Jump to ContentJump to Main Navigation

Open Physics

formerly Central European Journal of Physics

1 Issue per year

IMPACT FACTOR increased in 2013: 1.077

SCImago Journal Rank (SJR): 0.455
Source Normalized Impact per Paper (SNIP): 0.724

Open Access
VolumeIssuePage

Issues

Spatially extended populations reproducing logistic map

1Institute of Comuter Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Krakow, Poland

© 2010 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Open Physics. Volume 8, Issue 1, Pages 33–41, ISSN (Online) 2391-5471, DOI: 10.2478/s11534-009-0089-6, November 2009

Publication History

Published Online:
2009-11-15

Abstract

We discuss here the conditions that the spatially extended systems (SES) must satisfy to reproduce the logistic map. To address this dilemma we define a 2-D coupled map lattice with a local rule mimicking the logistic formula. We show that for growth rates of k⩽k ∞ (k ∞ is the accumulation point) the global evolution of the system exactly reproduces the cascade of period doubling bifurcations. However, for k > k ∞, instead of chaotic modes, the cascade of period halving bifurcations is observed. Consequently, the microscopic states at the lattice nodes resynchronize producing dynamically changing spatial patterns. By downscaling the system and assuming intense mobility of individuals over the lattice, the spatial correlations can be destroyed and the local rule remains the only factor deciding the evolution of the whole colony. We found the class of “atomistic” rules for which uncorrelated spatially extended population matches the logistic map both for pre-chaotic and chaotic modes. We concluded that the global logistic behavior can be expected for a spatially extended colony with high mobility of individuals whose microscopic behavior is governed by a specific semi-logistic rule in the closest neighborhood. Conversely, the populations forming dynamically changing spatial clusters behave in a different way than the logistic model and reproduce at least the steady-state fragment of the logistic map.

Keywords: logistic map; spatially extended systems; cellular automata; chaos

PACS: 82.40.Bj; 89.75.Fb; 89.75.Da

  • [1] K. Kaneko, I. Tsuda, Complex Systems: Chaos and beyond (Springer Verlag, Berlin, 2001) 273

  • [2] B. E. Kendall, Theor. Popul. Biol. 54, 11 (1998) http://dx.doi.org/10.1006/tpbi.1998.1365 [CrossRef]

  • [3] A. L. Lloyd, J. Theor. Biol. 173, 217 (1995) http://dx.doi.org/10.1006/jtbi.1995.0058 [CrossRef]

  • [4] R. Law, D. J. Murrell, U. Dieckmann, Ecology 84, 252 (2003) http://dx.doi.org/10.1890/0012-9658(2003)084[0252:PGISAT]2.0.CO;2 [CrossRef]

  • [5] A. Bejan, Shape and Structure, from Engineering to Nature (Cambridge University Press, 2000) 324

  • [6] E. Ben-Jacob, I. Cohen, H. Levine, Adv. Phys. 49, 395 (2000) http://dx.doi.org/10.1080/000187300405228 [CrossRef]

  • [7] I. Cohen, I. Golding, Y. Kozlovsky, E. Ben-Jacob, Fractals 7, 235 (1999) http://dx.doi.org/10.1142/S0218348X99000244 [CrossRef]

  • [8] E. E. Holmes, M. A. Lewis, J. E. Banks, R. R. Veit, Ecology 75, 17 (1994) http://dx.doi.org/10.2307/1939378 [CrossRef]

  • [9] B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems (Cambridge University Press, Cambridge, 1998) 341

  • [10] S. A. Wolfram, New Kind of Science (Wolfram Media Incorporated, 2002) 1263

  • [11] Yang Xin-She, Y. Young, In: S. Olariu, A. Y. Zomaya (Eds.), Handbook of Bioinspired Algorithms and Applications (Chapman & Hall/CRC, Boca Raton, London, New York, 2006) 273

  • [12] W. Dzwinel, D.A. Yuen, Int. J. Mod. Phys. C 16, 357 (2005) http://dx.doi.org/10.1142/S0129183105007182 [CrossRef]

  • [13] K. Krawczyk, W. Dzwinel, D.A. Yuen, Int. J. Mod. Phys. C 14, 1385 (2003) http://dx.doi.org/10.1142/S0129183103006199 [CrossRef]

  • [14] V. Grimm, S. F. Railsback, Individual-Based Modelling and Ecology (Princeton University Press: Princeton, NJ, 2005) 480

  • [15] D. J. Murrell, U. Dieckmann, R. Law, J. Theor. Biol. 229, 421 (2004) http://dx.doi.org/10.1016/j.jtbi.2004.04.013 [CrossRef]

  • [16] S. P. Ellner, J. Theor. Biol. 210, 435 (2001) http://dx.doi.org/10.1006/jtbi.2001.2322 [CrossRef]

  • [17] A. G. Schuster, Deterministic chaos, Polish edition (Wydawnictwo Naukowe PWN, Warszawa, 1993) 274

  • [18] P. J. S. Franks, Limnol. Oceanogr. 42, 2997 (1997)

  • [19] P. Topa. W. Dzwinel, D. A. Yuen, Int. J. Mod. Phys. C 17, 1437, (2006) http://dx.doi.org/10.1142/S0129183106009898 [CrossRef]

  • [20] H. R. Thompson, Ecology 37, 391 (1956) http://dx.doi.org/10.2307/1933159 [CrossRef]

  • [21] K. Kaneko, Physica D 34, 1 (1989) http://dx.doi.org/10.1016/0167-2789(89)90227-3 [CrossRef]

  • [22] G. Pizarro, D. Griffeath, D. R. Noguera, Journal of Environmental Engineering 127, 782 (2001) http://dx.doi.org/10.1061/(ASCE)0733-9372(2001)127:9(782) [CrossRef]

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Jeffrey R. Groff
American Journal of Physics, 2013, Volume 81, Number 10, Page 725
[2]
RAMÓN ALONSO-SANZ
International Journal of Bifurcation and Chaos, 2011, Volume 21, Number 01, Page 101

Comments (0)

Please log in or register to comment.