Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

Mathematical Biosciences

Volumn 216, Issue 1, 2008, Pages 90-99

  Erguler, Kamil ^a   Stumpf, Michael P H ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Circadian clock; Population dynamics; Quadratic map; Stochastic systems

Indexed keywords

ABSORBING STATES; BIOLOGICAL MECHANISMS; CANONICAL MODELS; CIRCADIAN CLOCK; GAUSSIAN NOISES; GENERIC MODELS; INTRINSIC NOISES; LINEAR DYNAMICS; LOGISTIC MAPS; MOLECULAR SPECIES; NOISY FEEDBACKS; QUADRATIC MAP; RELIABLE; SIMPLE SYSTEMS; STATISTICAL INTERPRETATIONS; STOCHASTIC; STOCHASTIC DYNAMICAL SYSTEMS; STOCHASTIC SYSTEMS;

CONTROL THEORY; DYNAMICAL SYSTEMS; DYNAMICS; FEEDBACK; GAUSSIAN NOISE (ELECTRONIC); MAPS; POPULATION STATISTICS; SCHRODINGER EQUATION; STOCHASTIC CONTROL SYSTEMS; STOCHASTIC PROGRAMMING; TRELLIS CODES;

CHAOTIC SYSTEMS;

CANONICAL ANALYSIS; CIRCADIAN RHYTHM; GAUSSIAN METHOD; LOGISTICS; MAP; POPULATION DYNAMICS; POPULATION MODELING; STATISTICAL ANALYSIS; STOCHASTICITY;

ARTICLE; CIRCADIAN RHYTHM; DENSITY FUNCTIONAL THEORY; HISTOGRAM; NOISE; NONLINEAR SYSTEM; STATISTICAL MODEL; STATISTICS; STOCHASTIC MODEL;

DATA INTERPRETATION, STATISTICAL; LOGISTIC MODELS; NONLINEAR DYNAMICS; POPULATION DYNAMICS; STOCHASTIC PROCESSES;

EID: 54249089721     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2008.08.012     Document Type: Article

References (28)

1. May R. Simple mathematical models with very complicated dynamics. Nature 261 (1976) 459
2. Mueller L., and Joshi A. Stability in Model Populations (2000), Princeton University
3. Johnson N., and Kotz S. Urn Models and their Applications: An Approach to Modern Discrete Probability Theory (1977), Wiley, New York
4. May R. Irreproducible results. Nature 262 (1976) 646
5. Li T., and Yorke J. Period three implies chaos. Am. Math. Mon. (1975)
6. Crutchfield J., Farmer J., and Huberman B. Fluctuations and simple chaotic dynamics. Phys. Rev. Lett. 92 2 (1982) 45
7. Rim S., Hwang D., Kim I I., and Kim C. Chaotic transition of random dynamical systems and chaos synchronization by common noises. Phys. Rev. Lett. 85 11 (2000) 2304 ISSN 0031-90Print
8. Elezovic N., Zupanovic V., and Zubrinic D. Box dimension of trajectories of some discrete dynamical systems. Chaos Solitons Fract. 34 (2007) 244 10.1016/j.chaos.2006.03.060
9. U. Tirnakli, C. Beck, C. Tsallis, Central limit behavior of deterministic dynamical systems, Phys. Rev. E 75 (2007), Art. No. 040106.
10. Savi M.A. Effects of randomness on chaos and order of coupled logistic maps. Phys. Lett. A 364 (2007) 389 10.1016/j.physleta.2006.11.095
11. Mayer-Kress G., and Haken H. The influence of noise on the logistic model. J. Stat. Phys. 26 1 (1981) 149
12. Dennis B., Desharnais R., Cushing J., Henson S., and Costantino R. Can noise induce chaos?. OIKOS-Copenhagen 102 (2003) 329
13. Crutchfield J., and Huberman B. Fluctuations and the onset of chaos. Phys. Lett. A 77 (1980) 407
14. Hofbauer J., and Sigmund K. Evolutionary Games and Population Dynamics (1998), Cambridge University, Cambridge, UK
15. Grossmann S., and Thomae S. Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturforsch. A 32 (1977) 1353
16. Jensen R., and Myers C. Images of the critical points of nonlinear maps. Phys. Rev. A 32 (1985) 1222
17. D. Gonze, J. Halloy, J. Leloup, A. Goldbeter, Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour, C. R. Biol. (2003). Available from: .
18. Kanz H., and Schreiber T. Nonlinear Time-Series Analysis (1997), Cambridge University, Cambridge, UK
19. Sornette D. Critical Phenomena in Natural Sciences (2003), Springer
20. Stollenwerk N., and Jansen V. Evolution towards criticality in an epidemiological model for meningococcal disease. Phys. Lett. A 317 (2003) 87
21. Fogedby H., and Jensen M. Weak noise approach to the logistic map. J. Stat. Phys. 121 (2005) 759 10.1007/s10955-005-5457-z
22. Ancel L.W. A quantitative model of the simpson-baldwin effect. J. Theor. Biol. 196 2 (1999) 197
23. Stumpf M., Laidlaw Z., and Jansen V. Herpes viruses hedge their bets. Proc. Natl. Acad. Sci. USA 99 (2002) 15234
24. Kussel E., and Leibler S. Phenotypic diversity, population growth, and information in fluctuating environments. Science 309 (2005) 2075
25. L. Chen, R. Wang, T. Kobayashi, K. Aihara, Dynamics of gene regulatory networks with cell division cycle, Phys. Rev. E 70 (1) (2004), Art. No. 011909.
26. Axelrod R., Axelrod D.E., and Pienta K.J. Evolution of cooperation among tumor cells. Proc. Natl. Acad. Sci. USA 103 36 (2006) 13474
27. Barrio M., Burrage K., Leier A., and Tian T. Oscillatory regulation of hes1: discrete stochastic delay modelling and simulation. PLoS Comput. Biol. (2006)
28. K. Yang, W. Ma, H. Liang, Q. Ouyang, C. Tang, L. Lai, Dynamic simulations on the arachidonic acid metabolic network, PLoS Comput. Biol. 3 (2007) 523-530, Art. No. e55.