar 

  4. N. Lehman, N. Vaidya, and M. L. Manapat, “Spontaneous network formation among cooperative RNA replications,” Nature (London, U.K.) 491 (7422), 72 (2012).

    Article  Google Scholar 

  5. A. S. Bratus, A. S. Novozhilov, and A. P. Platonov, Dynamical Systems and Models in Biology (Fizmatlit, Moscow, 2010) [in Russian]; A. S. Bratus and A. S. Novozhilov, “A note on the replicator equation with explicit space and global regulation,” Math. Biosci. Eng. 8 (3) (2011).

  6. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, 1998).

    Book  Google Scholar 

  7. W. Jansen, “A permanence theorem for replicator and Lotka-Volterra systems,” Math. Biol. 25, 411–427 (1987).

    Article  MathSciNet  Google Scholar 

  8. A. K. Volosova, “Mathematical modeling of nonlinear dynamics of an open hypercycle system,” Extended Abstract of Cand. Sci. (Phys. Math.) Dissertation (Russ. Transp. Univ. MIIT, Moscow, 2011). http://www.dissercat.com.

  9. N. N. Bautin and E. A. Leontonovich, Methods and Techniques for the Qualitative Study of Dynamical Systems on the Plane, 2nd ed. (Fizmatlit, Moscow, 2003) [in Russian].

    Google Scholar 

  10. K. A. Volosov, “Methodology for the analysis of evolutionary systems with distributed parameters,” Extended Abstract of Cand. Sci. (Phys. Math.) Dissertation (MIEM, Moscow, 2007). http://eqworld.ipmnet.ru/dis/volosov/Doc2007.pdf.

  11. V. G. Danilov, V. P. Maslov, and K. A. Volosov, Mathematical Modeling of Heat and Mass Transfer Processes (Nauka, Moscow, 1987; Kluwer Academic, Dordrecht, Boston, London, 1995).

  12. G. P. Karev, The HKV Method of Solving of Replications and Models of Biological Populations and Communities (Natl. Center Biotechnol. Inform., Natl. Inst. Health, Rockville Pike, Bethesda, MD, USA).

  13. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Fizmatlit, Moscow, 2001; Chapman and Hall, CRC, Boca Raton, London, 2004).

  14. F. K. Berezovskaya, L. G. Kareva, and G. P. Karev, “Is it possible to prevent the 'Tragedy of Common Resource’?,” Math. Biolog. Bioinform. 7, 30–44 (2012).

    Article  Google Scholar 

  15. N. N. Moiseev, To Be or not to Be: Humanity’s Dilemma? (Ul’yanovsk. Dom Pechati, Moscow, 1999) [in Russian].

    Google Scholar 

  16. V. P. Maslov, Quantum Economics (Nauka, Moscow, 2006) [in Russian].

    MATH  Google Scholar 

  17. A. V. Shcherbakov, “Economics of Chernavsky,” Kompyut. Issled. Modelir. 9, 397–417 (2017).

    Google Scholar 

  18. N. K. Volosova, A. K. Volosova, K. A. Volosov, and S. P. Vakulenko, “To the question of 'the tragedy of exhaustion of the common resource',” in Proceedings of the 71st International Conference Herzen Readings, Apr. 9–13,2018 (RGPU Gertsena, St. Petersburg, 2018), pp. 60–74.

  19. N. K. Volosova, A. K. Volosova, K. A. Volosov, and S. P. Vakulenko, “To the question of the tragedy of exhaustion of the common resource,” in Proceedings of the International Conference on Differential Equations and Dynamical Systems, Suzdal, July 6–11,2018, p. 63.

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ACKNOWLEDGMENTS

The authors thank V.G. Danilov, A.S. Bratus, O.S. Rozanova, and V.P. Maslov for their helpful discussions.

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Authors and Affiliations

  1. Russian University of Transport (MIIT), Moscow, Russia

    N. K. Volosova, A. K. Volosova, K. A. Volosov & S. P. Vakulenko

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  1. N. K. Volosova
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  2. A. K. Volosova
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  3. K. A. Volosov
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  4. S. P. Vakulenko
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Correspondence to N. K. Volosova, A. K. Volosova, K. A. Volosov or S. P. Vakulenko.

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Translated by S. Avodkova

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Volosova, N.K., Volosova, A.K., Volosov, K.A. et al. Graphs for the Replicator Equations and the “Tragedy of the Exhaustion of the Common Resource”. Math Models Comput Simul 12, 246–256 (2020). https://doi.org/10.1134/S2070048220020167

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  • DOI: https://doi.org/10.1134/S2070048220020167

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