Skip to main content
SpringerLink
Log in

Two competing species in super-diffusive dynamical regimes

  • Interdisciplinary Physics
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract.

The dynamics of two competing species within the framework of the generalized Lotka-Volterra equations, in the presence of multiplicative α-stable Lévy noise sources and a random time dependent interaction parameter, is studied. The species dynamics is characterized by two different dynamical regimes, exclusion of one species and coexistence of both, depending on the values of the interaction parameter, which obeys a Langevin equation with a periodically fluctuating bistable potential and an additive α-stable Lévy noise. The stochastic resonance phenomenon is analyzed for noise sources asymmetrically distributed. Finally, the effects of statistical dependence between multiplicative noise and additive noise on the dynamics of the two species are studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Brillouin, Am. Sci. 37, 554 (1949)

    Google Scholar 

  2. L. von Bertalanffy, Science 111, 23 (1950)

    Article  ADS  Google Scholar 

  3. I. Prigogine, G. Nicolis, Self-Organization in Nonequilibrium Systems (John Wiley, New York, 1977)

  4. F. Mallamace, H.E. Stanley, The Physics of Complex Systems (New Advances and Perspectives), Proceedings of the Intern School of Physics Enrico Fermi, Course CLX, Società Italiana di Fisica (Bologna, and IOS Press, Amsterdam, 2004)

  5. A. Provata, I.M. Sokolov, B. Spagnolo, Eur. Phys. J. B 65, 307 (2008)

    Article  MATH  ADS  Google Scholar 

  6. see also The topical issue on Ecological Complex Systems, A. Provata, I.M. Sokolov, B. Spagnolo, Eur. Phys. J. B 65, 307 (2008)

    Article  MATH  ADS  Google Scholar 

  7. P. Romanczuk, I.D. Couzin, L. Schimansky-Geier, Phys. Rev. Lett. 102, 010602 (2009)

    Article  ADS  Google Scholar 

  8. A.N. Kolmogorov, Mathematische Annalen 104, 415 (1931)

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Lévy, Calcul des Probabilités (Gauthier-Villars, Paris, 1925)

  10. C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2004)

  11. L. Arnold, W. Horsthemke, R. Lefever, Z. Physik B 29, 367 (1978)

    Article  ADS  Google Scholar 

  12. J.R. Krebs, N.B. Davies, Behavioural Ecology: An Evolutionary Approach, 4th edn. (Blackwell, Oxford, 1997)

  13. D. Tilman, P.M. Kareiva, Spatial Ecology: The Role of Space in Population Dynamic and Interspecific Interactions (Princeton University Press, Princeton, 1997)

  14. P. Lévy, Théorie de l’addition des variables aléatoires (Gauthier-Villars, Paris, 1954)

  15. B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Welsey Reading, MA, 1954)

  16. G.M. Viswanathan, V. Afanasyev, S.V. Buldyrev, E.J. Murphy, P.A. Prince, H.E. Stanley, Nature 381, 413 (1996)

    Article  ADS  Google Scholar 

  17. G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. da Luz, E.P. Raposo, H.E. Stanley, Nature 401, 911 (1999)

    Article  ADS  Google Scholar 

  18. A.M. Reynolds, J. Phys. A: Math. Theor. 42, 434006 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  19. F. Bartumeus, F. Peters, S. Pueyo, C. Marrasé, J. Catalan, Proc. Natl. Acad. Sci. 100, 12771 (2003)

    Article  ADS  Google Scholar 

  20. A.M. Reynolds, Phys. Rev. E 78, 011906 (2008)

    Article  ADS  Google Scholar 

  21. A.M. Reynolds, M.A. Frye, PLoS One 2, e354 (2007)

  22. D.W. Sims et al., Nature 451, 1098 (2008)

    Article  ADS  Google Scholar 

  23. A.M. Reynolds, A.D. Smith, R. Menzel, U. Greggers, D.R. Reynolds, J.R. Riley, Ecology 88, 1955 (2007)

    Article  Google Scholar 

  24. J.C. Stout, D. Goulson, Anim. Behav. 62, 183 (2001)

    Article  Google Scholar 

  25. A.M. Reynolds, Europhys. Lett. 79, 30006 (2007)

    Article  ADS  Google Scholar 

  26. D. Boyer et al., Proc. R. Soc. B 273, 1743 (2006)

    Article  Google Scholar 

  27. S. Bertrand, J.M. Burgos, F. Gerlotto, J. Atiquipa, J. Mar. Sci. 62, 477 (2005)

    Google Scholar 

  28. P. Marchal, J.J. Poos, F. Quirijns, Fish. Res. 83, 33 (2007)

    Article  Google Scholar 

  29. A.M. Edwards et al., Nature 449, 1044 (2007)

    Article  ADS  Google Scholar 

  30. D. Boyer, O. Miramontes, G. Ramos-Fernandez, Evidence for biological Lévy flights stands, e-print arXiv:0802.1762v1 (2007)

