On the uniqueness of a positive stationary state in the dynamics of a population with asymmetric competition

  • A. A. Davydov
  • Amer Fadhel Nassar


For a nonlinear model of the dynamics of a size-structured (exploited) population with asymmetric form of competition, we prove a uniqueness theorem for a positive stationary solution under sufficiently general assumptions on the parameters of the model.


STEKLOV Institute Concave Function Competition Level Nonincreasing Function Asymmetric Competition 
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  1. 1.
    V. I. Arnol’d, “Hard” and “Soft” Mathematical Models (MTsNMO, Moscow, 2000) [in Russian].Google Scholar
  2. 2.
    A. A. Davydov and A. F. Nassar, “On a stationary state in the dynamics of a population with hierarchical competition,” Usp. Mat. Nauk 69 (6), 179–180 (2014) [Russ. Math. Surv. 69, 1126–1128 (2014)].CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. A. Davydov and A. F. Nassar, “Stationary regime of exploitation of size-structured population with hierarchical competition,” J. Math. Sci. 205 (2), 199–204 (2015).CrossRefMathSciNetGoogle Scholar
  4. 4.
    A. A. Davydov and A. S. Platov, “Optimal stationary solution in forest management model by accounting intra-species competition,” Moscow Math. J. 12 (2), 269–273 (2012).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. A. Davydov and A. S. Platov, “Optimal stationary solution for a model of exploitation of a population under intraspecific competition,” Sovrem. Mat., Fundam. Napr. 46, 44–48 (2012) [J. Math. Sci. 201 (6), 746–750 (2014)].Google Scholar
  6. 6.
    H. Von Foerster, “Some remarks on changing populations,” in The Kinetics of Cellular Proliferation, Ed. by F. Stohlman (Grune & Stratton, New York, 1959), pp. 382–407.Google Scholar
  7. 7.
    N. Hritonenko, Yu. Yatsenko, R.-U. Goetz, and A. Xabadia, “Optimal harvesting in forestry: Steady-state analysis and climate change impact,” J. Biol. Dyn. 7 (1), 41–58 (2013).CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGill-Hill, New York, 1955; LKI, Moscow, 2007).Google Scholar
  9. 9.
    T. R. Malthus, An Essay on the Principle of Population, or a View of Its Past and Present Effects on Human Happiness, with an Inquiry into Our Prospects Respecting the Future Removal or Mitigation of the Evils Which It Occasions (J. Murray, London, 1826).Google Scholar
  10. 10.
    A. G. McKendrick, “Applications of mathematics to medical problems,” Proc. Edinburgh Math. Soc. 44, 98–130 (1926).CrossRefGoogle Scholar
  11. 11.
    L. F. Murphy, “A nonlinear growth mechanism in size structured population dynamics,” J. Theor. Biol. 104 (4), 493–506 (1983).CrossRefGoogle Scholar
  12. 12.
    A. A. Panesh and A. S. Platov, “Optimization of size-structured population with interacting species,” J. Math. Sci. 188 (3), 293–298 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1964; Mir, Moscow, 1976).zbMATHGoogle Scholar
  14. 14.
    S. Tuljapurkar and H. Caswell, Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, (Chapman & Hall, New York, 1997).CrossRefGoogle Scholar
  15. 15.
    P.-F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” Correspondance Math. Phys. 10, 113–121 (1838).Google Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Vladimir State University Named after Alexander and Nikolay StoletovsVladimirRussia

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