The European Physical Journal B

, Volume 66, Issue 1, pp 97–106 | Cite as

Dissipative oscillations in spatially restricted ecosystems due to long range migration

Interdisciplinary Physics

Abstract

An ecosystem containing three interacting species is studied using both Mean Field approach and Kinetic Monte Carlo simulations on a lattice substrate. The so called 3rd order LLV model involves birth, death and reaction processes with 3rd order nonlinearities and feedbacks. At the mean field level this system exhibits conservative oscillations; the analytic form of the constant of motion is presented. The stochastic simulations show that the density oscillations disappear for sufficiently large lattices, while they are present locally, on small lattice windows. Introduction of mixing via long range migration in the two reacting species changes this picture. For small migration rates p, the behavior remains as with p = 0 and the system is divided into local asynchronous oscillators. As p increases the system passes through a phase transition and exhibits a weak disorder limit cycle through a supercritical Hopf-like bifurcation. The amplitude of the limit cycle depends on the rate p, on the range of migration r and on the system kinetic rates k1, k2 and k3.

PACS

82.40.Bj Oscillations, chaos, and bifurcations 05.45.Xt Synchronization; coupled oscillators 92.20.jp Ecosysystems, structure, dynamics and modeling 02.70.Uu Applications of Monte Carlo methods 05.65.+b Self-organized systems 05.45.-a Nonlinear dynamics and chaos 

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References

  1. A.J. Lotka, Proc. Natl. Acad. Sci. USA 6, 410 (1920) Google Scholar
  2. V. Volterra, Variation and fluctuation of a number of individuals in animal species living together, Translation In: R.N. Chapman: Animal Ecology (McGraw Hill, New York, 1931), pp. 409–448 Google Scholar
  3. J.D. Murray, Mathematical Biology (Springer, 1993) Google Scholar
  4. R.M. May, Stability and Complexity in Model Ecosystems (Cambridge University Press, Cambridge, 1974) Google Scholar
  5. R.M. May, Nature 269, 471 (1976) Google Scholar
  6. L. Stone, Nature 365, 617 (1993) Google Scholar
  7. B. Blasius, A. Huppert, L. Stone, Nature 399, 354 (1999) Google Scholar
  8. B. Blasius, R. Neff, F. Beck, U. Luttge, Proc. R. Soc. London B 266, 93 (1999) Google Scholar
  9. D.J. Murrell, American Naturalist 166, 354 (2005) Google Scholar
  10. F. Saffre, J.L. Deneubourg, J. Theor. Biol. 214, 441 (2002) Google Scholar
  11. J.L. Deneubourg, A. Lioni, C. Detrain, Biol. Bull. 202, 262 (2002) Google Scholar
  12. G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977) Google Scholar
  13. G. Ertl, R. Norton, J. Rustig, Phys. Rev. Lett. 49, 117 (1982) Google Scholar
  14. G. Ertl, Science 254, 1750 (1991) Google Scholar
  15. A. Provata, G. Nicolis, F. Baras, J. Chem. Phys. 110, 8361 (1999) Google Scholar
  16. L. Frachebourg, P.L. Krapivsky, E. Ben-Naim, Phys. Rev. E 54, 6186 (1996) Google Scholar
  17. G.A. Tsekouras, A. Provata, Phys. Rev. E 65, 056602 (2001) Google Scholar
  18. A. Provata, G.A. Tsekouras, Phys. Rev. E 67, 056602 (2003) Google Scholar
  19. A. Tretyakov, A. Provata, G. Nicolis, J. Phys. Chem. 99, 2770 (1995) Google Scholar
  20. R. Imbihl, G. Ertl, Chem. Rev. 95, 697 (1995) Google Scholar
  21. V.P. Zhdanov, Phys. Rev. E 59, 6292 (1999) Google Scholar
  22. H. Rose, H. Hempel, L. Schimanksy-Geier, Physica A 206, 421 (1994) Google Scholar
  23. G. Nicolis, I. Prigogine, Exploring Complexity (Freeman, New York, 1989) Google Scholar
  24. G. Nicolis, Introduction to Nonlinear Science (Cambridge University Press, Cambridge, 1995) Google Scholar
  25. E. Ben-Jacob, I. Cohena, I. Goldinga, D.L. Gutnickb, M. Tcherpakovb, D. Helbinga, I.G. Rona, Physica A 282, 247 (2000) Google Scholar
  26. E. Ben-Jacob, H. Levine, Nature 409, 985 (2001) Google Scholar
  27. T. Reichenbach, M. Mobilia, E. Frey, Phys. Rev. E 74, 051907 (2006) Google Scholar
  28. D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977) Google Scholar
  29. R.M. Ziff, E. Gulari, Y. Barshad, Phys. Rev. Lett. 56, 2553 (1968) Google Scholar
  30. E.V. Albano, Comp. Chem. Phys. 113, 10279 (2000) Google Scholar
  31. A. Efimov, A. Shabunin, A. Provata (submitted) Google Scholar
  32. W.G. Wilson, E. McCauley, A.M. De Roos, Bull. Math. Biol. 57, 507 (1995) Google Scholar
  33. A.M. De Roos, E. McCauley, W.G. Wilson, Theor. Pop. Biol. 53, 108 (1998) Google Scholar
  34. G. Szabó, A. Szolnoki, R. Izsák, J. Phys. A: Math. Gen. 37, 2599 (2004) Google Scholar
  35. D.H. Zanette, Phys. Rev. E 64, 050901 (2001) Google Scholar
  36. M. Kuperman, G. Abramson, Phys. Rev. Lett. 86, 2909 (2001) Google Scholar
  37. D.H. Zanette, M. Kuperman, Physica A 309, 445 (2002) Google Scholar
  38. S.H. Strogatz, Non-linear Dynamics and Chaos (West-view, New York, 1994); A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization (Cambridge University Press, Cambridge, 2001) Google Scholar
  39. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76, 1804 (1996) Google Scholar
  40. A. Shabunin, V. Astakov, J. Kurths, Phys. Rev. E 72, 016218 (2005) Google Scholar
  41. A. Neiman, A. Silchenko, V. Anishchenko, L. Schimansky-Geier Phys. Rev. E 58, 7118 (1998) Google Scholar
  42. K. Wood, C. Van de Broeck, R. Kawai, K. Lindenberg, Phys. Rev. Lett. 96, 145701 (2006); K. Wood, C. Van de Broeck, R. Kawai, K. Lindenberg, Phys. Rev. E 75, 041107 (2007) Google Scholar
  43. A. Shabunin, V. Astakhov, V. Demidov, A. Provata, F. Baras, G. Nicolis, V. Anishchenko, Chaos, Solitons and Fractals 15, 395 (2003) Google Scholar
  44. S. Clar, B. Drossel, F. Schwabl, J. Phys.: Cond. Mat. 8, 6803 (1996) Google Scholar
  45. K. Schenk, B. Drossel, F. Schwabl, Phys. Rev. E 65, 026135 (2002) Google Scholar
  46. N. Kouvaris, A. Provata, Nonlinear Phenomena in Complex Systems 11, 259 (2008) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Institute of Physical ChemistryAthensGreece
  2. 2.Department of Mathematical, Physical and Computational ScienceFaculty of Engineering, Aristotle University of ThessalonikiThessalonikiGreece

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