  31. A.M. Reynolds, Ecology 89, 2357 (2008)

    Article  Google Scholar 

  32. M.A. Lomholt, K. Tal, R. Metzler, J. Klafter, Proc. Natl. Acad. Sci. USA. 105, 11055 (2008)

    Article  ADS  Google Scholar 

  33. H.E. Hurst, Trans. Am. Soc. Civ. Eng. 116, 770 (1951)

    Google Scholar 

  34. J.A. Viecelli, J. Atmos. Sci. 55, 677 (1998)

    Article  ADS  Google Scholar 

  35. P.D. Ditlevsen, Geophys. Res. Lett. 26, 1441 (1999)

    Article  ADS  Google Scholar 

  36. W. Feller, An introduction to Probability Theory and Its Apllications (John Wiley and Sons, Inc., New York, 1971), Vol. 2

  37. R.N. Mantegna, H.E. Stanley, Nature 383, 587 (1996)

    Article  ADS  Google Scholar 

  38. R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000)

  39. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. V.V. Uchaikin, Physics-Uspekhi 46, 821 (2003)

    Article  ADS  Google Scholar 

  41. A.V. Chechkin, V. Yu. Gonchar, J. Klafter, R. Metzler, Adv. Chem. Phys. 133, 439 (2006)

    Google Scholar 

  42. B. Dybiec, E. Gudowska-Nowak, I.M. Sokolov, Phys. Rev. E 76, 041122 (2007)

    Article  ADS  Google Scholar 

  43. A.A. Dubkov, B. Spagnolo, V.V. Uchaikin, Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, 2649 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. A.A. Dubkov, B. Spagnolo, Eur. Phys. J. B 65, 361 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. A.A. Dubkov, A. La Cognata, B. Spagnolo, J. Stat. Mech.: Theory Exp. P01002 (2009)

  46. B. Dybiec, E. Gudowska-Nowak, New J. Phys. 9, 452 (2007)

    Article  ADS  Google Scholar 

  47. B. Dybiec, E. Gudowska-Nowak, J. Stat. Mech.: Theory Exp. P05004 (2009)

  48. L. Zhang, A. Song, J. He, J. Phys. A: Math. Theor. 42, 475003 (2009)

    Article  ADS  Google Scholar 

  49. B. Dybiec, Phys. Rev. E 80, 041111 (2009)

    Article  ADS  Google Scholar 

  50. A. Upadhyayaa, J.P. Rieub, J.A. Glazier, Y. Sawadac, Physica A 293, 549 (2001)

    Article  ADS  Google Scholar 

  51. J. Bascompte, R. Sole, Modelling Spatiotemporal Dynamics in Ecology (Springer, Berlin, 1998)

  52. G. Nicolis, C. Nicolis, Foundations of Complex Systems (World Scientific, Singapore, 2007)

  53. D. Valenti, A. Fiasconaro, B. Spagnolo, Physica A 331, 477 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  54. A.J. Lotka, Proc. Nat. Acad. Sci USA 6, 410 (1920)

    Article  ADS  Google Scholar 

  55. V. Volterra, Mem. R. Accad. Nazionale Lincei, Series 6, 31 (1926)

    Google Scholar 

  56. A.D. Bazykin, Nonlinear dynamics of interacting populations (World Scientific, Singapore, 1998)

  57. J.M.G. Vilar, R.V. Solé, Phys. Rev. Lett. 80, 4099 (1998)

    Article  ADS  Google Scholar 

  58. P. Jung, P. Hänggi, Phys. Rev. A 44, 8032 (1991)

    Article  ADS  Google Scholar 

  59. P. Jung, P. Hänggi, Europhys. Lett. 8, 505 (1989)

    Article  ADS  Google Scholar 

  60. L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)

    Article  ADS  Google Scholar 

  61. R.N. Mantegna, B. Spagnolo, M. Trapanese, Phys. Rev. E 63, 011101 (2001)

    Article  ADS  Google Scholar 

  62. P. Talkner et al., New J. Phys. 7, 14 (2005)

    Article  Google Scholar 

  63. A. Caruso, M.E. Gargano, D. Valenti, A. Fiasconaro, B. Spagnolo, Fluct. Noise Lett. 5, L349 (2005)

    Article  Google Scholar 

  64. D. Valenti, L. Tranchina, M. Brai, A. Caruso, C. Cosentino, B. Spagnolo, Ecol. Model. 213, 449 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica e Tecnologie Relative, Group of Interdisciplinary Physics, Università di Palermo and CNISM-INFM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128, Palermo, Italy

    A. La Cognata, D. Valenti & B. Spagnolo

  2. N.I. Lobachevsky State University, 23 Gargarin Ave., Nizhniy Novgorod, 603950, Russia

    A. A. Dubkov

Authors
  1. A. La Cognata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Valenti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Spagnolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. A. Dubkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. La Cognata, D. Valenti or B. Spagnolo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

La Cognata, A., Valenti, D., Spagnolo, B. et al. Two competing species in super-diffusive dynamical regimes. Eur. Phys. J. B 77, 273–279 (2010). https://doi.org/10.1140/epjb/e2010-00239-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-00239-6

Keywords

Search

Navigation