Advertisement

Turing Patterns in a Cross Diffusive System

  • Nishith Mohan
  • Nitu KumariEmail author
Conference paper
  • 49 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 307)

Abstract

In this paper we investigate the role of cross diffusion in pattern formation for a tritrophic food chain model. In the formulated model the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley Martin functional response. We have proved that the stationary uniform solution of the system is stable in the presence of diffusion and absence of cross diffusion but unstable in the presence of cross diffusion. Moreover we carry out numerical simulations to understand the Turing pattern formation for various self and cross diffusivity coefficients of the top level predator.

Notes

Acknowledgements

This work has further been extended and published in [24].

References

  1. 1.
    Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predatorprey system. J. Math. Biol. 36(4), 389–406 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L.A. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology (Cambridge University Press, Cambridge, 1984)Google Scholar
  3. 3.
    P.W. Price et al., Interactions among three trophic levels: influence of plants on interactions between insect herbivores and natural enemies. Annu. Rev. Ecol. Syst. 11(1), 41–65 (1980)CrossRefGoogle Scholar
  4. 4.
    A. Hastings, T. Powell, Chaos in a three-species food chain. Ecology 72(3), 896–903 (1991)CrossRefGoogle Scholar
  5. 5.
    K. McCann, P. Yodzis, Biological conditions for chaos in a three-species food chain. Ecology 75(2), 561–564 (1994)CrossRefGoogle Scholar
  6. 6.
    S. Gakkhar, R.K. Naji, Chaos in three species ratio dependent food chain. Chaos Solitons Fractals 14(5), 771–778 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Rai, R.K. Upadhyay, Chaotic population dynamics and biology of the top-predator. Chaos Solitons Fractals 21(5), 1195–1204 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    P.H. Crowley, E.K. Martin, Functional responses and interference within and between year classes of a dragonfly population. J. N. Am. Benthol. Soc. 8(3), 211–221 (1989)CrossRefGoogle Scholar
  9. 9.
    R.K. Upadhyay, R.K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response. Chaos Solitons Fractals 42(3), 1337–1346 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G.T. Skalski, J.F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092 (2001)CrossRefGoogle Scholar
  11. 11.
    Y. Dong et al., Qualitative analysis of a predator-prey model with crowley-martin functional response. Int. J. Bifurc. Chaos 25(09), 1550110 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.M. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B: Biol. Sci. 237(641), 37–72 (1952)Google Scholar
  13. 13.
    S. Kondo, T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329(5999), 1616–1620 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    S. Kondo, The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin. Genes Cells 7(6), 535–541 (2002)CrossRefGoogle Scholar
  15. 15.
    B. Dubey, N. Kumari, R.K. Upadhyay, Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach. J. Appl. Math. Comput. 31(1–2), 413–432 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    N. Kumari, Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator. ISRN Biomath. 2013, 12 (2013)Google Scholar
  17. 17.
    K. Kuto, Stability of steady-state solutions to a preypredator system with cross-diffusion. J. Differ. Equ. 197(2), 293–314 (2004)CrossRefGoogle Scholar
  18. 18.
    K. Kuto, Y. Yamada, Multiple coexistence states for a preypredator system with cross-diffusion. J. Differ. Equ. 197(2), 315–348 (2004)CrossRefGoogle Scholar
  19. 19.
    P.Y.H. Pang, M. Wang, Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200(2), 245–273 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Wang, Stationary patterns caused by cross-diffusion for a three-species prey-predator model. Comput. Math. Appl. 52(5), 707–720 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    A.B. Medvinsky et al., Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    C. Tian, Z. Ling, Z. Lin, Spatial patterns created by cross-diffusion for a three-species food chain model. Int. J. Biomath. 7(02), 1450013 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    C. Tian, Turing patterns created by cross-diffusion for a Holling II and Leslie-Gower type three species food chain model. J. Math. Chem. 49(6), 1128–1150 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    N. Kumari, N. Mohan, Cross diffusion induced turing patterns in a tritrophic food chain model with crowley-martin functional response. Mathematics 7(3), 229 (2019)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology MandiMandiIndia

Personalised recommendations