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A toy model for life at the “edge of chaos”
Comput.& Graphics,Vol. 20, No. 6, pp. 921-923, 1996 CopyrightQ 1996 Ekvier Science Ltd Printedin GreatBritain. All rights rcmved
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0097-8493/96
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PII: soo97-8493(96)ooo61-1
Chaos & Graphics
A TOY MODEL
FOR LIFE AT THE “EDGE OF CHAOS” N. VANDEWALLE+
and M. AUSLOOS
SUPBAS, Institute of Physics BS, Sart Tilman, University of Li&ge,B-4000 Li%ge,Belgium e-mail: [email protected] this short paper, we &strati the emergenceof complex structures from a simple model of life at the “edge of chaos”. The visualization of the generated patterns provides a good way for understanding the self-organizedcritical dynamics of the model. Copyright 0 1996Elsevier ScienceLtd
Abstract-In
Self-organized criticality (SOC) has received much attention since it was proposed to be a paradigm for the description of a wide variety of dynamical processes [I, 21. This behavior is the tendency of large systems to evolve spontaneously towards a critical state, i.e. a state which presents long-range correlations in space and time (the so-called “edge of chaos”). This behavior is in contrast with classical phase transitions for which the fine tuning of an external parameter (a temperature-like parameter) is needed to get a critical situation. Numerous experimental and numerical evidences of SOC [3] can be found: cellular automata, dynamics of sandpiles [l], earthquakes, vortex motion in type-II high-T, superconductors, invasion percolation, fractal growth, interface pinning, economics [4], . . . Some toy models exhibiting SOC were also proposed for biological evolution [5, 61. Beside the physical interests of the latter SOC models, they can mimic some features of evolution. The aim of this paper is to illustrate the emergence of complex structures from a simple model of life at the edge of chaos. The model considers the growth of trees which look similar to the family trees (the socalled phylogenetic trees) used by paleontologists. On such a branched structure, all the existing species correspond to the leaves of the tree. A random scalar number fj between zero and one is associated to each i-species. This number represents the “fitness” of the species, i.e. the degreeof adaptation of the speciesto its environment as discussedby Wright [7]. At each step of the process,a branchingevent (a speciation) is assumedto take place for the specieshaving the lowestf;: value, i.e. the weakestspeciesof the system. The latter speciationevent leadsto two new species eachreceiving a new random fitnessvalue. Correlations are introduced in the processby assumingthat each speciationalso perturbs the fitnessvalue f/ of other speciesi which are separatedby a distancedg
+ Author for correspondence. 921
lessthan a parameterk from the branching point. The distancedo betweentwo speciesi and i of the tree is definedas the minimumnumber of segments neededto connect thesespecies.The latter j-species receive new random fitness values. The selectionbranching-perturbation processdescribedabove is repeateda desirednumber, t, of timesleadingto the growth of phylogenetic-liketrees. The model is relatively simple but generates complex patterns. Figure 1 presentsthe two-dimensional projection of a resulting tree made of 5000 species for the k = 2 case.Threedifferent stagesof the growth are illustrated. The phylogenetic-like trees generatedare found to be fractal structures. The fractal dimensionof thesetrees,which is Df~l.89 for the k=2 case, seemsto be k-dependent [6]. This fractality is a signatureof criticality in the spatial self-organizationof the tree growth. The processalsoself-organizesthe fitnessdistribution in a step-likedistribution wherethe majority of fitnessvalues are above a thresholdfc which is kdependent. For k=2, f,=0.445f0.001 [8]. This meansthat all speciationstake place through species having a fitnessbelowfc. The formation of such a gapf, allowsus to describethe growth processasan intennittency of avalancheslike for sandpile dynamics[l]. An avalancheis defined as a connected sequenceof activity below fc [5, 61. The avalanche sizes s are distributed on power laws n(s)=:s+ reflectinga critical self-organizationin the dynamics of the tree growth. The value of the exponent ‘5, which is r ~312 in the k= 2 case, seemsto be dependentof the parameter k [6]. A power law avalanchedistribution impliesthat avalanchesof all sizesare generatedincluding catastrophicones. For the particular growth of Fig. 1, about 110successive avalanchesof various sizeshave beenenumerated. This intermittency of avalanchesis similar to the dynamicsof biological evolution. Indeed, bursts of life activity of all sizesappearin the paleontological record [9], the most important one being the Cambrian explosion 600 million years ago. The latter observation is in opposition to the Darwin
Fig. I. The two-dimensional
projection of three growth stages of a fractal phylogenetic-like tree. The number of species is 5000 for the last stage of the growth Each color (from blue to redj represents 12 successive avalanches.
