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Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics

Gorban, A.N. (2004) Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics. [Preprint]

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Abstract

If we find a representation of an infinite-dimensional dynamical system as a nonlinear kinetic system with {\it conservation of supports} of distributions, then (after some additional technical steps) we can state that the asymptotics is finite-dimensional. This conservation of support has a {\it quasi-biological interpretation, inheritance} (if a gene was not presented initially in a isolated population without mutations, then it cannot appear at later time). These quasi-biological models can describe various physical, chemical, and, of course, biological systems. The finite-dimensional asymptotic demonstrates effects of {\it ``natural" selection}. The estimations of asymptotic dimension are presented. The support of an individual limit distribution is almost always small. But the union of such supports can be the whole space even for one solution. Possible are such situations: a solution is a finite set of narrow peaks getting in time more and more narrow, moving slower and slower. It is possible that these peaks do not tend to fixed positions, rather they continue moving, and the path covered tends to infinity at $t \rightarrow \infty$. The {\it drift equations} for peaks motion are obtained. Various types of stability are studied. In example, models of cell division self-synchronization are studied. The appropriate construction of notion of typicalness in infinite-dimensional spaces is discussed, and the ``completely thin" sets are introduced.

Item Type:Preprint
Keywords:Selection, Limit sets, Kinetic equations, Self-synchronization, Finite-dimensional asymptotic
Subjects:Computer Science > Dynamical Systems
ID Code:3815
Deposited By: Gorban, Prof Alexander N.
Deposited On:09 Sep 2004
Last Modified:11 Mar 2011 08:55

References in Article

Select the SEEK icon to attempt to find the referenced article. If it does not appear to be in cogprints you will be forwarded to the paracite service. Poorly formated references will probably not work.

1. Darwin, Ch., On the origin of species by means of natural selection, or preservation of favoured races in the struggle for life: A Facsimile of the First Edition, Harvard, 1964.

http://www.literature.org/authors/darwin-charles/the-origin-of-species/

2. Haldane, J.B.S., The Causes of Evolution, Princeton Science Library, Princeton University Press, 1990.

3. Mayr, E., Animal Species and Evolution. Cambridge, MA: Harvard University Press, 1963.

4. Ewens, W.J., Mathematical Population Genetics. Springer-Verlag, Berlin, 1979.

5. Rozonoer, L.I., Sedyh, E.I., On the mechanisms of of evolution of self-reproduction systems, 1,

Automation and Remote Control, {\bf 40}, 2 (1979), 243--251; 2,ibid., {\bf 40}, 3 (1979), 419--429; 3, ibid, {\bf 40}, 5 (1979), 741-749.

6. Gorban, A.N., Dynamical systems with inheritance, in: Some problems of community dynamics, R.G. Khlebopros (ed.), Institute of Physics RAS, Siberian Branch, Krasnoyarsk, 1980 [in Russian].

7. Gorban, A.\ N., Equilibrium encircling. Equations of chemical kinetics and their

thermodynamic analysis, Nauka, Novosibirsk, 1984 [in Russian].

8. Semevsky, F.N., Semenov S.M., Mathematical modeling of ecological processes, Gidrometeoizdat, Leningrad, 1982 [in Russian].

9. Gorban, A.N., Khlebopros, R.G., Demon of Darwin: Idea of optimality and natural selection, Nauka (FizMatGiz), Moscow, 1988 [in Russian].

10. Zakharov, V.E., L'vov, V.S., Starobinets, S.S.,

Turbulence of spin-waves beyond threshold of their

parametric-excitation, Uspekhi Fizicheskikh Nauk {\bf 114}, 4 (1974), 609--654; English translation Sov. Phys. - Usp. {\bf 17}, 6 (1975), 896--919.

11. Zakharov, V.E., L'vov, V.S., Falkovich, G.E., Kolmogorov spectra of turbulence, vol. 1 Wave Turbulence. Springer, Berlin, 1992.

12. L'vov, V.S., Wave turbulence under parametric excitation applications to magnets, Springer, Berlin, Heidelberg, 1994.

13. Ezersky A.B., Rabinovich M.I., Nonlinear-wave competition and anisotropic spectra of spatiotemporal chaos of Faraday ripples, Europhysics Letters {\bf 13}, 3 (1990), 243--249.

14. Krawiecki, A, Sukiennicki, A., Marginal synchronization of spin-wave amplitudes in a model for chaos in parallel pumping, Physica Status Solidi B--Basic Research {\bf 236}, 2 (2003), 511--514.

15. Vorobev, V.M., Selection of normal variables for unstable conservative media, Zhurnal Tekhnicheskoi Fiziki, {\bf 62}, 8 (1992), 172--175.

16. Seminozhenko, V.P., Kinetics of interacting quasiparticles in strong external fields. Phys. Reports, {\bf 91}, 3 (1982), 103--182.

17. Haken, H., Synergetics, an introduction. Nonequilibrium phase transitions and self--organization in

physics, chemistry and biology, Springer, Berlin, Heidelberg, New York, 1978.

18. Bourbaki, N., Elements of mathematics - Integration I, Springer, Berlin, Heidelberg, New York, 2003.

19. Smale, S., Structurally stable systems are not dense,

Amer. J. Math., {\bf 88} (1966), 491--496.

20. Birkhoff, G.D., Dynamical systems, AMS Colloquium Publications, Providence, 1927. Online:

http://www.ams.org/online \_bks/coll9/

21. Hasselblatt, B., Katok, A. (Eds.), Handbook of Dynamical Systems, Vol. 1A, Elsevier, 2002.

22. Katok, A., Hasselblat, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math. and its Applications, Vol. 54, Cambridge University Press, 1995.

23. Levin, L.A., Randomness conservation inequalities; Information and independence in mathematical theories, Information and Control, {\bf 61}, 1 (1984), 15--37.

24. Gause, G.F., The struggle for existence, Williams and Wilkins, Baltimore, 1934.

Online: http://www.ggause.com/Contgau.htm.

25. Volterra, V., Lecons sur la th\'eorie mathematique de la lutte pour la vie, Gauthier-Villars, Paris, 1931.

26. Gromov, M., A dynamical model for synchronisation and for inheritance in micro-evolution: a survey of papers of A.Gorban, The talk given in the IHES seminar, ``Initiation to functional genomics: biological, mathematical and algorithmical aspects", Institut Henri Poincar\'e, November 16, 2000.

27. Kuzenkov, O.A., Weak solutions of the Cauchy problem in the set of Radon probability measures, Differential Equations, {\bf 36}, 11 (2000), 1676--1684.

28. Kuzenkov, O.A., A dynamical system on the set of Radon probability measures, Differential Equations, {\bf 31}, 4 (1995), 549--554.

29. Diekmann, O. A beginner's guide to adaptive dynamics, in: Mathematical modelling of population dynamics, Banach Center Publications, V.~63, Institute of Mathematics Polish Academy of Sciences, Warszawa, 2004, 47--86.

30. Blekhman, I.I., Synchronization in science and technology. ASME Press, N.Y., 1988.

31. Pikovsky, A., Rosenblum, M., Kurths, J., Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, 2002.

32. Josi\'c, K., Synchronization of chaotic systems and invariant manifolds, Nonlinearity 13 (2000)

1321-1336.

33. Mosekilde, E., Maistrenko, Yu., Postnov, D.,

Chaotic synchronization: Applications to living systems, World Scientific, Singapore, 2002.

34. Cooper, S., Minimally disturbed, multi-cycle, and reproducible synchrony using a eukaryotic ``baby machine", Bioessays {\bf 24} (2002), 499--501.

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