Spatiotemporal pattern formation in a diffusive predator-prey system: An analytical approach. (English) Zbl 1181.35022

Summary: We propose and analyse a mathematical model to study the mathematical aspect of the 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 an aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in a 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 a marine ecosystem.


35B36 Pattern formations in context of PDEs
35B35 Stability in context of PDEs
92C15 Developmental biology, pattern formation
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Abraham, E.R.: The generation of plankton patchiness by turbulent stirring. Nature 391, 577–580 (1998)
[2] Ahmed, S., Rao, M.R.M.: Theory of Ordinary Differential Equations with Applications in Biology and Engineering. East-West Press, New Delhi (1999)
[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)
[4] Chen, B., Wang, M.: Qualitative analysis for a diffusive predator-prey model. Comput. Math. Appl. 55(3), 339–355 (2008) · Zbl 1155.35390
[5] Denman, K.L.: Covariability of chlorophyll and temperature in the sea. Deep-Sea Res. 23, 539–550 (1976)
[6] Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross- diffusion. Ecol. Model. 171, 67–76 (2001)
[7] Dubey, B., Hussain, J.: Modelling the interaction of two biological species in polluted environment. J. Math. Anal. Appl. 246, 58–79 (2000) · Zbl 0952.92030
[8] Dubois, D.M.: A model of patchiness for prey-predator plankton populations. Ecol. Model. 1, 67–80 (1975)
[9] Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006) · Zbl 1142.35022
[10] Fasham, M.J.R.: The statistical and mathematical analysis of plankton patchiness. Oceanogr. Mar. Biol. Annu. Rev. 16, 43–79 (1978)
[11] Freedman, H.I., So, J.H.W.: Global stability and persistence of simple food chains. Math. Biosci. 76, 69–86 (1985) · Zbl 0572.92025
[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)
[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) · Zbl 1155.34361
[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) · Zbl 1156.35479
[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 · Zbl 1144.35029
[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) · Zbl 1155.37046
[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)
[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)
[20] Malchow, H.: Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics. Proc. R. Soc. Lond. B 251, 103–109 (1993)
[21] Malchow, H.: Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system. J. Mar. Syst. 7, 193–202 (1996)
[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)
[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) · Zbl 1001.92050
[24] Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) · Zbl 0682.92001
[25] Platt, T.: Local phytoplankton abundance and turbulence. Deep-Sea Res. 19, 183–187 (1972)
[26] Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. OIKOS 62, 271–282 (1991)
[27] Segel, L.A., Jackson, J.L.: Dissipative structures: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)
[28] Stamov, G.T.: Almost periodic models in impulsive ecological systems with variable diffusion. J. Appl. Math. Comput. 27, 243–255 (2008) · Zbl 1160.34074
[29] Thomas, J.: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics. Springer, New York (1995) · Zbl 0831.65087
[30] Turing, A.M.: On the chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952) · Zbl 1403.92034
[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 · Zbl 1197.37121
[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) · Zbl 1337.35080
[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) · Zbl 01835461
[34] Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969)
[35] Wolpert, L.: The development of pattern and form in animals. Carol. Biol. Read. 1(5), 1–16 (1977)
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.