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Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 736–752 | Cite as

Mathematical Model of Nicholson’s Experiment

  • S. D. Glyzin
Article

Abstract

Considered is a mathematical model of insects population dynamics and an attempt is made to explain classical experimental results of Nicholson based on it. In the first section of the paper Nicholson’s experiment is described and dynamic equations for its modeling are chosen. A priori estimates for model parameters can be made more precise by means of local analysis of the dynamical system, that is carried out in the second section. For parameter values found there stability loss of the equilibrium of the problem leads to the bifurcation of stable two-dimensional torus. Numerical simulations based on the estimates from the second section allows to explain classical Nicholson’s experiment, which detailed theoretical rationale is given in the last section. There for an attractor of the system the largest Lyapunov exponent is computed. The nature of change of this exponent allows to additionally narrow the area of model parameters search. Justification of this experiment was made possible only due to combination of analytical and numerical methods in studying of equations of insects population dynamics. At the same time, the analytical approach made it possible to perform numerical analysis in a rather narrow region of the parameter space. It is not possible to get into this area, based only on general considerations.

Keywords

differential-difference equations asymptotic behavior stability Lyapunov exponents insect population dynamics 

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References

  1. 1.
    Nicholson, A.J., An outline of the dynamics of animal populations, Aust. J. Zool., 1954, vol. 2, no. 1, pp. 9–65.CrossRefGoogle Scholar
  2. 2.
    Nicholson, A.J., The self-adjustment of populations to change, Cold Spring Harbor Symp. Quant. Biol., 1958, vol. 22, pp. 153–173.CrossRefGoogle Scholar
  3. 3.
    May, R.M., Conway, G.R., Hassel, M.P., and Southwood, T.R.E., Time delay, density dependence and single oscillations, J. Anim. Ecol., 1974, vol. 43, pp. 747–770.CrossRefGoogle Scholar
  4. 4.
    Oster, G. and Guckenheimer, J., Bifurcation phenomena in population models, in The Hopf Bifurcation, Marsden, J. and Mccracken, M., Eds., Berlin: Spring-Verlag, 1976, pp. 327–345CrossRefGoogle Scholar
  5. 5.
    Kolesov, Yu.S., Insect population modeling, Biofizika, 1983, vol. 28, no. 3, pp. 513–514.Google Scholar
  6. 6.
    Kolesov, Yu. and Kubyshkin, Ye.P., Some properties of solutions to differential equations simulating the insect population dynamics, in Issledovaniya po ustoychivosti i teorii kolebaniy. Mezhvuz. sb. (Studies on Sustainability and Oscillation Theory. Inter-university Collection of Papers), Yaroslavl, 1983, pp. 64–86Google Scholar
  7. 7.
    Kubyshkin, Ye.P., Local methods in the study of a system of differential-difference equations modeling the dynamics of changes in the number of populations of insects, in Nelineinye kolebaniya v zadachakh ekologii. Mezhvuz. sb. (Nonlinear Oscillations in Problems of Ecology. Inter-university Collection of Papers), Yaroslavl, 1985, pp. 70–82Google Scholar
  8. 8.
    Glyzin, S.D., Two-frequency oscillations of the fundamental equation of the insect population dynamics, in Nelineinye kolebaniya v zadachakh ekologii. Mezhvuz. sb. (Nonlinear Oscillations in Problems of Ecology. Interuniversity Collection of Papers), Yaroslavl, 1984, pp. 91–116Google Scholar
  9. 9.
    Kaschenko, S.A., Stationary states of a delay differentional equation of insect population’s dynamics, Model. Anal. Inf. Sist., 2012, vol. 19, no. 5, pp. 18–34.Google Scholar
  10. 10.
    Glyzin, S.D., A registration of age groups for the Hutchinson’s equation, Model. Anal. Inf. Sist., 2007, vol. 14, no. 3, pp. 29–42.MathSciNetGoogle Scholar
  11. 11.
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, 1983.CrossRefMATHGoogle Scholar
  12. 12.
    Glyzin, D.S., Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., The dynamic renormalization method for finding the maximum Lyapunov exponent of a chaotic attractor, Differ. Equations, 2005, vol. 41, no. 2, pp. 284–289.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Crombie, A.C., On competition between different species of graminivoros insects, Proc. R. Soc. (B), 1946, vol. 133, pp. 362–395.CrossRefGoogle Scholar
  14. 14.
    Crombie, A.C., Further experiments on insect competition, Proc. R. Soc. (B), 1946, vol. 133, pp. 76–109.CrossRefGoogle Scholar
  15. 15.
    Birch, L.C., Experimental background to study of the distribution and abundance of insects. 1. The influence of temperature, moisture and food on the innate capacity for increase of three grain beetles, Ecology, 1953, vol. 34, pp. 608–611.Google Scholar

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© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia
  2. 2.Scientific Center in Chernogolovka RASChernogolovka, Moscow oblastRussia

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