Journal of Applied Mathematics and Computing

, Volume 31, Issue 1–2, pp 413–432 | Cite as

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

  • Balram Dubey
  • Nitu Kumari
  • Ranjit Kumar Upadhyay


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.


Prey-predator system Reaction-diffusion equations Marine ecosystem Chaos Spatiotemporal pattern 

Mathematics Subject Classification (2000)

35B35 92C15 35K57 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, E.R.: The generation of plankton patchiness by turbulent stirring. Nature 391, 577–580 (1998) CrossRefGoogle Scholar
  2. 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. 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) CrossRefGoogle Scholar
  4. 4.
    Chen, B., Wang, M.: Qualitative analysis for a diffusive predator-prey model. Comput. Math. Appl. 55(3), 339–355 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Denman, K.L.: Covariability of chlorophyll and temperature in the sea. Deep-Sea Res. 23, 539–550 (1976) Google Scholar
  6. 6.
    Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross- diffusion. Ecol. Model. 171, 67–76 (2001) CrossRefGoogle Scholar
  7. 7.
    Dubey, B., Hussain, J.: Modelling the interaction of two biological species in polluted environment. J. Math. Anal. Appl. 246, 58–79 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dubois, D.M.: A model of patchiness for prey-predator plankton populations. Ecol. Model. 1, 67–80 (1975) CrossRefGoogle Scholar
  9. 9.
    Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 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. 11.
    Freedman, H.I., So, J.H.W.: Global stability and persistence of simple food chains. Math. Biosci. 76, 69–86 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 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. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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 Google Scholar
  18. 18.
    Levin, S.A., Segel, L.A.: Hypothesis for origin of planktonic patchiness. Nature 259, 659 (1976) CrossRefGoogle Scholar
  19. 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) CrossRefGoogle Scholar
  20. 20.
    Malchow, H.: Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics. Proc. R. Soc. Lond. B 251, 103–109 (1993) CrossRefGoogle Scholar
  21. 21.
    Malchow, H.: Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system. J. Mar. Syst. 7, 193–202 (1996) CrossRefGoogle Scholar
  22. 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) CrossRefGoogle Scholar
  23. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) zbMATHGoogle Scholar
  25. 25.
    Platt, T.: Local phytoplankton abundance and turbulence. Deep-Sea Res. 19, 183–187 (1972) Google Scholar
  26. 26.
    Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. OIKOS 62, 271–282 (1991) CrossRefGoogle Scholar
  27. 27.
    Segel, L.A., Jackson, J.L.: Dissipative structures: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972) CrossRefGoogle Scholar
  28. 28.
    Stamov, G.T.: Almost periodic models in impulsive ecological systems with variable diffusion. J. Appl. Math. Comput. 27, 243–255 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Thomas, J.: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics. Springer, New York (1995) zbMATHGoogle Scholar
  30. 30.
    Turing, A.M.: On the chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952) CrossRefGoogle Scholar
  31. 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. 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) CrossRefMathSciNetGoogle Scholar
  33. 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) CrossRefMathSciNetGoogle Scholar
  34. 34.
    Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969) CrossRefGoogle Scholar
  35. 35.
    Wolpert, L.: The development of pattern and form in animals. Carol. Biol. Read. 1(5), 1–16 (1977) MathSciNetGoogle Scholar
  36. 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) CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2008

Authors and Affiliations

  • Balram Dubey
    • 1
  • Nitu Kumari
    • 2
  • Ranjit Kumar Upadhyay
    • 2
  1. 1.Department of MathematicsBirla Institute of Technology and SciencePilaniIndia
  2. 2.Department of Applied MathematicsIndian School of Mines UniversityDhanbadIndia

Personalised recommendations