Journal of Mathematical Sciences

, Volume 229, Issue 4, pp 439–454 | Cite as

Explicitly Solvable Models of Redistribution of the Conflict Space

  • T. V. Karataeva
  • V. D. Koshmanenko
  • S. M. Petrenko

We analyze a class of explicitly solvable models for the problems of redistribution of the conflict space between two alternative opponents. The existence of equilibrium state is proved for a complex nonlinear system whose time evolution is generated by the conflict interaction between its components. Explicit formulas are obtained for the limit compromise distributions in terms of the densities of probability measures. We consider a number of specific model examples of the dynamics of redistribution of the conflict territory and formation of an equilibrium (compromise) distribution of the space. We propose an interpretation of the results for the case of social and territorial conflicts.


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  1. 1.
    V. D. Koshmanenko, “Theorem on conflict for a pair of stochastic vectors,” Ukr. Mat. Zh., 55, No. 4, 555–560 (2003); English translation: Ukr. Math. J., 55, No. 4, 671–678 (2003).Google Scholar
  2. 2.
    V. Koshmanenko, “The theorem of conflicts for probability measures,” Math. Meth. Oper. Res., 59, No. 2, 303–313 (2004).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    V. D. Koshmanenko and N. V. Kharchenko, “Invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures,” Ukr. Math. J., 56, No. 7, 927–938 (2004); English translation: Ukr. Math. J., 56, No. 7, 1102–1116 (2004).Google Scholar
  4. 4.
    V. Koshmanenko and N. Kharchenko, “Spectral properties of image measures after conflict interactions,” Theory Stochast. Process., 10 (26), No. 3-4, 73–81 (2004).Google Scholar
  5. 5.
    M. V. Bondarchuk, V. D. Koshmanenko, and N. V. Kharchenko, “Properties of the limit states of a dynamical conflict system,” Nelin. Kolyv., 7, No. 4, 446–461 (2004); English translation: Nonlin. Oscillat., 7, No. 4, 432–447 (2004).Google Scholar
  6. 6.
    M. V. Bondarchuk, V. D. Koshmanenko, and I. V. Samoilenko, “Dynamics of conflict interaction between systems with internal structure,” Nelin. Kolyv., 9, No. 4, 435–450 (2006); English translation: Nonlin. Oscillat., 9, No. 4, 423–437 (2006).Google Scholar
  7. 7.
    S. Albeverio, V. Koshmanenko, and I. Samoilenko, “The conflict interaction between two complex systems: cyclic migration,” J. Interdiscipl. Math., 11, No. 2, 163–185 (2008).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    V. Koshmanenko and I. Samoilenko, “The conflict triad dynamical system,” Comm. Nonlin. Sci. Numer. Simulat., 16, 2917–2935 (2011).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    V. D. Koshmanenko and Karataeva, “Cyclic model of fire-water conflict,” in: Thesis of the XVII Internat. Conf. “Dynamical System Modeling and Stability Investigation” (Kyiv, May, 2015) (2015), p. 180.Google Scholar
  10. 10.
    T. V. Karataeva and V. D. Koshmanenko, “Model of a dynamical system of the “fire–water” conflict type,” Nelin. Kolyv., 17, No. 2, 228–247 (2014) English translation: J. Math. Sci., 208, No. 5, 551–570 (2015).Google Scholar
  11. 11.
    E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge Univ. Press, Cambridge (1991).CrossRefMATHGoogle Scholar
  12. 12.
    J. M. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science, Addison-Wesley Publ. Comp. (1997).Google Scholar
  13. 13.
    Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator–prey system,” J. Math. Biol., 36, 389–406 (1998).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    M. Maron, “Modelling populations: from malthus to the threshold of artificial life, evolutionary and adaptive systems,” Univ. Sussex (2003), pp. 1–17.Google Scholar
  15. 15.
    T. K. Kar, “Modelling and analysis of a harvested prey–predator system incorporating a prey refuge,” J. Comput. Appl. Math., 185, 19–33 (2006).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    P. T. Coleman, R. Vallacher, A. Nowak, and L. Bui-Wrzosinska, “Interactable conflict as an attractor: presenting a dynamical-systems approach to conflict, escalation, and interactability,” IACM Meeting Paper (2007).Google Scholar
  17. 17.
    V. D. Koshmanenko, “Existence theorems of the ω-limit states for conflict dynamical systems,” Meth. Funct. Anal. Topol., 20, No.4, 379–390 (2014).MathSciNetMATHGoogle Scholar
  18. 18.
    V. D. Koshmanenko and S. M. Petrenko, “Hahn–Jordan decomposition as an equilibrium state in the conflict system,” Ukr. Mat. Zh., 68, No. 1, 64–77 (2016); English translation: Ukr. Math. J., 68, No. 1, 67–82 (2016).Google Scholar
  19. 19.
    J. D. Murray, Mathematical Biology I: An Introduction, Springer, New York (2002).MATHGoogle Scholar
  20. 20.
    J. D. Murray, Mathematical Biology II: Spatial Models and Biometrical Applications, Springer, New York (2004).CrossRefGoogle Scholar
  21. 21.
    J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Princeton Univ. Press, Princeton (1960).MATHGoogle Scholar
  22. 22.
    T. C. Schelling, The Strategy of Conflict, Harvard Univ. Press, Cambridge (1980).MATHGoogle Scholar
  23. 23.
    S. Md. M. Khan and K. I. Takahashi, “Mathematical model of conflict and cooperation with non-annihilating multi-opponent,” J. Interdiscipl. Math., 9, No. 3, 459–473 (2006).MATHGoogle Scholar
  24. 24.
    S. Md. M. Khan and K. I. Takahashi, “Segregation through conflict,” Adv. Appl. Soc., 3, No. 8, 315–319 (2013).CrossRefGoogle Scholar
  25. 25.
    V. D. Koshmanenko, “On dynamical system of conflict with fair redistribution of vital resources,” in: Abstr. of the Internat. Conf. “Dynamical Systems and Their Applications” (Kyiv, 2015, June 22–26) (2015), p. 31.Google Scholar
  26. 26.
    V. Koshmanenko and I. Verygina, “Dynamical systems of conflict in terms of structural measures,” Meth. Funct. Anal. Topol., 22, No. 1, 81–93 (2016).MathSciNetMATHGoogle Scholar
  27. 27.
    I. V. Veryhina, “Comparison of strategies of two opponents in the problem of “occupation” of territory,” Dopov. Nats. Akad. Nauk Ukr. No. 5, 7–12 (2016).CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • T. V. Karataeva
    • 1
  • V. D. Koshmanenko
    • 1
  • S. M. Petrenko
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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