Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

  • Per Arne RikvoldEmail author


We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.


Evolution Self-optimization Community stability Predator–prey model Mutualism 

Mathematics Subject Classification (2000)

92D15 92D25 60K35 


  1. 1.
    Alonso D. and McKane A.J. (2004). Sampling Hubbell’s neutral theory of biodiversity. Ecol. Lett. 7: 901–10 CrossRefGoogle Scholar
  2. 2.
    Armstrong R.A. and McGehee R. (1980). Competitive exclusion. Am. Nat. 115: 151–70 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bak P. and Sneppen K. (1993). Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71: 4083–086 CrossRefGoogle Scholar
  4. 4.
    Bascompte J., Jordano P. and Olesen J.M. (2006). Asymmetric coevolutionary networks facilitate biodiversity maintenance. Science 312: 431–33 CrossRefGoogle Scholar
  5. 5.
    den Boer P.J. (1986). The present status of the competitive exclusion principle. Trends Ecol. Evol. 1: 25–8 CrossRefGoogle Scholar
  6. 6.
    Bronstein J.L. (1994). Our current understanding of mutualism. Quart. Rev. Biol. 69: 31–1 CrossRefGoogle Scholar
  7. 7.
    Caldarelli G., Higgs P.G. and McKane A.J. (1998). Modelling coevolution in multispecies communities. J. Theor. Biol. 193: 345–58 CrossRefGoogle Scholar
  8. 8.
    Chowdhury, D., Stauffer, D.: Evolutionary ecology in-silico: does mathematical modelling help in understanding the generic trends? J. Biosci. 30, 277–87 (references therein) (2005)Google Scholar
  9. 9.
    Chowdhury D., Stauffer D. and Kunwar A. (2003). Unification of small and large time scales for biological evolution: Deviations from power law. Phys. Rev. Lett. 90: 068101 CrossRefGoogle Scholar
  10. 10.
    Christensen K., di Collobiano S.A., Hall M. and Jensen H.J. (2002). Tangled-nature: a model of evolutionary ecology. J. Theor. Biol. 216: 73–4 CrossRefMathSciNetGoogle Scholar
  11. 11.
    di Collobiano S.A., Christensen K. and Jensen H.J. (2003). The tangled nature model as an evolving quasi-species model. J. Phys. A 36: 883–91 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Crawford J.D. (1991). Introduction to bifurcation theory. Rev. Mod. Phys. 63: 991–037 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Crosby J.L. (1970). The evolution of genetic discontinuity: computer models of the selection of barriers to interbreeding between subspecies. Heredity 25: 253–97 CrossRefGoogle Scholar
  14. 14.
    Doebeli, M., Dieckmann, U.: Evolutionary branching and sympatric speciation caused by different types of ecological interactions. Am. Nat. 156, S77–S101 (references therein) (2000)Google Scholar
  15. 15.
    Dorogovtsev S.N., Mendes J.F.F. and Pogorelov Yu.G. (2000). Bak-Sneppen model near zero dimension. Phys. Rev. E 62: 295–98 CrossRefGoogle Scholar
  16. 16.
    Drossel B., Higgs P.G. and McKane A.J. (2001). The influence of predator–prey population dynamics on the long-term evolution of food web structure. J. Theor. Biol. 208: 91–07 CrossRefGoogle Scholar
  17. 17.
    Drossel B., McKane A. and Quince C. (2004). The impact of non-linear functional responses on the long-term evolution of food web structure. J. Theor. Biol. 229: 539–48 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Dunne J., Williams R.J. and Martinez N.D. (2002). Network structure and diversity loss in food webs: robustness ineases with connectance. Ecol. Lett. 5: 558–67 CrossRefGoogle Scholar
  19. 19.
    Eigen M. (1971). Selforganization of matter and evolution of biological maomolecules. Naturwissenschaften 58: 465 CrossRefGoogle Scholar
  20. 20.
    Eigen M., McCaskill J. and Schuster P. (1988). Molecular quasi-species. J. Phys. Chem. 92: 6881–891 CrossRefGoogle Scholar
  21. 21.
    Garlaschelli, D.: Universality in food webs. Eur. Phys. J. B 38, 277–85 (references therein) (2004)Google Scholar
