Journal of the Korean Society for Industrial and Applied Mathematics

  • 제16권4호
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  • Pages.249-256
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  • 2012
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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PATTERN FORMATION FOR A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH CROSS DIFFUSION

  • Sambath, M. (DEPARTMENT OF MATHEMATICS, BHARATHIAR UNIVERSITY) ;
  • Balachandran, K. (DEPARTMENT OF MATHEMATICS, BHARATHIAR UNIVERSITY)
  • 투고 : 2012.09.27
  • 심사 : 2012.12.07
  • 발행 : 2012.12.25
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초록

In this work, we analyze the spatial patterns of a predator-prey system with cross diffusion. First we get the critical lines of Hopf and Turing bifurcations in a spatial domain by using mathematical theory. More specifically, the exact Turing region is given in a two parameter space. Our results reveal that cross diffusion can induce stationary patterns which may be useful in understanding the dynamics of the real ecosystems better.

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참고문헌

  1. D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology. 83 (2002), 28-34. https://doi.org/10.1890/0012-9658(2002)083[0028:MIBPCG]2.0.CO;2
  2. M. Banerjee, Self-replication of spatial patterns in a ratio-dependent predator-prey model, Math. Comput. Modelling. 51 (2010), 44-52. https://doi.org/10.1016/j.mcm.2009.07.015
  3. F. Bartumeus, D. Alonso and J. Catalan, Self-organized spatial structures in a ratio-dependent predator-prey model, Physica A. 295 (2001), 53-57. https://doi.org/10.1016/S0378-4371(01)00051-6
  4. D.L. Benson, J. Sherratt and P.K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull.Math.Biol. 55 (1993), 365-384. https://doi.org/10.1007/BF02460888
  5. J.M. Chung and E. Peacock-Lpez, Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion, J. Chem. Phys. 127 (2007), 174903. https://doi.org/10.1063/1.2784554
  6. J.M. Chung and E. Peacock-Lpez, Cross-diffusion in the templator model of chemical self-replication, Phys. Let. A. 371 (2007), 41-47. https://doi.org/10.1016/j.physleta.2007.04.114
  7. D.L. DeAngelis, R.L. Goldstein and R.V. O'Neill, A model for trophic interaction, Ecology. 56 (1975), 881-892. https://doi.org/10.2307/1936298
  8. B. Dubey , B. Das and J. Hussain, A predator - prey interaction model with self and cross-diffusion, Ecol. Model. 141 (2001), 67-76. https://doi.org/10.1016/S0304-3800(01)00255-1
  9. Y.H. Fan and W.T. Li, Global asymptotic stability of a ratio-dependent predator prey system with diffusion, J.Comput.Appl.Math. 188 (2006), 205-227. https://doi.org/10.1016/j.cam.2005.04.007
  10. M.R. Garvie, Finite-Difference schemes for reaction diffusion equations modelling predator prey interactions in MATLAB, Bull.Math.Biol. 69 (2007), 931-956. https://doi.org/10.1007/s11538-006-9062-3
  11. C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey model, Bull.Math.Biol. 61 (1999), 19-32. https://doi.org/10.1006/bulm.1998.0072
  12. S.A. Levin, T.M. Powell and J.H. Steele, Patch Dynamics, Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1993.
  13. P.P. Liu and Z. Jin, Pattern formation of a predator-prey model, Nonlinear Anal. Hybrid Syst. 3 (2009), 177-183. https://doi.org/10.1016/j.nahs.2008.12.004
  14. Y. Lou andW. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations. 154 (1999), 157-190 . https://doi.org/10.1006/jdeq.1998.3559
  15. E.A. McGehee and E. Peacock-Lpez, Turing patterns in a modified Lotka Volterra model, Phys. Lett. A. 342 (2005), 90-98. https://doi.org/10.1016/j.physleta.2005.04.098
  16. A. Morozov, S.Petrovskii and B.L. Li, Bifurcations and chaos in a predator-prey system with the allee effect, Proc. R. Soc. Lond. B 271 (2004), 1407-1414. https://doi.org/10.1098/rspb.2004.2733
  17. J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
  18. M. Sambath, S. Gnanavel and K. Balachandran, Stability and Hopf bifurcation of a diffusive predator-prey model with predator saturation and competition, Anal. Appl. 1-18 (2012) DOI:10.1080/00036811.2012.742185
  19. J.A. Sherratt, Turing bifurcations with a temporally varying diffusion coefficient, J.Math.Biol. 33 (1995), 295-308.
  20. X.K. Sun, H.F. Huo and H. Xiang, Bifurcation and stability analysis in predator-prey model with a stagestructure for predator, Nonlinear Dyn. 58 (2009), 497-513. https://doi.org/10.1007/s11071-009-9495-y
  21. A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B: Biol. Sci. 237 (1952), 37-72. https://doi.org/10.1098/rstb.1952.0012

피인용 문헌

  1. SPATIOTEMPORAL DYNAMICS OF A PREDATOR-PREY MODEL INCORPORATING A PREY REFUGE vol.3, pp.1, 2012, https://doi.org/10.11948/2013006