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Mathematical Model of Nicholson’s Experiment

https://doi.org/10.18255/1818-1015-2017-3-365-386

Abstract

Considered  is a mathematical model of insects  population dynamics,  and  an attempt is made  to explain  classical experimental results  of Nicholson with  its help.  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  the stability loss of the  problem  equilibrium  of the  leads to the  bifurcation of a stable  two-dimensional torus.   Numerical simulations  based  on the  estimates  from the  second section  allows to explain  the  classical Nicholson’s experiment, whose detailed  theoretical substantiation is given in the last section.  There for an atrractor of the  system  the  largest  Lyapunov  exponent is computed. The  nature of this  exponent change allows to additionally narrow  the area of model parameters search.  Justification of this experiment was made possible  only  due  to  the  combination of analytical and  numerical  methods  in studying  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.

About the Author

Sergey D. Glyzin
P.G. Demidov Yaroslavl State University; Scientific Center in Chernogolovka RAS
Russian Federation

Doctor, Professor YSU.

14 Sovetskaya str., Yaroslavl 150003; 9 Lesnaya str., Chernogolovka, Moscow  region,  142432



References

1. Nicholson A. J., “An outline of the dinamics of animal populations”, Aust. J. Zool., 2:1 (1954), 9-65.

2. Nicholson A. J., “The self-adjustment of populations to change”, Cold Spring Harbor Symp. Quant. Biol., 22 (1958), 153-173.

3. May R.M., Conway G.R., Hassel M.P., Southwood T.R.E., “Time delay, density dependence and single oscillations”, J. Anim. Ecology, 43 (1974), 747-770.

4. Oster G., Guckenheimer J., “Bifurcation fenomena in population models”, The Hopf Bifurcation, eds. J. Marsden, M. McCracken, Spring-Verlag, Berlin, 1976, 327-345.

5. Kolesov Yu. S., “Modelirovanie populyatsii nasekomykh”, Biofizika, 28:3 (1983), 513-514, (in Russian).

6. Kolesov U.S., Kubyshkin Ye.P., “Nekotoryye svoystva resheniy differentsialnoraznostnykh uravneniy, modeliruyushchikh dinamiku izmeneniya chislennosti populyatsiy nasekomykh”, Issledovaniya po ustoychivosti i teorii kolebaniy, 1983, 64 - 86, (in Russian).

7. Kubyshkin Ye.P., “Lokalnyye metody v issledovanii sistemy differentsialnoraznostnykh uravneniy, modeliruyushchikh dinamiku izmeneniya chislennosti populyatsiy nasekomykh”, Nelineynyye kolebaniya v zadachakh ekologii, 1985, 70-82, (in Russian).

8. Glyzin S. D., “Dvukhchastotnyye kolebaniya fundamental’nogo uravneniya dinamiki populyatsiy nasekomykh”, Nelineynyye kolebaniya i ekologiya, 1984, 91 - 116, (in Russian).

9. Kaschenko S. A., “Stationary States of a Delay Differentional Equation of Insect Population’s Dynamics”, Model. Anal. Inform. Sist., 19:5 (2012), 18-34, (in Russian).

10. Glyzin S. D., “A registration of age groups for the Hutchinson’s equation”, Model. Anal. Inform. Sist., 14:3 (2007), 29-42, (in Russian).

11. Guckenheimer J., Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer, 1983.

12. Glyzin D. S., Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “The Dynamic Renormalization Method for Finding the Maximum Lyapunov Exponent of a Chaotic Attractor”, Differ. Equ., 41:2 (2005), 284-289.

13. Crombie A.C., “On competition between different species of graminivoros insects”, Proc. R. Soc. (B), 133 (1946), 362-395.

14. Crombie A.C., “Further experiments on insect competition”, Proc. R. Soc. (B), 133 (1946), 76-109.

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 grane beatles”, Ecology, 34 (1953), 608-611.


Review

For citations:


Glyzin S.D. Mathematical Model of Nicholson’s Experiment. Modeling and Analysis of Information Systems. 2017;24(3):365-386. (In Russ.) https://doi.org/10.18255/1818-1015-2017-3-365-386

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ISSN 1818-1015 (Print)
ISSN 2313-5417 (Online)