ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Efimov A. V., Shabunin A. V. Mixing and diffusion effect on spatial-temporal dynamics in stochastic Lotka–Volterra system with discrete phase space. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 1, pp. 57-76. DOI: 10.18500/0869-6632-2009-17-1-57-76

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 209)
Article type: 

Mixing and diffusion effect on spatial-temporal dynamics in stochastic Lotka–Volterra system with discrete phase space

Efimov Anton Viktorovich, Saratov State University
Shabunin Aleksej Vladimirovich, Saratov State University

The influence of two types of diffusion on dynamics of stochastic lattice Lotka– Volterra model is considered in this work. The simulations were carried out by means of Kinetic Monte-Carlo algorithm. It is shown that the local diffusion considerably changes the dynamics of the model and accelerates the interaction processes on the lattice, while the mixing results in appearance of global periodic oscillations. The global oscillations appear due to phenomenon of phase synchronization. Various characteristics of the system and their dependence on parameters have been considered in detail. Submitted results form the basis for further researches of the control possibilities for systems with competitive dynamics. They also demonstrate one of the plausible reasons of species diversity and stability of population dynamics in ecosystems.

  1. Ziff RM, Gulari E, Barshad Y. Kinetic phase transitions in irreversible surface-reaction model. Phys. Rev. Lett. 1986;56(24):2553–2556. DOI: 10.1103/PhysRevLett.56.2553.
  2. Albano EV, Marro J. Monte Carlo study of the CO-poisoning dynamics in a model for the catalytic oxidation of CO. J. Chem. Phys. 2000;113(22):10279–10283. DOI: 10.1063/1.1323508.
  3. Tammaro M, Evans JW. Chemical diffusivity and wave propagation in surface reactions: lattice-gas model mimicking CO-oxidation with high CO-mobility. J. Chem. Phys. 1998;108(2):762. DOI: 10.1063/1.475436.
  4. Liu DJ, Evans JW. Symmetry-breaking and percolation transitions in a surface reaction model with superlattice ordering. Phys. Rev. Lett. 2000;84(5):955–958. DOI: 10.1103/PhysRevLett.84.955.
  5. De Decker Y, Baras F, Kruse N, Nicolis G. Modeling the NO+H2 reaction on a Pt field emitter tip: Mean-field analysis and Monte-Carlo simulations. J. Chem. Phys. 2002;117(22):10244–10257. DOI: 10.1063/1.1518961.
  6. Zhdanov VP. Surface restructuring, kinetic oscillations and chaos in heterogeneous catalytic reactions. Phys. Rev. E. 1999;59:6292–6309.
  7. Provata A, Nicolis G, Baras F. Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model. J. Chem. Phys. 1999;110(17):8361–8368. DOI: 10.1063/1.478746.
  8. Tsekouras GA, Provata A. Fractal properties of the lattice Lotka-Volterra model. Phys. Rev. E. 2002;65(1):016204. DOI: 10.1103/PHYSREVE.65.016204.
  9. Shabunin AV, Efimov AV, Tsekouras GA, Provata A. Scalling, cluster dynamics and complex oscillations in a multispecies lattice Lotka–Volterra model. Physica A. 2005;347:117–136. DOI: 10.1016/J.PHYSA.2004.09.021.
  10. Monetti R, Rozenfeld A, Albano E. Study of interacting particle systems: The transition to the oscillatory behavior of a prey-predator model. Physica A. 2000;283(1):52–58. DOI: 10.1016/S0378-4371(00)00127-8.
  11. Antal T, Droz M, Lipowski A, Odor G. Critical behavior of a lattice prey-predator model. Phys. Rev. E. 2001;64(3):036118. DOI: 10.1103/PhysRevE.64.036118.
  12. Droz M, Pekalski A. Different strategies of evolution in a predator-prey system. Physica A. 2001;298(3-4):545–552. DOI: 10.1016/S0378-4371(01)00271-0.
  13. Satulovsky JE, Tome T. Spatial instabilities and local oscillations in a lattice gas Lotka–Volterra model. J. Math. Biology. 1997;35(3):344–358.
  14. Spagnolo B, Cirone M, La Barbera A, De Pasquale F. Noise-induced effects in population dynamics. J. Phys.: Condensed Matter. 2002;14(9):2247–2255. DOI: 10.1088/0953-8984/14/9/313.
  15. Ertl G. Oscillatory kinetics and spatiotemporal selforganization in reactions at solid surfaces. Science. 1991;254(5039):1750–1755. DOI: 10.1126/science.254.5039.1750.
  16. Ertl G, Norton PR, Rustig J. Kinetic oscillations in the platinum-catalyzed oxidation of CO. Phys. Rev. Lett. 1982;49(2):177–180. DOI: 10.1103/PhysRevLett.49.177.
  17. Voss C, Kruse N. Chemical wave propagation and rate oscillations during the NO2/H2 reaction over Pt. Ultramicroscopy. 1998;73(1-4):211–216. DOI: 10.1016/S0304-3991(97)00158-7.
  18. Theraulaz G, Bonabeau E, Nicolis SC, Sole RV, Fourcassie V, Blanco S, Fournier R, Jolly JL, Fernandez P, Grimal A, Dalle P, Deneubourg JL. Spatial patterns in ant colonies. Proceedings of National Academy of Sciences USA. 2002;99(15):9645–9649. DOI: 10.1073/pnas.152302199.
  19. Ben-Jacob E, Shochet O, TenenBaum A, Cohen I, Czirok A, Vicsek T. Generic modelling of cooperative growth patterns in bacterial colonies, Nature. 1994;368(6466):46–49. DOI: 10.1038/368046a0.
  20. Deneubourg JL, Lioni A, Detrain C. Dynamics of aggregation and emergence of cooperation. Biological Bulletin. 2002;202(3):262–267. DOI: 10.2307/1543477.
  21. Saffre F, Deneubourg JL. Swarming strategies for cooperative species. J. Theoretical Biology. 2002;214(3):441–451. DOI: 10.1006/jtbi.2001.2472.
  22. Reichenbach T, Mobilia M, Frey E. Coexistence versus extinction in the stochastic cyclic Lotka–Volterra model. Phys. Rev. E. 2006;74(5):051907. DOI: 10.1103/PhysRevE.74.051907.
  23. Tokita K. Statistical mechanics of relative species abundance. Ecological Informatics. 2006;1(3):315–324. DOI: 10.1016/j.ecoinf.2005.12.003.
  24. Washenberger MJ, Mobilia M, Tauber UC. Influence of local carrying capacity restrictions on stochastic predator-prey models. J. Phys.: Condensed Matter. 2007;19(6):065139. DOI: 10.1088/0953-8984/19/6/065139.
  25. Valenti D, Schimansky-Geier L, Sailer X, Spagnolo B, Iacomi M. Moment equations in a Lotka–Volterra extended system with time correlated noise. Acta Physica Polonica B. 2007;38(5):1961–1972.
  26. Refael A, Schiffer M, Shnerb NM. Amplitude-dependent frequency, desynchronization, and stabilization in noisy metapopulation dynamics. Phys. Rev. L. 2007;98(9):098104. DOI: 10.1103/PhysRevLett.98.098104.
  27. Turing AM. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B. Biological Sciences. 1952;237(641):37–72.
  28. Murray JD. A pre-pattern formation mechanism for animal coat marking. Journal of Theoretical Biology. 1981;88(1):161–199. DOI: 10.1016/0022-5193(81)90334-9.
  29. Murray JD, Maini PK. A new approach то the generation of pattern and form in embryology. Science Progress. 1986;70(280):539–553.
Short text (in English):
(downloads: 73)