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Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

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Abstract

In this paper, we propose and analyse a mathematical model to study the mathematical aspect of reaction diffusion pattern formation mechanism in a predator-prey system. An attempt is made to provide an analytical explanation for understanding plankton patchiness in a minimal model of aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in marine system. Our analytical findings, supported by the results of numerical experiments, suggest that an unstable diffusive system can be made stable by increasing diffusivity constant to a sufficiently large value. It is also observed that the solution of the system converges to its equilibrium faster in the case of two-dimensional diffusion in comparison to the one-dimensional diffusion. The ideas contained in the present paper may provide a better understanding of the pattern formation in marine ecosystem.

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References

  1. Abraham, E.R.: The generation of plankton patchiness by turbulent stirring. Nature 391, 577–580 (1998)

    Article  Google Scholar 

  2. Ahmed, S., Rao, M.R.M.: Theory of Ordinary Differential Equations with Applications in Biology and Engineering. East-West Press, New Delhi (1999)

    Google Scholar 

  3. Brentnall, S.J., Richards, K.J., Brindley, J., Murphy, E.: Plankton patchiness and its effect on large-scale productivity. J. Plankton Res. 25(2), 121–140 (2003)

    Article  Google Scholar 

  4. Chen, B., Wang, M.: Qualitative analysis for a diffusive predator-prey model. Comput. Math. Appl. 55(3), 339–355 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Denman, K.L.: Covariability of chlorophyll and temperature in the sea. Deep-Sea Res. 23, 539–550 (1976)

    Google Scholar 

  6. Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross- diffusion. Ecol. Model. 171, 67–76 (2001)

    Article  Google Scholar 

  7. Dubey, B., Hussain, J.: Modelling the interaction of two biological species in polluted environment. J. Math. Anal. Appl. 246, 58–79 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dubois, D.M.: A model of patchiness for prey-predator plankton populations. Ecol. Model. 1, 67–80 (1975)

    Article  Google Scholar 

  9. Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fasham, M.J.R.: The statistical and mathematical analysis of plankton patchiness. Oceanogr. Mar. Biol. Annu. Rev. 16, 43–79 (1978)

    Google Scholar 

  11. Freedman, H.I., So, J.H.W.: Global stability and persistence of simple food chains. Math. Biosci. 76, 69–86 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grieco, L., Tremblay, L.-B., Zambianchi, E.: A hybrid approach to transport processes in the Gulf of Naples: an application to phytoplankton and zooplankton population dynamics. Cont. Shelf Res. 25, 711–728 (2005)

    Article  Google Scholar 

  13. Huo, H.-F., Li, W.-T., Nieto, J.J.: Periodic solutions of delayed predator-prey model with the Beddington-DeAngelis functional response. Chaos Solitons Fractals 33(2), 505–512 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ko, W., Ryu, K.: Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment. J. Math. Anal. Appl. 327, 539–549 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ko, W., Ryu, K.: A qualitative study on general Gauss-type predator-prey models with non-monotonic functional response. Nonlinear Anal.: Real World Appl. (2008). doi:10.1016/j.nonrwa.2008.05.012

    Google Scholar 

  16. Li, W.-T., Wu, S.-L.: Traveling waves in a diffusive predator-prey model with Holling type-III functional response. Chaos Solitons Fractals 37, 476–486 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu, Q., Li, B., Jin, Z.: Resonance and frequency-locking phenomena in spatially extended phytoplankton-zooplankton system with additive nose and periodic forces. J. Stat. Mech.: Theory Exp. (2008). Article no. po5011

  18. Levin, S.A., Segel, L.A.: Hypothesis for origin of planktonic patchiness. Nature 259, 659 (1976)

    Article  Google Scholar 

  19. Ludwig, D., Jones, D., Holling, C.: Qualitative analysis of an insect outbreak system: the spruce budworm and forest. J. Anim. Ecol. 47, 315–332 (1978)

    Article  Google Scholar 

  20. Malchow, H.: Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics. Proc. R. Soc. Lond. B 251, 103–109 (1993)

    Article  Google Scholar 

  21. Malchow, H.: Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system. J. Mar. Syst. 7, 193–202 (1996)

    Article  Google Scholar 

  22. Medvinsky, A.B., Tikhonova, I.A., Aliev, R.R., Li, B.L., Lin, Z.S., Malchow, H.: Patchy environment as a factor of complex plankton dynamics. Phys. Rev. E 64, 021915-7 (2001)

    Article  Google Scholar 

  23. Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    MATH  Google Scholar 

  25. Platt, T.: Local phytoplankton abundance and turbulence. Deep-Sea Res. 19, 183–187 (1972)

    Google Scholar 

  26. Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. OIKOS 62, 271–282 (1991)

    Article  Google Scholar 

  27. Segel, L.A., Jackson, J.L.: Dissipative structures: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)

    Article  Google Scholar 

  28. Stamov, G.T.: Almost periodic models in impulsive ecological systems with variable diffusion. J. Appl. Math. Comput. 27, 243–255 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Thomas, J.: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics. Springer, New York (1995)

    MATH  Google Scholar 

  30. Turing, A.M.: On the chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952)

    Article  Google Scholar 

  31. Upadhyay, R.K., Kumari, N., Rai, V.: Wave of chaos in a diffusive system: generating realistic patterns of patchiness in plankton-fish dynamics. Chaos Solitons Fractals (2007). doi:10.1016/j.chaos.2007.07.078

    Google Scholar 

  32. Upadhyay, R.K., Kumari, N., Rai, V.: Wave of chaos and pattern formation in spatial predator-prey systems with Holling type IV predator response. Math. Model. Nat. Phenom. 3(4), 71–95 (2008)

    Article  MathSciNet  Google Scholar 

  33. Vilar, J.M.G., Sole, R.V., Rubi, J.M.: On the origin of plankton patchiness. Phys. A: Stat. Mech. Appl. 317, 239–246 (2003)

    Article  MathSciNet  Google Scholar 

  34. Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969)

    Article  Google Scholar 

  35. Wolpert, L.: The development of pattern and form in animals. Carol. Biol. Read. 1(5), 1–16 (1977)

    MathSciNet  Google Scholar 

  36. Xiao, J.-H., Li, H.-H., Yang, J.-Z., Hu, G.: Chaotic Turing pattern formation in spatiotemporal systems. Front. Phys. China 1, 204–208 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science, Pilani, 333031, India

    Balram Dubey

  2. Department of Applied Mathematics, Indian School of Mines University, Dhanbad, 826004, India

    Nitu Kumari & Ranjit Kumar Upadhyay

Authors
  1. Balram Dubey
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  2. Nitu Kumari
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  3. Ranjit Kumar Upadhyay
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Correspondence to Ranjit Kumar Upadhyay.

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Dubey, B., Kumari, N. & Upadhyay, R.K. Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach. J. Appl. Math. Comput. 31, 413–432 (2009). https://doi.org/10.1007/s12190-008-0221-6

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  • DOI: https://doi.org/10.1007/s12190-008-0221-6

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