zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.
MSC:
35B36Pattern formation in solutions of PDE
35B35Stability of solutions of PDE
92C15Developmental biology, pattern formation
35K57Reaction-diffusion equations
References:
[1]Abraham, E.R.: The generation of plankton patchiness by turbulent stirring. Nature 391, 577–580 (1998) · doi:10.1038/35361
[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) · doi:10.1093/plankt/25.2.121
[4]Chen, B., Wang, M.: Qualitative analysis for a diffusive predator-prey model. Comput. Math. Appl. 55(3), 339–355 (2008) · Zbl 1155.35390 · doi:10.1016/j.camwa.2007.03.020
[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) · doi:10.1016/S0304-3800(01)00255-1
[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 · doi:10.1006/jmaa.2000.6741
[8]Dubois, D.M.: A model of patchiness for prey-predator plankton populations. Ecol. Model. 1, 67–80 (1975) · doi:10.1016/0304-3800(75)90006-X
[9]Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006) · Zbl 1142.35022 · doi:10.1016/j.jde.2006.01.013
[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 · doi:10.1016/0025-5564(85)90047-1
[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) · doi:10.1016/j.csr.2004.10.014
[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 · doi:10.1016/j.chaos.2005.12.045
[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 · doi:10.1016/j.jmaa.2006.04.077
[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
[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 · doi:10.1016/j.chaos.2006.09.039
[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) · doi:10.1038/259659a0
[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) · doi:10.2307/3939
[20]Malchow, H.: Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics. Proc. R. Soc. Lond. B 251, 103–109 (1993) · doi:10.1098/rspb.1993.0015
[21]Malchow, H.: Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system. J. Mar. Syst. 7, 193–202 (1996) · doi:10.1016/0924-7963(95)00012-7
[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) · doi:10.1103/PhysRevE.64.021915
[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 · doi:10.1137/S0036144502404442
[24]Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)
[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) · doi:10.2307/3545491
[27]Segel, L.A., Jackson, J.L.: Dissipative structures: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972) · doi:10.1016/0022-5193(72)90090-2
[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 · doi:10.1007/s12190-008-0054-3
[29]Thomas, J.: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics. Springer, New York (1995)
[30]Turing, A.M.: On the chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952) · doi:10.1098/rstb.1952.0012
[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
[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) · doi:10.1051/mmnp:2008071
[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 · doi:10.1016/S0378-4371(02)01322-5
[34]Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969) · doi:10.1016/S0022-5193(69)80016-0
[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) · doi:10.1007/s11467-006-0014-z