zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
A minimal model of pattern formation in a prey-predator system. (English) Zbl 0990.92040
Summary: The spatio-temporal dynamics of a prey-predator community is described by two reaction-diffusion equations. It is shown that for a class of initial conditions the spatio-temporal system dynamics resembles a “phase transition” between a regular and an irregular phase, separated by a moving boundary. A simple approach to specify spatio-temporal chaos is proposed.

35K57Reaction-diffusion equations
37N25Dynamical systems in biology
Full Text: DOI
[1] Turing, A. M.: On the chemical basis of morphogenesis. Phil. trans. R. soc. Lond. 237, 37-72 (1952)
[2] Segel, L. A.; Jackson, J. L.: Dissipative structure: an explanation and an ecological example. J. theor. Biol. 37, 545-559 (1972)
[3] Malchow, M.: Spatio-temporal pattern formation in nonlinear nonequilibrium plankton dynamics. Proc. R. Soc. lond. 251, 103-109 (1993)
[4] Wroblewski, J. S.; O’brien, J. J.: A spatial model of plankton patchiness. Marine biology 35, 161-176 (1976)
[5] Vastano, J. A.; Pearson, J. E.; Horsthemke, W.; Swinney, H. L.: Chemical pattern formation with equal diffusion coefficients. Phys. lett. 124, 320-324 (1987)
[6] Barenblatt, G. I.; Vinogradov, M. E.; Gorbunov, A. E.; Petrovskii, S. V.: Modeling impact waves in complex ecological systems. Oceanology 33, 5-12 (1993)
[7] Pascual, M.: Diffusion-induced chaos in a spatial predator-prey system. Proc. R. Soc. lond. 251, 1-7 (1993)
[8] Sherratt, J. A.; Lewis, M. A.; Fowler, A. C.: Ecological chaos in the wake of invasion. Proc. natl. Acad. sci. USA 92, 2524-2528 (1995) · Zbl 0819.92024
[9] Lewis, M. A.: A tale of two tails: the mathematical links between dispersal, patchiness and variability in a biological invasion. Abstracts of the 3rd European conf. On math, in biology and medicine (1996)
[10] Murray, J. D.: Mathematical biology. (1989) · Zbl 0682.92001
[11] Shigesada, N.; Kawasaki, K.: Biological invasions: theory and practice. (1997)
[12] S.V. Petrovskii and H. Malchow, Critical phenomena in plankton communities: KISS model revisited (to appear). · Zbl 0996.92037
[13] Vandenhouten, R.; Goebbels, G.; Rasche, M.; Tegtmeier, H.: Third edition SANTIS--A tool for signal analysis and time series processing, version 1.1. User manual. SANTIS--A tool for signal analysis and time series processing, version 1.1. User manual (1996)
[14] Shibata, H.: Quantitative characterization of spatiotemporal chaos. Physica A 252, 428-449 (1998)
[15] Sherratt, J. A.; Eagan, B. T.; Lewis, M. A.: Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?. Phil. trans. R. soc. Lond. B 352, 21-38 (1997)
[16] Vinogradov, M. E.: Open-ocean ecosystems. Marine ecology 5 (1983)