DOI QR코드

DOI QR Code

TURING INSTABILITY IN A PREDATOR-PREY MODEL IN PATCHY SPACE WITH SELF AND CROSS DIFFUSION

  • Aly, Shaban (Department of Mathematics, King Khalid University)
  • Received : 2013.01.23
  • Accepted : 2013.02.28
  • Published : 2013.06.25

Abstract

A spatio-temporal models as systems of ODE which describe two-species Beddington - DeAngelis type predator-prey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one's density, i.e. there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

Keywords

References

  1. Aly S., Bifurcations in a predator-prey model with diffusion and memory, Int. Journal of Bifurcation and Chaos (IJBC), Vol. 16 No. 6(2006)1855-1863. https://doi.org/10.1142/S0218127406015751
  2. Aly S., Farkas M., Bifurcation in a predator-prey model in patchy Environment with diffusion, Nonlinear Analysis: Real World Applications, 5(2004) 519-526. https://doi.org/10.1016/j.nonrwa.2003.11.004
  3. Aly S., Farkas M., Bifurcation in a predator-prey model in patchy environment with cross-diffusion, Annales Univ. Sci. Budapest, 47(2004) 143-153.
  4. Turing A. M. The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B237, (1953) 37-72, reprinted: Bull. Math. Biol. 52 (1990) 153-197.
  5. Dieckmann O., Law R. and Metz J. A. J, The Geometry of Ecological Interaction: Simplifying Spatial Complexity, Cambridge University Press, 2000.
  6. Farkas M. Two ways of modeling cross diffusion, Nonlinear Analysis, TMA., 30 (1997) 1225-1233. https://doi.org/10.1016/S0362-546X(96)00161-7
  7. Farkas M. Dynamical Models in Biology, Academic Press, 2001.
  8. Huang Y., Diekmann O. Interspecific influence on mobility and Turing instability, Bull. Math. Biol. 65 (2003) 143-156. https://doi.org/10.1006/bulm.2002.0328
  9. Murray J. D. Mathematical Biology, Berlin, Springer-Verlag, 1989.
  10. Takeuchi Y. Global Dynamical Properties of Lotka-Volterra system, World Scientific, 1996.