No Access

STOCHASTICS OF ORDER n IN BIOLOGICAL SYSTEMS: APPLICATIONS TO POPULATION DYNAMICS, THERMODYNAMICS, NONEQUILIBRIUM PHASE AND COMPLEXITY

GUY JUMARIE

Department of Mathematics, University of Quebec at Montreal, P.O. Box 8888, Downtown Station, Montreal, QC H3C 3P8, Canada

Search for more papers by this author

https://doi.org/10.1142/S021833900300083XCited by:9 (Source: Crossref)

Abstract

In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

Invited Paper

Keywords:

References

A. B. Allen and N. Kamiya (eds.) , Primitive Motile Systems in Cell Biology ( Academic Press , New York , 1964 ) . Google Scholar
H. Atlan , L'organisation biologique et la théorie de l'information ( Herman , Paris , 1972 ) . Google Scholar
M. Caputo , Elasticita e Dissipazione ( Zanichelli , Bologna , 1969 ) . Google Scholar
M. M. Djrbashian and A. B. Nersesian, Izv. Acad. Nauk Armjanskoi SSR 3(1), 3 (1968). Google Scholar
L. L. Gatlin, J. Theoret. Biol. 10, 281 (1966), DOI: 10.1016/0022-5193(66)90127-5. Crossref, Web of Science, Google Scholar
L. L. Gatlin, J. Theoret. Biol. 18, 181 (1968), DOI: 10.1016/0022-5193(68)90160-4. Crossref, Web of Science, Google Scholar
I. I. Gikhman and A. W. Skorohod , Stochastic Differential Equations ( Springer Verlag , Berlin , 1972 ) . Crossref, Google Scholar
H. Haken , Advanced Synergetics ( Springer Verlag , Berlin , 1987 ) . Google Scholar
H. Haken , Information and Self-Organisation ( Springer Verlag , Berlin , 1988 ) . Crossref, Google Scholar
G. Hochberg, Ann. Probability 6(3), 433 (1978), DOI: 10.1214/aop/1176995529. Crossref, Web of Science, Google Scholar
E. T. Jaynes, Phys. Rev. 106, 620 (1957), DOI: 10.1103/PhysRev.106.620. Crossref, Web of Science, Google Scholar
G. Jumarie , Relative Information, Theories and Applications ( Springer Verlag , Berlin , 1990 ) . Crossref, Google Scholar
G. Jumarie, J. Biol. Phys. 23(2), 101 (1997), DOI: 10.1023/A:1004932712919. Crossref, Web of Science, Google Scholar
G. Jumarie, Chaos, Solitons and Fractals 3(6), 881 (1998). Google Scholar
G. Jumarie, Chaos, Solitons and Fractals 10(7), 1193 (1999), DOI: 10.1016/S0960-0779(98)00103-9. Crossref, Web of Science, Google Scholar
G. Jumarie , Maximum Entropy, Information Without Probability and Complex Fractals ( Kluwer Academic , Dordrecht , 2000 ) . Crossref, Google Scholar
G. Jumarie, Int. J. Mod. Phys. A 16(31), 5061 (2001), DOI: 10.1142/S0217751X01005468. Link, Web of Science, Google Scholar
E. Pinel , Les fondements de la biologie mathématique non statistique. Biométrie leucocytaire ( Maloine , Paris , 1973 ) . Google Scholar
E. Pinel , La relativité en biologie ( Maloine , Paris , 1975 ) . Google Scholar
R. W. Poole , Introduction to Quantitative Ecology ( McGraw Hill , New York , 1974 ) . Google Scholar
H. Risken , The Fokker–Planck Equation ( Springer Verlag , Berlin , 1984 ) . Crossref, Google Scholar
H. G. Shuster , Deterministic Chaos: An Introduction ( Physik Verlag , Weihhein , 1984 ) . Google Scholar
Yu. M. Svirezhev and D. O. Logofet , Stability of Biological Community ( Mir Publ. , Moscow , 1983 ) . Google Scholar