Differential Equations

, Volume 55, Issue 9, pp 1164–1173 | Cite as

Application of the Leray-Schauder Principle to the Analysis of a Nonlinear Integral Equation

  • M. V. NikolaevEmail author
  • A. A. Nikitin
Integral and Integro-Differential Equations


We study a nonlinear integral equation arising from the parametric closure for the third spatial moment in the Dieckmann-Law model of stationary biological communities. The existence of a fixed point of the integral operator defined by this equation is analyzed. The noncompactness of the resulting operator is proved. Conditions are stated under which the equation in question has a nontrivial solution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors express their gratitude to Ulf Dieckmann for posing this problem and for his interest in this work.


A.A. Nikitin’s research was supported by Russian Science Foundation, project no. 17-11-01168.


  1. 1.
    Law, R., Murrell, D.J., and Dieckmann, U., Population growth in space and time: spatial logistic equations, Ecology, 2003, vol. 84, no. 1, pp. 252–262.CrossRefGoogle Scholar
  2. 2.
    Raghib, M., Nicholas, A.H., and Dieckmann, U., A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics, J. Math. Biol., 2011, vol. 62, pp. 605–653.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Law, R. and Plank, M.J., Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions, Bull. Math. Biol., 2015, vol. 77, pp. 586–613.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Murrell, D.J. and Dieckmann, U., On moment closures for population dynamics in continuous space, J. Theor. Biol., 2004, vol. 229, pp. 421–432.CrossRefGoogle Scholar
  5. 5.
    Davydov, A.A., Danchenko, V.I., and Nikitin, A.A., On the integral equation for stationary distributions of biological communities, in Problemy dinamicheskogo upravleniya. Sb. nauchn. tr. (Dynamic Control Problems. Coll. Sci. Pap.), Moscow: Fac. Comput. Math. Cybern., Moscow State Univ., 2009, no. 3, pp. 15–29.Google Scholar
  6. 6.
    Davydov, A.A., Danchenko, V.I., and Zvyagin, M.Yu., Existence and uniqueness of a stationary distribution of a biological community, Proc. Steklov Inst. Math., 2009, vol. 267, no. 1, pp. 40–49.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bodrov, A.G. and Nikitin, A.A., Qualitative and numerical analysis of an integral equation arising in a model of stationary communities, Dokl. Math., 2014, vol. 89, no. 2, pp. 210–213.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bodrov, A.G. and Nikitin, A.A., Examining the biological species steady-state density equation in spaces with different dimensions, Moscow Univ. Comput. Math. Cybern., 2015, vol. 39, no. 4, pp. 157–162.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nikitin, A.A. and Nikolaev, M.V., Equilibrium integral equations with kurtosian kernels in spaces of various dimensions, Moscow Univ. Comput. Math. Cybern., 2018, vol. 42, no. 3, pp. 105–113.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nikitin, A.A., On the closure of spatial moments in a biological model and the integral equations it leads to, Int. J. Open Inf. Technol., 2018, vol. 6, no. 10, pp. 1–8.Google Scholar
  11. 11.
    Smirnov, V.I., Kurs vysshei matematiki (A Course in Higher Mathematics), Moscow: Nauka, 1974, Vol. 2.Google Scholar
  12. 12.
    Krasnosel’skii, M.A., Topologicheskie metody v teorii nelineinykh integral’nykh uravnenii (Topological Methods in the Theory of Nonlinear Integral Equations), Moscow: Gostekhizdat, 1956.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.RUDN UniversityMoscowRussia

Personalised recommendations