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)
Global asymptotic behavior in single-species discrete diffusion systems. (English) Zbl 0799.92014
A single-species dynamical system is considered which is composed of several patches connected by discrete diffusion. Based on cooperative system theory and the property of a cooperative matrix, sufficient and necessary conditions are obtained for the system with linear diffusion to become extinct and for one with nonlinear diffusion to be globally stable. The authors obtain a critical patch number in the system with linear diffusion for the species to become extinct.

92D25Population dynamics (general)
Full Text: DOI
[1] Allen, L. J. S.: Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biol. 45, 209-227 (1983) · Zbl 0543.92020
[2] Allen, L. J. S.: Persistence, extinction, and critical patch number for island populations. J. Math. Biol. 24, 617-625 (1987) · Zbl 0603.92019
[3] Berman, A., Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. New York London: Academic Press 1979 · Zbl 0484.15016
[4] Butler, G., Freedman, H. I., Waltman, P.: Uniformly persistent system. Proc. Am. Math. Soc. 96, 425-430 (1986) · Zbl 0603.34043 · doi:10.1090/S0002-9939-1986-0822433-4
[5] Freedman, H. I., Takeuchi, Y.: Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear Anal., Theory Methods Appl 13, 993-1002 (1989) · Zbl 0685.92018 · doi:10.1016/0362-546X(89)90026-6
[6] Freedman, H. I., Waltman, P.: Persistence in models of three competitive populations. Math. Biosci. 73, 89-101 (1985) · Zbl 0584.92018 · doi:10.1016/0025-5564(85)90078-1
[7] Gard, T. C., Hallam, T. G.: Persistence in food webs. I. Lotka-Volterra food chains. Bull. Math. Biol. 41, 877-891 (1979) · Zbl 0422.92017
[8] Hofbauer, J., Sigmund, K.: The theory of evolution and dynamical systems. Cambridge: Cambridge University Press 1988 · Zbl 0678.92010
[9] Kamke, E.: Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II. Acta Math. 58, 57-85 (1932) · Zbl 58.0449.02 · doi:10.1007/BF02547774
[10] Ludwig, D., Aronson, D. G., Weinberger, H. F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217-258 (1979) · Zbl 0412.92020
[11] Selgrade, J. F.: Asymptotic behavior of solutions to single loop positive feedback systems. J. Differ. Equations 38, 80-103 (1980) · Zbl 0438.34052 · doi:10.1016/0022-0396(80)90026-1
[12] Selgrade, J. F.: On the existence and uniqueness of connecting orbits. Nonlinear Anal., Theory Methods Appl. 7, 1123-1125 (1983) · Zbl 0524.58039 · doi:10.1016/0362-546X(83)90021-4
[13] Smith, H. L.: On the asymptotic behavior of a class of deterministic models of cooperating species. SIAM J. Appl. Math. 46, 368-375 (1986) · Zbl 0607.92023 · doi:10.1137/0146025
[14] Smith, H. L.: Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal., Theory Methods Appl. 10, 1037-1052 (1986) · Zbl 0612.34035 · doi:10.1016/0362-546X(86)90087-8
[15] Takeuchi, Y.: Cooperative system theory and global stability of diffusion models. Acta Appl. Math. 14, 49-57 (1989) · Zbl 0665.92017 · doi:10.1007/BF00046673