Life at the “edge of chaos”
923
interest. We thank also D.StauBer, P.Bak, K. Sneppen, H. gradualism for which speciation events takes place Flyvbjerg, D. Somettc and L. Peliti for comments and continuously and slowly [lo]. discussions. The color scale from blue to red in Fig. 1 is used for the visualization of the particular growth of the phylogenetic-like trees. Each color represents12 successive avalanches. One should remark that the REFERENcFs growth process jumps from leaves to leaves which 1. Bale, P., Tang, C. and Wiesenfeld, K., Self-organized criticality: an explanation of l/f Noise. Physical Review could be far away from each other. A careful Letters, 1987, 59, 381-384. numerical analysis shows that this “squirrel walk” 2. Bak, P., Tang, C. and Wiesenfeld, K., Self-organized at the end of the tree branchesis a L.&y flight criticality. Physical Review A, 1988, 38, 364-371. characterizing long range correlations in the process, 3. Vandewalle, N. and Ausloos, M., in Annual Reviews of i.e. characterizing the self-organized critical dyComputational Physics, Vol. 3, ed. D. Stauffer. World Scientific, Singapore, 1996, pp. 45-85. namics of the model. Moreover, one should observe Edwards, S. F., Role of theory in the future of in Fig. 1 that someregionsnear the root are no more 4. Technology. Public lecture, MITI,- Tokyo, 1986. visited by the walk. This means that some species are 5. Bak, P. and Sneppen, K., Punctuated equilibrium and screened from the process and are frozen for further criticality in a simple model of evolution. Physical Review Letters, 1993, 71, 4083-4086. evolution in this model. The screening is intimately 6. Vandewalle, N. and Ausloos, M., Self-organized related to SOC [8] and is responsible for the criticality in phylogenetic tree growths. Journal de complexity of the pattern as for e.g. in DiffusionPhysiqu; I (France)~ 1995,5, loll-1025. Limited Aggregation [l 11. 7. Wright, S., Character change, speciation and the higher The model of tree-like evolution has not the taxa. Evolution, 1982, 36,427-443. 8. Vandewalle, N. and Ausloos, M., The screening of ambition to simulate real evolution, but it suggests species in a Darwinistic tree-like model of evolution. interestingly that nature works at the “edge of Physica D, 1996,98,262-270. chaos” [2]. As emphasized here, complex patterns 9. Eldregde, N. and Gould, S. J., Punctuated equilibrium emerge from SOC. Further developmentsconcern prevails. Nature, 1988, 332, 211. the extension of the model in order to test the 10. Darwin, C., The Origin of Species, John Murray, London, 1859. robustness of SOC against more realistic rules, 11. Stanley, H. E., Fractals and multifractals: the interplay including, e.g. extinctions of species [12, 131, correlaof physics and geometry. In Fractal and Disordered tions betweenfitnessvalues,etc. Systems, Vol. 1, eds. A. Bunde and S. Havlin. Springer, Berlin, 1991, pp. l-52. Acknowledgements-‘Lois work was financially supported 12. Vandewalle, N. and Ausloos, M., The robustness of through the ARC (94-99/174) contract of the University of self-organized criticality against extinctions in a treeLiege. Part of this work was also supported through the like model of evolution. Europhysics Letters, 1995, 32, Im&lse Program on Hi&Tc Superconductors ofSSTC 613-618. under contract SU/O2/013. The Belxium Research Funds 13. Kramer, M., Vandewalle, N. and Ausloos, M., Spcciafor Industry and ‘A&ulture (FR%, Brussels) is also tions and extinctions in a self-organized critical model acknowledged. We thank Soumya Pastoret for the choice of of Journal de Physique I (France), _ tree-like ___ ___ evolution. .___ the colors in Fie. - 1., and Marcel Kramer for his continued 6, 599-606, lYY6.
