A study on mathematical models describing population changes of biological species

생물 종의 개체 수 변화를 기술하는 수학적 모델에 대한 고찰

  • Shim, Seong-A (Department of Mathematics, Sungshin women's University)
  • 심성아 (성신여자대학교 수학과)
  • Received : 2011.04.03
  • Accepted : 2011.05.10
  • Published : 2011.05.31

Abstract

Various mathematical models have been widely studied recently in both fields of mathematics and ecology since they help us understand the dynamical process of population changes in biological species living in a certain habitat and give useful predictions. The world population model proposed by Malthus, a British economist, in his work 'An Essay on the Principle of Population' published in the period of 1789~1826 is one of the early mathematical models on population changes. Malthus' models and the carrying capacity models of Verhulst in 1845 were based on exponential type functions. The independent research field of mathematical ecology has been started from Lotka's works in 1920's. Since then various different mathematical models has been proposed and examined. This article mainly deals with single species population change models expressed in terms of ordinary differential equations.

일정 영역에 서식하는 생물 종의 개체 수가 변화하는 역학적 과정을 이해하고 실질적인 예측을 하는데 도움을 주는 여러가지 수학적 모델이 현재 수학과 생태학 분야에서 활발하게 연구되고 있다. 영국의 경제학자 Malthus가 1798년부터 시작하여 1826년까지 출간한 An Essay on the Principle of Population에서 제안했던 세계인구 변화 모델과 1845년 Verhulst의 한계수용모델은 개체 수 변화에 대한 초기 수학적 모델로서 지수적 형태에 기초한 것이었다. 수리생물학으로 불리는 학문분야는 1920년경 Lotka의 연구에서 본격적으로 시작되었다고 할 수 있다. 이때부터 여러 가지 다양한 수학적 모델들이 제안되어지고 검증되어져 왔다. 이 논문에서는 주로 상미분방정식(ordinary differential equations)으로 표현되는 단일 생물종에 대한 개체 수 변화모델들을 살펴본다.

Acknowledgement

Supported by : Sungshin Women's University

References

  1. J.R. Beddington and R.M. may, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. https://doi.org/10.1126/science.197.4302.463
  2. F. Brauer and D.A. Sanchez, Constant rate population harvesting: equilibrium and stability, Theor. Population Biol., 8 (1975). 12-30. https://doi.org/10.1016/0040-5809(75)90036-2
  3. W.A. Calder, III, An allometric approach to population cycles of mammals, Journal of Theoretical Biology, Volume 100 (1983). Issue 2, 275-282. https://doi.org/10.1016/0022-5193(83)90351-X
  4. G. Caselli, J. Vallin, and G. Wunsch, Demography: Analysis and Synthesis, Academic Press, 2005.
  5. B. Charlesworth, Evolution in Age-structured Populations, Cambridge University Press, Cambridge, 1980.
  6. P.M.G. Harris, The history of human populations, Indianapolis. Volume 1 (2000).
  7. G.A. De Leo and A.P. Dobson, Allometry and simple epidemic models for microparasites, Nature 379 (1996), 720-722. https://doi.org/10.1038/379720a0
  8. P.H. Leslie, On the use of matrices in certain population mathematics, Biometrika, 33 (1945), 183-212. https://doi.org/10.1093/biomet/33.3.183
  9. A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.
  10. D. Ludwig, D.D. Jones, and C.S. Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and forest, J. Anim. Ecol., 47 (1978), 315-332. https://doi.org/10.2307/3939
  11. T.R. Malthus, An Essay on the Principle of Population, Penguin Books, 1970, Originally published in 1798.
  12. R.M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, second edition, 1975.
  13. J.A.J. Metz and O. Diekmann, The dynamics of physiologically structured populations, Lecture Notes in Biomathematics, volume 68, Springer-Verlag, Berlin-Heidelberg-New York, 1986.
  14. J.H. Myers and C.J. Krebs, Population cycles in rodents, Sci. Amer., June 1974, 38-46.
  15. A.J. Nicholson, The self adjustment of populations to change, Cold Spring Harbor Symposium on Quant, Biol., 22 (1957), 153-173.
  16. R.O. PETERSON, R.E. PAGE and K.M. DODGE, Wolves, Moose, and the Allometry of Population Cycles, Science 22 June 1984: Vol. 224, no. 4655, 1350-1352. https://doi.org/10.1126/science.224.4655.1350
  17. P.-F. Verhulst, Recherche mathemathiques sur le loi d'accroissement de la population, Nouveau Memoires de l'Academie Royale des Sciences et Belles Lettres de Bruxelles, 18 (1845), 3-38.
  18. P.-F. Verhulst, Deuxieme memoire sur la loi d'accroissement de la population, Memoires de l'Academie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 20 (1847), 1-32.