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The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations: : where *''x'' is the number of prey (for example, rabbits); *''y'' is the number of some predator (for example, foxes); * and represent the growth rates of the two populations over time; * represents time; and * are positive real parameters describing the interaction of the two species. The Lotka–Volterra system of equations is an example of a Kolmogorov model,〔Freedman, H.I., ''Deterministic Mathematical Models in Population Ecology'', Marcel Dekker, (1980)〕〔Brauer, F. and Castillo-Chavez, C., ''Mathematical Models in Population Biology and Epidemiology'', Springer-Verlag, (2000)〕〔Hoppensteadt, F., ("Predator-prey model" ), ''Scholarpedia'', 1(10), 1563, (2006)〕 which is a more general framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease, and mutualism. ==History== The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910.〔Lotka, A.J., "Contribution to the Theory of Periodic Reaction", ''J. Phys. Chem.'', 14 (3), pp 271–274 (1910)〕〔Goel, N.S. et al., “On the Volterra and Other Non-Linear Models of Interacting Populations”, ''Academic Press Inc.'', (1971)〕 This was effectively the logistic equation,〔Berryman, A.A., ("The Origins and Evolution of Predator-Prey Theory" ), ''Ecology'', 73(5), 1530–1535, (1992)〕 which was originally derived by Pierre François Verhulst.〔Verhulst, P.H., ("Notice sur la loi que la population poursuit dans son accroissement" ). ''Corresp. mathématique et physique'' 10, 113–121, (1838)〕 In 1920 Lotka extended, via Kolmogorov (see above), the model to "organic systems" using a plant species and a herbivorous animal species as an example 〔Lotka, A.J., ( "Analytical Note on Certain Rhythmic Relations in Organic Systems” ), ''Proc. Natl. Acad. Sci. U.S.'', 6, 410–415, (1920)〕 and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics.〔Lotka, A.J., ''Elements of Physical Biology'', Williams and Wilkins, (1925)〕 The same set of equations were published in 1926 by Vito Volterra, a mathematican and physicist who had become interested in mathematical biology.〔〔Volterra, V., “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”, ''Mem. Acad. Lincei Roma'', 2, 31–113, (1926)〕〔Volterra, V., ''Variations and fluctuations of the number of individuals in animal species living together'' in Animal Ecology, Chapman, R.N. (ed), McGraw–Hill, (1931)〕 Volterra's enquiry was inspired through his interactions with the marine biologist Umberto D'Ancona who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him as the fishing effort had been very much reduced during the war years. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.〔Kingsland, S., "Modeling Nature: Episodes in the History of Population Ecology", University of Chicago Press, 1995.〕 The model was later extended to include density dependent prey growth and a functional response of the form developed by C.S. Holling; a model that has become known as the Rosenzweig-McArthur model.〔Rosenzweig, M. L., & MacArthur, R. H. (1963). Graphical representation and stability conditions of predator-prey interactions. American Naturalist, 209-223.〕 Both the Lotka–Volterra and Rosenzweig-MacArthur models have been used to explain the dynamics natural populations of predators and prey, such as the lynx and snowshoe hare data of the Hudson Bay Company〔Gilpin, M. E. (1973). Do hares eat lynx?. American Naturalist, 727-730.〕 and the moose and wolf populations in Isle Royale National Park,.〔Jost, C., Devulder, G., Vucetich, J.A., Peterson, R., and Arditi, R., ("The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose" ), ''J. Anim. Ecol.'', 74(5), 809–816 (2005)〕 In the late 1980s an alternative to the Lotka–Volterra predator-prey model (and its common prey dependent generalizations) emerged, the ratio dependent or Arditi–Ginzburg model.〔Arditi, R. and Ginzburg, L.R. (1989) ("Coupling in predator-prey dynamics: ratio dependence" ) ''Journal of Theoretical Biology'', 139: 311–326.〕 The validity of prey or ratio dependent models has been much debated.〔Abrams, P. A., & Ginzburg, L. R. (2000). The nature of predation: prey dependent, ratio dependent or neither?. Trends in Ecology & Evolution, 15(8), 337-341.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lotka–Volterra equations」の詳細全文を読む スポンサード リンク
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