Pergamon
0097-8493/96
$15.00 + 0.00
PII: soo97-8493(96)ooo61-1
Chaos & Graphics
A TOY MODEL
FOR LIFE AT THE “EDGE OF CHAOS” N. VANDEWALLE+
and M. AUSLOOS
SUPBAS, Institute of Physics BS, Sart Tilman, University of Li&ge,B-4000 Li%ge,Belgium e-mail: [email protected] this short paper, we &strati the emergenceof complex structures from a simple model of life at the “edge of chaos”. The visualization of the generated patterns provides a good way for understanding the self-organizedcritical dynamics of the model. Copyright 0 1996Elsevier ScienceLtd
Abstract-In
Self-organized criticality (SOC) has received much attention since it was proposed to be a paradigm for the description of a wide variety of dynamical processes [I, 21. This behavior is the tendency of large systems to evolve spontaneously towards a critical state, i.e. a state which presents long-range correlations in space and time (the so-called “edge of chaos”). This behavior is in contrast with classical phase transitions for which the fine tuning of an external parameter (a temperature-like parameter) is needed to get a critical situation. Numerous experimental and numerical evidences of SOC [3] can be found: cellular automata, dynamics of sandpiles [l], earthquakes, vortex motion in type-II high-T, superconductors, invasion percolation, fractal growth, interface pinning, economics [4], . . . Some toy models exhibiting SOC were also proposed for biological evolution [5, 61. Beside the physical interests of the latter SOC models, they can mimic some features of evolution. The aim of this paper is to illustrate the emergence of complex structures from a simple model of life at the edge of chaos. The model considers the growth of trees which look similar to the family trees (the socalled phylogenetic trees) used by paleontologists. On such a branched structure, all the existing species correspond to the leaves of the tree. A random scalar number fj between zero and one is associated to each i-species. This number represents the “fitness” of the species, i.e. the degreeof adaptation of the speciesto its environment as discussedby Wright [7]. At each step of the process,a branchingevent (a speciation) is assumedto take place for the specieshaving the lowestf;: value, i.e. the weakestspeciesof the system. The latter speciationevent leadsto two new species eachreceiving a new random fitnessvalue. Correlations are introduced in the processby assumingthat each speciationalso perturbs the fitnessvalue f/ of other speciesi which are separatedby a distancedg
+ Author for correspondence. 921
lessthan a parameterk from the branching point. The distancedo betweentwo speciesi and i of the tree is definedas the minimumnumber of segments neededto connect thesespecies.The latter j-species receive new random fitness values. The selectionbranching-perturbation processdescribedabove is repeateda desirednumber, t, of timesleadingto the growth of phylogenetic-liketrees. The model is relatively simple but generates complex patterns. Figure 1 presentsthe two-dimensional projection of a resulting tree made of 5000 species for the k = 2 case.Threedifferent stagesof the growth are illustrated. The phylogenetic-like trees generatedare found to be fractal structures. The fractal dimensionof thesetrees,which is Df~l.89 for the k=2 case, seemsto be k-dependent [6]. This fractality is a signatureof criticality in the spatial self-organizationof the tree growth. The processalsoself-organizesthe fitnessdistribution in a step-likedistribution wherethe majority of fitnessvalues are above a thresholdfc which is kdependent. For k=2, f,=0.445f0.001 [8]. This meansthat all speciationstake place through species having a fitnessbelowfc. The formation of such a gapf, allowsus to describethe growth processasan intennittency of avalancheslike for sandpile dynamics[l]. An avalancheis defined as a connected sequenceof activity below fc [5, 61. The avalanche sizes s are distributed on power laws n(s)=:s+ reflectinga critical self-organizationin the dynamics of the tree growth. The value of the exponent ‘5, which is r ~312 in the k= 2 case, seemsto be dependentof the parameter k [6]. A power law avalanchedistribution impliesthat avalanchesof all sizesare generatedincluding catastrophicones. For the particular growth of Fig. 1, about 110successive avalanchesof various sizeshave beenenumerated. This intermittency of avalanchesis similar to the dynamicsof biological evolution. Indeed, bursts of life activity of all sizesappearin the paleontological record [9], the most important one being the Cambrian explosion 600 million years ago. The latter observation is in opposition to the Darwin
Fig. I. The two-dimensional
projection of three growth stages of a fractal phylogenetic-like tree. The number of species is 5000 for the last stage of the growth Each color (from blue to redj represents 12 successive avalanches.
