For citation:
Pankratova I. N. Representation of many-group population model as one-species population model with many parameters. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 6, pp. 135-142. DOI: 10.18500/0869-6632-2005-13-5-135-142
Representation of many-group population model as one-species population model with many parameters
We propose a dynamic system determined by a many-dimensional logistic map as a variant of a nonlinear model for dynamics of a biological many-group population. In some parts of a compact phase space the map displays a behavior which is atypical for a oneparameter one-dimensional logistic map. For a many-group population model it means stepwise changes of a total population density and densities of population age groups. We have an opportunity of getting a total population age groups changing periodically with the same period in many various parts of a phase space. A mechanism of dynamics originated in this manner for such a many-group population is discussed.
- Leslie PH. On the use of matrices in certain population mathematics. Biometrika. 1945;33(3):183–212. DOI: 10.2307/2332297.
- Geramita JM, Pullman MJ. An introduction to the application of nonnegative matrices to biological systems. In: Queen’s Papers in Pure and Applied Mathematics. No. 68. Kingston, Ontario, Canada: Queen’s Univ.; 1984.
- Caswell H. Matrix Population Models: Construction, Analysis and Interpretation. Sunderland, Massachusettes, USA: Sunauer Associates Inc.; 1989. 328 p.
- Logofet DO. Once more on the nonlinear Leslie model: the asymptotic behavior of trajectories in primitive and imprimitive cases. Dokl. Math. 1991;43(3):861–865.
- Pankratova IN, Rakhimberdiev MI. On the limit sets of a system of discrete equations with scalar nonlinearity. Bulletin of the National Academy of Sciences of the Republic of Kazakhstan. Physical and Mathematical Series. 1993;(5):56 (in Russian).
- Pankratova IN. On limit sets of a multidimensional analogue of a nonlinear logistic difference equation. Differential Equations. 1996;32(7):1006–1008.
- Sharkovsky AN, Kolyada SF, Sivak AG, Fedorenko VV. Dynamics of One-Dimensional Maps. Dordrecht: Springer; 1997. 262 p. DOI: 10.1007/978-94-015-8897-3.
- Feigenbaum M. Universality in the behavior of nonlinear systems. Sov. Phys. Usp. 1983;141(2):343–374 (in Russian). DOI: 10.3367/UFNr.0141.198310e.0343.
- Pankratova IN. Dynamic properties of a multidimensional analogue of a nonlinear logistic difference equation for typical cases of one-parameter dynamics. Bulletin of the National Academy of Sciences of the Republic of Kazakhstan. Physical and Mathematical Series. 2001;(5):55 (in Russian).
- Pankratova IN. One-dimensional representations of a multidimensional analogue of a nonlinear logistic difference equation. Kazakh Mathematical Journal. 2004;4(1):62 (in Russian).
- Pankratova IN. Reduction of a multidimensional analogue of a nonlinear logistic difference equation to a one-dimensional one. Differential Equations. 2004;40(11):1514–1515 (in Russian).
- Poston T, Stewart I. Catastrophe Theory and Its Applications. New York: Dover Publications; 1978. 479 p.
- Gantmacher FR. The Theory of Matrices. New York: Chelsea Publishing; 1959. 374 p.
- Pankratova IN, Rakhimberdiev MI. Canonical form of multidimensional analogue of nonlinear logistic difference equation. Kazakh Mathematical Journal. 2003;3(1):54 (in Russian).
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