Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 15, 2009

Spatially extended populations reproducing logistic map

Witold Dzwinel
From the journal Open Physics


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.

[1] K. Kaneko, I. Tsuda, Complex Systems: Chaos and beyond (Springer Verlag, Berlin, 2001) 273 10.1007/978-3-642-56861-9Search in Google Scholar

[2] B. E. Kendall, Theor. Popul. Biol. 54, 11 (1998) in Google Scholar PubMed

[3] A. L. Lloyd, J. Theor. Biol. 173, 217 (1995) in Google Scholar

[4] R. Law, D. J. Murrell, U. Dieckmann, Ecology 84, 252 (2003)[0252:PGISAT]2.0.CO;2Search in Google Scholar

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

[6] E. Ben-Jacob, I. Cohen, H. Levine, Adv. Phys. 49, 395 (2000) in Google Scholar

[7] I. Cohen, I. Golding, Y. Kozlovsky, E. Ben-Jacob, Fractals 7, 235 (1999) in Google Scholar

[8] E. E. Holmes, M. A. Lewis, J. E. Banks, R. R. Veit, Ecology 75, 17 (1994) in Google Scholar

[9] B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems (Cambridge University Press, Cambridge, 1998) 341 10.1017/CBO9780511549755Search in Google Scholar

[10] S. A. Wolfram, New Kind of Science (Wolfram Media Incorporated, 2002) 1263 Search in Google Scholar

[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 Search in Google Scholar

[12] W. Dzwinel, D.A. Yuen, Int. J. Mod. Phys. C 16, 357 (2005) in Google Scholar

[13] K. Krawczyk, W. Dzwinel, D.A. Yuen, Int. J. Mod. Phys. C 14, 1385 (2003) in Google Scholar

[14] V. Grimm, S. F. Railsback, Individual-Based Modelling and Ecology (Princeton University Press: Princeton, NJ, 2005) 480 10.1515/9781400850624Search in Google Scholar

[15] D. J. Murrell, U. Dieckmann, R. Law, J. Theor. Biol. 229, 421 (2004) in Google Scholar

[16] S. P. Ellner, J. Theor. Biol. 210, 435 (2001) in Google Scholar

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

[18] P. J. S. Franks, Limnol. Oceanogr. 42, 2997 (1997) 10.4319/lo.1997.42.5_part_2.1273Search in Google Scholar

[19] P. Topa. W. Dzwinel, D. A. Yuen, Int. J. Mod. Phys. C 17, 1437, (2006) in Google Scholar

[20] H. R. Thompson, Ecology 37, 391 (1956) in Google Scholar

[21] K. Kaneko, Physica D 34, 1 (1989) in Google Scholar

[22] G. Pizarro, D. Griffeath, D. R. Noguera, Journal of Environmental Engineering 127, 782 (2001) in Google Scholar

Published Online: 2009-11-15
Published in Print: 2010-2-1

© 2010 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Scroll Up Arrow