MIXING AND DIFFUSION EFFECT ON SPATIAL-TEMPORAL DYNAMICS IN STOCHASTIC LOTKA–VOLTERRA SYSTEM WITH DISCRETE PHASE SPACE
Cite this article as:
Efimov А. V., Shabunin А. 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: https://doi.org/10.18500/0869-6632-2009-17-1-57-76
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 75the 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 R.M., Gulari E., Barshad Y. Kinetic phase transitions in irreversible surface-reaction model // Phys. Rev. Lett. 1986. Vol. 56. P. 2553.
2. Albano E.V. and Marro J. Monte Carlo study of the CO-poisoning dynamics in a model for the catalytic oxidation of CO // J. Chem. Phys. 2000. Vol. 113. P. 10279.
3. Tammaro M. and Evans J.W. Chemical diffusivity and wave propagation in surface reactions: lattice-gas model mimicking CO-oxidation with high CO-mobility // J. Chem. Phys. 1998. Vol. 108. P. 762.
4. Liu D.J. and Evans J.W. Symmetry-breaking and percolation transitions in a surface reaction model with superlattice ordering // Phys. Rev. Lett. 2000. Vol. 84. P. 955.
5. De Decker Y., Baras F., Kruse N. and Nicolis G. Modeling the NO+H2 reaction on a Pt field emitter tip: Mean-field analysis and Monte-Carlo simulations // J. Chem. Phys. 2002. Vol. 117. P. 10244.
6. Zhdanov V.P. Surface restructuring, kinetic oscillations and chaos in heterogeneous catalytic reactions // Phys. Rev. E. 1999. Vol. 59. P. 6292.
7. Provata A., Nicolis G. and Baras F. Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model // J. Chem. Phys. 1999. Vol. 110. P. 8361.
8. Tsekouras G.A. and Provata A. Fractal properties of the lattice Lotka-Volterra model // Phys. Rev. E. 2002. Vol. 65. art. no 016204.
9. Shabunin A.V., Efimov A.V., Tsekouras G.A. and Provata A. Scalling, cluster dynamics and complex oscillations in a multispecies lattice Lotka–Volterra model // Physica A. 2005. Vol. 347. P. 117.
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. Vol. 283. P. 52.
11. Antal T., Droz M., Lipowski A. and Odor G. Critical behavior of a lattice prey-predator model // Phys. Rev. E. 2001. Vol. 64. P. 036118.
12. Droz M. and Pekalski A. Different strategies of evolution in a predator-prey system // Physica A. 2001. Vol. 298. P. 545.
13. Satulovsky J.E. and Tome T. Spatial instabilities and local oscillations in a lattice gas Lotka–Volterra model // J. Math. Biology. 1997. Vol. 35. P. 344.
14. Spagnolo B., Cirone M., La Barbera A. and De Pasquale F. Noise-induced effects in population dynamics // J. Phys.: Condensed Matter. 2002. Vol. 14. P. 2247.
15. Ertl G. Oscillatory kinetics and spatiotemporal selforganization in reactions at solid surfaces // Science. 1991. Vol. 254. P. 1750.
16. Ertl G., Norton P.R. and Rustig J. Kinetic oscillations in the platinum-catalyzed oxidation of CO // Phys. Rev. Lett. 1982. Vol. 49. P. 177.
17. Voss C. and Kruse N. Chemical wave propagation and rate oscillations during the NO2/H2 reaction over P t // Ultramicroscopy. 1998. Vol. 73. P. 211.
18. Theraulaz G., Bonabeau E., Nicolis S.C., Sole R.V., Fourcassie V., Blanco S., Fournier R., Jolly J.L., Fernandez P., Grimal A., Dalle P. and Deneubourg J.L. Spatial patterns in ant colonies // Proceedings of National Academy of Sciences USA. 2002. Vol. 99, No 15. P. 9645.
19. Ben-Jacob E., Shochet O., TenenBaum A., Cohen I., Czirok A. and Vicsek T. Generic modelling of cooperative growth patterns in bacterial colonies // Nature. 1994. Vol. 368. P. 46.
20. Deneubourg J.L., Lioni A. and Detrain C. Dynamics of aggregation and emergence of cooperation // Biological Bulletin. 2002. Vol. 202, No 3. P. 262.
21. Saffre F. and Deneubourg J.L. Swarming strategies for cooperative species // J. Theoretical Biology. 2002. Vol. 214, No 3. P. 441.
22. Reichenbach T., Mobilia M. and Frey E. Coexistence versus extinction in the stochastic cyclic Lotka–Volterra model. // Phys. Rev. E. 2006. Vol. 74. P. 051907.
23. Tokita K. Statistical mechanics of relative species abundance // Ecological Informatics. 2006. Vol. 1, No 3. P. 315.
24. Washenberger M.J., Mobilia M. and Tauber U.C. ̈ Influence of local carrying capacity restrictions on stochastic predator-prey models // J. Phys.: Condensed Matter. 2007. Vol. 19. P. 065139.
25. Valenti D., Schimansky-Geier L., Sailer X., Spagnolo B. and Iacomi M. Moment equations in a Lotka–Volterra extended system with time correlated noise // Acta Physica Polonica B. 2007. Vol. 38, No 5. P. 1961.
26. Refael A., Schiffer M. and Shnerb N.M. Amplitude-dependent frequency, desynchronization, and stabilization in noisy metapopulation dynamics // Phys. Rev. L. 2007. Vol. 98. P. 098104.
27. Turing A.M. The chemical basis of morphogenesis // Philosophical Transactions of the Royal Society of London. Series B. Biological Sciences. 1952. Vol. 237, No 641. P. 37.
28. Murray J.D. A pre-pattern formation mechanism for animal coat marking // Journal of Theoretical Biology. 1981. Vol. 88, No 1. P. 161.
29. Murray J.D. and Maini P.K. A new approach то the generation of pattern and form in embryology // Science Progress. 1986. Vol. 70, No 280. Part 4. P. 539.
BibTeX
author = {А. V. Efimov and А. V. Shabunin},
title = {MIXING AND DIFFUSION EFFECT ON SPATIAL-TEMPORAL DYNAMICS IN STOCHASTIC LOTKA–VOLTERRA SYSTEM WITH DISCRETE PHASE SPACE},
year = {2009},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {17},number = {1},
url = {https://old-andjournal.sgu.ru/en/articles/mixing-and-diffusion-effect-on-spatial-temporal-dynamics-in-stochastic-lotka-volterra},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2009-17-1-57-76},pages = {57--76},issn = {0869-6632},
keywords = {spatial structures,synchronization,self-organization.},
abstract = {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 75the 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. }}