ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Frisman E. Y., Neverova G. P., Revutskaya O. L., Kulakov M. P. Dynamic modes of two­-age population model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 2, pp. 113-130. DOI: 10.18500/0869-6632-2010-18-2-113-130

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Article type: 

Dynamic modes of two­-age population model

Frisman Efim Yakovlevich, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch
Neverova Galina Petrovna, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch
Revutskaya Oksana Leonidovna, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch
Kulakov Matvej Pavlovich, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

In this paper we research a mathematical model of dynamics for the population number. We considered the population of the two-age classes by the beginning of the next season: the younger, one including not reproductive individuals, and the senior class, consisting of the individuals participating in reproduction. The model parameters (birth rate and survival rates) represent the exponential functions of the both age groups numbers. According to this supposition the density-dependent factors restrict the development of population. Analytical and numerical analysis of the model is made. We investigate the dynamic modes of the model. It is shown that density-dependent factors of regulation for the population number can lead to generation of fluctuations and chaotic dynamics behavior of the population.

  1. Ricker WE. Stock and recruitment. J. Fish. Res. Board Can. 1954;11(5):559–623. DOI: 10.1139/f54-039.
  2. May RM. Stability and Complexity in Model Ecosystems. Princeton: Princeton Univ. Press; 1974.
  3. May RM. The Croonian Lecture, 1985 – When two and two make four: nonlinear phenomena in ecology. Proc. R. Soc. London. Series B. 1986;228(1252):241–266. DOI: 10.1098/rspb.1986.0054.
  4. Shapiro AP. On the question of cycles in return sequences. In: Management and Information. Issue 3. Vladivostok: DVNTs of the USSR Academy of Sciences. 1972:96 (in Russian).
  5. Shapiro AP, Luppov SP. Recurrent equations in the theory of population biology. Moscow: Nauka; 1983. 133 p. (in Russian).
  6. Leslie PH. On the use of matrices in certain population mathematics. Biometrika. 1945;33(3):183–212. DOI: 10.1093/biomet/33.3.183.
  7. Leslie PH. Some further notes on the use of matrices in population mathematics. Biometrica. 1948;35(3-4):213–245. DOI: 10.1093/biomet/35.3-4.213.
  8. Lefkovitch LP. The study of population growth in organisms grouped by stages. Biometrics Society. 1965;21(1):1–18. DOI: 10.2307/2528348.
  9. Svirezhev YuM, Logofet DO. Stability of biological communities. Moscow: Nauka; 1978. 352 p. (in Russian).
  10. Logofet DO. The theory of matrix models of population dynamics with age and additional structures. Zh. Obshch. Biol. 1991;52(6):793–804 (in Russian).
  11. Logofet DO, Belova IN. Nonnegative matrices as a tool to model population dynamics: Classical models and contemporary expansions.  J. Math. Sci. 2008;155(6):894–907.
  12. Hastings A. Age dependent dispersal is not a simple process: Density dependence, stability, and chaos. Theor. Popul. Biol. 1992;41(3):388–400. DOI: 10.1016/0040-5809(92)90036-S.
  13. Lebreton JD. Demographic models for subdivided populations: The renewal equation approach. Theor. Popul. Biol. 1996;49(3):291–313. DOI: 10.1006/tpbi.1996.0015.
  14. Kooi BW, Kooijman SA. Discrete event versus continuous approach to reproduction in structured population dynamics. Theor. Popul. Biol. 1999;56(1):91–105. DOI: 10.1006/tpbi.1999.1416.
  15. Shapiro AP. The role of density regulation in the occurrence of fluctuations in the abundance of a multi-aged population. Studies in mathematical population ecology. Vladivostok: DVNC AN SSSR. 1983:3–17 (in Russian).
  16. Frisman EYa, Luppov SP, Skokova IN, Tuzinkevich AV. Complex modes of population dynamics represented by two age classes. Mathematical studies in population ecology. Vladivostok: DVO of the USSR Academy of Sciences. 1988:4–8 (in Russian).
  17. Frisman EYa, Skaletskaya EI. Strange Attractors in Simplest Models of Biological Population Dynamics. The Review of Applied and Industrial Mathematics. 1994;1(6):988–1004 (in Russian).
  18. Nedorezov LV,  Neklyudova VL. A Continuous-discrete models of time course of the number of a two age population. Siberian Journal of Ecology. 1999;4:371–375 (in Russian).
  19. Nedorezov LV, Utyupin YuV. A discrete-continuous model for a bisexual population dynamics. Siberian Mathematical Journal. 2003;44(3):511–518. DOI: 10.1023/A:1023821016511.
  20. Dazho R. Fundamentals of Ecology. Moscow: Progress; 19752. 415 p. (in Russian).
  21. Neimark YuI, Landa PS. Stochastic and Chaotic Oscillations. Moscow: Nauka; 1987. 424 p. (in Russian).
  22. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  23. Chernyavsky FB, Lazutkin AN. The Cycles Lemmings and Voles in the North. Magadan: IBPN FEB RAS; 2004. 150 p. (in Russian).
  24. Nikolsky GV. Ecology of fishes. Moscow: Vysshaya Shkola; 1974. 357 p. (in Russian).
  25. Inchausti P, Ginzburg LR. Small mammals cycles in northern Europe: patterns and evidence for the maternal effect hypothesis. Journal of Animal Ecology 1998;67:180–194. DOI: 10.1046/j.1365-2656.1998.00189.x.
  26. Charlesworth B. Natural selection on multivariate traits in age-structured populations. Proc. R. Soc. Lond. B. 1993;251:47-52. DOI: 10.1098/rspb.1993.0007.
  27. Ferriere R, Gatto M. Chaotic population dynamics can result from natural selection. Proc. R. Soc. Lond. B. 1993;251:33-38. DOI: 10.1098/rspb.1993.0005.
  28. Frisman EY, Zhdanova OL. Evolutionary transition to complex population dynamic patterns in a two-age population. Russian Journal of Genetics. 2009;45(9):1124-1133. DOI: 10.1134/S1022795409090142.
Short text (in English):
(downloads: 86)