Life at the “edge of chaos”
923
interest. We thank also D.StauBer, P.Bak, K. Sneppen, H. gradualism for which speciation events takes place Flyvbjerg, D. Somettc and L. Peliti for comments and continuously and slowly [lo]. discussions. The color scale from blue to red in Fig. 1 is used for the visualization of the particular growth of the phylogenetic-like trees. Each color represents12 successive avalanches. One should remark that the REFERENcFs growth process jumps from leaves to leaves which 1. Bale, P., Tang, C. and Wiesenfeld, K., Self-organized criticality: an explanation of l/f Noise. Physical Review could be far away from each other. A careful Letters, 1987, 59, 381-384. numerical analysis shows that this “squirrel walk” 2. Bak, P., Tang, C. and Wiesenfeld, K., Self-organized at the end of the tree branchesis a L.&y flight criticality. Physical Review A, 1988, 38, 364-371. characterizing long range correlations in the process, 3. Vandewalle, N. and Ausloos, M., in Annual Reviews of i.e. characterizing the self-organized critical dyComputational Physics, Vol. 3, ed. D. Stauffer. World Scientific, Singapore, 1996, pp. 45-85. namics of the model. Moreover, one should observe Edwards, S. F., Role of theory in the future of in Fig. 1 that someregionsnear the root are no more 4. Technology. Public lecture, MITI,- Tokyo, 1986. visited by the walk. This means that some species are 5. Bak, P. and Sneppen, K., Punctuated equilibrium and screened from the process and are frozen for further criticality in a simple model of evolution. Physical Review Letters, 1993, 71, 4083-4086. evolution in this model. The screening is intimately 6. Vandewalle, N. and Ausloos, M., Self-organized related to SOC [8] and is responsible for the criticality in phylogenetic tree growths. Journal de complexity of the pattern as for e.g. in DiffusionPhysiqu; I (France)~ 1995,5, loll-1025. Limited Aggregation [l 11. 7. Wright, S., Character change, speciation and the higher The model of tree-like evolution has not the taxa. Evolution, 1982, 36,427-443. 8. Vandewalle, N. and Ausloos, M., The screening of ambition to simulate real evolution, but it suggests species in a Darwinistic tree-like model of evolution. interestingly that nature works at the “edge of Physica D, 1996,98,262-270. chaos” [2]. As emphasized here, complex patterns 9. Eldregde, N. and Gould, S. J., Punctuated equilibrium emerge from SOC. Further developmentsconcern prevails. Nature, 1988, 332, 211. the extension of the model in order to test the 10. Darwin, C., The Origin of Species, John Murray, London, 1859. robustness of SOC against more realistic rules, 11. Stanley, H. E., Fractals and multifractals: the interplay including, e.g. extinctions of species [12, 131, correlaof physics and geometry. In Fractal and Disordered tions betweenfitnessvalues,etc. Systems, Vol. 1, eds. A. Bunde and S. Havlin. Springer, Berlin, 1991, pp. l-52. Acknowledgements-‘Lois work was financially supported 12. Vandewalle, N. and Ausloos, M., The robustness of through the ARC (94-99/174) contract of the University of self-organized criticality against extinctions in a treeLiege. Part of this work was also supported through the like model of evolution. Europhysics Letters, 1995, 32, Im&lse Program on Hi&Tc Superconductors ofSSTC 613-618. under contract SU/O2/013. The Belxium Research Funds 13. Kramer, M., Vandewalle, N. and Ausloos, M., Spcciafor Industry and ‘A&ulture (FR%, Brussels) is also tions and extinctions in a self-organized critical model acknowledged. We thank Soumya Pastoret for the choice of of Journal de Physique I (France), _ tree-like ___ ___ evolution. .___ the colors in Fie. - 1., and Marcel Kramer for his continued 6, 599-606, lYY6.