  22. 22.
    Gavrilets S. (1999). Dynamics of clade diversification on the morphological hypercube. Proc. R. Soc. Lond. B 266: 817–24 CrossRefGoogle Scholar
  23. 23.
    Gavrilets S. (2004). Fitness Landscapes and The Origin of Species. Princeton University Press, Princeton and Oxford Google Scholar
  24. 24.
    Gavrilets S. and Boake C.R.B. (1998). On the evolution of premating isolation after a founder event. Am. Nat. 152: 706–16 CrossRefGoogle Scholar
  25. 25.
    Gavrilets S., Li H. and Vose M.D. (2000). Patterns of parapatric speciation. Evolution 54: 1126–134 Google Scholar
  26. 26.
    Gavrilets S. and Vose A. (2005). Dynamic patterns of adaptive radiation. Proc. Natl. Acad. Sci. USA 102: 18040–8045 CrossRefGoogle Scholar
  27. 27.
    Goldenfeld N. (1992). Lectures on Phase Transitions and the Renormalization Group. Addison–Wesley, Reading, MA Google Scholar
  28. 28.
    Haken H. (1977). Synergetics—An Introduction. Springer, Berlin zbMATHGoogle Scholar
  29. 29.
    Hall M., Christensen K., di Collobiano S.A. and Jensen H.J. (2002). Time-dependent extinction rate and species abundance in a tangled-nature model of biological evolution. Phys. Rev. E 66: 011904 CrossRefGoogle Scholar
  30. 30.
    Hardin G. (1960). The competitive exclusion principle. Science 131: 1292–297 CrossRefGoogle Scholar
  31. 31.
    Hohenberg P.C. and Halperin B. (1977). Theory of dynamic critical phenomena. Rev. Mod. Phys. 49: 435–79 CrossRefGoogle Scholar
  32. 32.
    Hubbell, S.P.: The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton, Chap. 5 (2001)Google Scholar
  33. 33.
    Kauffman S.A. (1993). The Origins of Order. Self-organization and selection in evolution. Oxford University Press, Oxford Google Scholar
  34. 34.
    Kauffman S.A. and Johnsen S. (1991). Coevolution to the edge of chaos: coupled fitness landscapes, poised states and coevolutionary avalanches. J. Theor. Biol. 149: 467–05 CrossRefGoogle Scholar
  35. 35.
    Kawanabe H., Cohen J.E. and Iwasaki K. (1993). Mutualism and Community Organization. Oxford University Press, Oxford Google Scholar
  36. 36.
    Krebs, C.J.: Ecological Methodology. Harper & Row, New York, Chap. 10 (1989)Google Scholar
  37. 37.
    Krebs, C.J.: Ecology. The Experimental Analysis of Distribution and Abundance, 5th edn. Benjamin Cummings, San Francisco, Chaps. 13, 14 (2001)Google Scholar
  38. 38.
    Metz J.A.J., Nisbet R.M. and Geritz S.A.H. (1992). How should we define ‘fitness’for general ecological scenarios?. Trends Ecol. Evol. 7: 198–02 CrossRefGoogle Scholar
  39. 39.
    Murray J.D. (1989). Mathematical Biology. Springer, Berlin zbMATHGoogle Scholar
  40. 40.
    Newman M.E.J. (2005). Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46: 323–51 CrossRefGoogle Scholar
  41. 41.
    Newman M.E.J. and Palmer R.G. (2003). Modeling Extinction. Oxford University Press, Oxford Google Scholar
  42. 42.
    Newman M.E.J. and Sibani P. (1999). Extinction, diversity and survivorship of taxa in the fossil record. Proc. R. Soc. Lond. B 266: 1583–599 Google Scholar
  43. 43.
    Paczuski M., Maslov S. and Bak P. (1996). Avalanche dynamics in evolution, growth, and depinning models. Phys. Rev. E 53: 414–43 CrossRefGoogle Scholar
  44. 44.
    Pathria, R.K.: Statistical Mechanics, 2nd edn. Butterworth-Heinemann, Oxford, Chaps. 11, 14 (1996)Google Scholar
  45. 45.
    Pigolotti S., Flammini A., Marsili M. and Maritan A. (2005). Species lifetime distribution for simple models of ecologies. Proc. Natl. Acad. Sci. USA 102: 15747–5751 zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (1992). Numerical Recipes, 2nd edn. Cambridge University Press, Cambridge Google Scholar
  47. 47.
    Rikvold, P.A., Sevim, V.: An individual-based predator–prey model for biological coevolution: fluctuations, stability, and community structure. Phys. Rev. E E-print arXiv:q-bio.PE/0611023 (in press)Google Scholar
