Abstract
One of the key problems related to the Bak-Sneppen evolution model is to compute the limit distribution of the fitnesses in the stationary regime, as the size of the system tends to infinity. Simulations in [3, 1, 4] suggest that the one-dimensional limit marginal distribution is uniform on (p c , 1), for some p c ∼ 0.667. In this paper we define three critical thresholds related to avalanche characteristics. We prove that if these critical thresholds are the same and equal to some p c (we can only prove that two of them are the same) then the limit distribution is the product of uniform distributions on (p c , 1), and moreover p c <0.75. Our proofs are based on a self-similar graphical representation of the avalanches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bak, P.: How Nature Works. New-York: Springer-Verlag, 1996
Bak, P., Sneppen, K.: Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Let. 74, 4083–4086 (1993)
Jensen, H.J.: Self-Organized Criticality. Cambridge: Cambridge Lecture Notes in Physics, 1998
Grassberger, P.: The Bak-Sneppen model for punctuated evolution. Phys. Lett. A 200(3), 277–282 (1995)
Maslov, S.: Infinite hierarchy of exact equations in the Bak-Sneppen model. Phys. Rev. Lett. 77, 1182–1186 (1996)
Moreno, Y., Vazquez: The Bak-Sneppen model on scale-free networks. Europhys. Lett. 57, 765–771 (2002)
Cafiero, R., De los Rios, P., Valleriani, A., Vega, J.L.: Levy-nearest-neighbors Bak-Sneppen model. Phys. Rev. E.60 (2 Part A), R1111-R1114 (1999)
De Los Rios, P., Marsili, M., Vendruscolo, M.: High dimensional Bak-Sneppen model. Phys. Rev. Lett. 80(26), 5746–5749 (2001)
Pis‘mak, Yu.M.: Self-organized criticality in simple model of evolution: exact description of scaling laws. Acta Physica Slovatica 52(6), 525–532 (2002)
de Boer, J., Derrida, B., Flyvbjerg, H., Jackson, A.D., Wettig, T.: Simple model of self-organized biological evolution. Phys. Rev. Lett. 73, 906–909 (1994)
Marsili, M., De Los Rios, P., Maslov, S.: Expansion around the mean-field solution of the Bak-Sneppen model. Phys. Rev. Lett. 80(7), 1457–1460 (1998)
Barbay, J., Kenyon, C.: On the discrete Bak-Sneppen model of self-organized criticality. In: Proceedings of the Twelth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Washington, DC, January 2001
Meester, R., Znamenski, D.: Non-triviality of a discrete Bak-Sneppen evolution model. J. Stat. Phys. 109(516), 987–1004 (2002)
Jovanovic, B., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Punctuated Equilibrium and ‘History Dependent’ percolation. Phys. Rev. E 50, R2403-R2406 (1994)
Meester, R., Znamenski, D.: On the limit behaviour of the Bak-Sneppen evolution model. To appear in Annals of Probability (2002)
Bose, I., Chaudhuri, I.: Bacterial evolution and the Bak-Sneppen model. Int. J. Mod. Phy. C 12(5), 675–683 (2001)
Kovalev, O.V., Pis’mak, Yu. M., Vechernin, V.V.: Self-organized criticality in the model of biological evolution describing interaction of ‘‘Coenophilous’’ and ‘‘Coenophobous’’ species. Europhys. Lett. 40, 471–476 (1997)
Donangelo, R., Fort, H.: A model for mutation in bacterial populations. To appear in Phys. Rev. Lett. (2002)
Cuniberti,G., Valleriani, A., Vega, J. L.: Effects of regulation on a self-organized market. Quantitative Finance 1, 332–338 (2001)
Yamano, T.: Regulation effects on market with Bak-Sneppen model in high dimensions. Int. J. Mod. Phys. C 12(9), 1329–1333 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
Meester, R., Znamenski, D. Critical Thresholds and the Limit Distribution in the Bak-Sneppen Model. Commun. Math. Phys. 246, 63–86 (2004). https://doi.org/10.1007/s00220-004-1044-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1044-4