  48. 48.
    Rikvold, P.A.: Complex behavior in simple models of biological coevolution. Int. J. Mod. Phys. C. E-print arXiv:q-bio.PE/0609013 (in press)Google Scholar
  49. 49.
    Rikvold, P.A.: Fluctuations in models of biological maoevolution. In: Kish, L.B., Lindenberg, K., Gingl, Z. (eds.) Noise in Complex Systems and Stochastic Dynamics III, pp. 148–55. SPIE, The International Society for Optical Engineering, Bellingham, WA (E-print arXiv:q-bio.PE/0502046) (2005)Google Scholar
  50. 50.
    Rikvold P.A. and Zia R.K.P. (2003). Punctuated equilibria and 1/f noise in a biological coevolution model with individual-based dynamics. Phys. Rev. E 68: 031913 CrossRefGoogle Scholar
  51. 51.
    Roberts A. (1974). The stability of a feasible random ecosystem. Nature (Lond) 251: 607–08 CrossRefGoogle Scholar
  52. 52.
    Sato K., Ito Y., Yomo T. and Kaneko K. (2003). On the relation between fluctuation and response in biological systems. Proc. Natl. Acad. Sci. USA 100: 14,086–4,090 Google Scholar
  53. 53.
    Sevim V. and Rikvold P.A. (2005). Effects of correlated interactions in a biological coevolution model with individual-based dynamics. J. Phys. A 38: 9475–489 zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–23, 628–56 (1948)Google Scholar
  55. 55.
    Shannon C.E. and Weaver W. (1949). The Mathematical Theory of Communication. University of Illinois Press, Urbana zbMATHGoogle Scholar
  56. 56.
    Solé R.V., Bascompte J. and Manrubia S. (1996). Extinction: bad genes or weak chaos?. Proc. R. Soc. Lond B 263: 1407–413 CrossRefGoogle Scholar
  57. 57.
    Strogatz S.H. (1994). Nonlinear Dynamics and Chaos. Westview Press, Boston Google Scholar
  58. 58.
    Thompson J.N. (1998). Rapid evolution as an ecological process. Trends Ecol. Evol. 13: 329–32 CrossRefGoogle Scholar
  59. 59.
    Thompson J.N. (1999). The evolution of species interactions. Science 284: 2116–118 CrossRefGoogle Scholar
  60. 60.
    Tokita K. and Yasutomi A. (2003). Emergence of a complex and stable network in a model ecosystem with extinction and mutation. Theor. Popul. Biol. 63: 131–46 zbMATHCrossRefGoogle Scholar
  61. 61.
    Verhulst P.F. (1838). Notice sur la loi que la population suit dans son acoissement. Corres. Math. Physique 10: 113–21 Google Scholar
  62. 62.
    Volkov I., Banavar J.R., He F., Hubbell S.P. and Maritan A. (2005). Density dependence explains tree species abundance and diversity in tropical forests. Nature 438: 658–61 CrossRefGoogle Scholar
  63. 63.
    Wills C., Harms K.E., Condit R., King D., Thompson J., He F., Muller-Landau H.C., Ashton P., Losos E., Comita L., Hubbell S., LaFrankie J., Bunyavejchevin S., Dattaraja H.S., Davies S., Esufali S., Foster R., Gunatilleke N., Gunatilleke S., Hall P., Itoh A., John R., Kiratiprayoon S., Massa M., Nath C., NurSupradi Noor M., Kassim A.R., Sukumar R., Suresh H.S., Sun I.F., Tan S., Yamakura T., Zimmerman J. and Lao S.L. (2006). Nonrandom processes maintain diversity in tropical forests. Science 311: 527–31 CrossRefGoogle Scholar
  64. 64.
    Yoshida T., Jones L.E., Ellner S.P., Fussmann G.F. and Hairston N.G. (2003). Rapid evolution drives ecological dynamics in a predator–prey system. Nature 424: 303–06 CrossRefGoogle Scholar
  65. 65.
    Zia R.K.P. and Rikvold P.A. (2004). Fluctuations and correlations in an individual-based model of biological coevolution. J. Phys. A 37: 5135–155 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Computational Science, Center for Materials Research and Technology, National High Magnetic Field Laboratory, and Department of PhysicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Fundamental Sciences, Faculty of Integrated Human StudiesKyoto UniversityKyotoJapan

Personalised recommendations