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)
Asymptotic behavior of a predator-prey system with diffusion and delays. (English) Zbl 0872.92019
Summary: The asymptotic behavior of solutions of a predator-prey system is determined. The model incorporates time delays due to gestation and assumes that the prey disperses between two patches of a heterogeneous environment with barriers between patches and that the predator disperses between the patches with no barrier. Conditions are established for the permanence of the populations and the global attractivity of a positive equilibrium.

34K25Asymptotic theory of functional-differential equations
34K99Functional-differential equations
34K20Stability theory of functional-differential equations
Full Text: DOI
[1] Freedman, H. I.; Takeuchi, Y.: Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear anal. 13, 993-1002 (1989) · Zbl 0685.92018
[2] Kuang, Y.; Takeuchi, Y.: Predator-prey dynamics in models of prey dispersal in two patch environments. Math. biosci. 120, 77-98 (1994) · Zbl 0793.92014
[3] Lu, Zhengyi; Takeuchi, Y.: Permanence and global stability for coopeative Lotka-Volterra diffusion systems. Nonlinear anal. 10, 963-975 (1992) · Zbl 0784.93092
[4] Beretta, E.; Takeuchi, Y.: Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delays. SIAM J. Appl. math. 48, 627-651 (1988) · Zbl 0661.92018
[5] Freedman, H. I.: Single species migration in two habitats: persistence and extinction. Math. model. 8, 778-780 (1987)
[6] Freedman, H. I.; Rai, B.; Waltman, P.: Mathematical models of population interactions with dispersal. II. differential survival in a change of habitat. J. math. Anal. appl. 115, 140-154 (1986) · Zbl 0588.92020
[7] Levin, S. A.: Dispersion and population interactions. Amer. natur. 108, 207-228 (1974)
[8] Levin, S. A.: Spatial patterning and the structure of ecological communities. (1976) · Zbl 0338.92017
[9] Beretta, E.; Takeuchi, Y.: Global stability of single species diffusion Volterra models with continuous time delays. Bull. math. Biol. 49, 431-448 (1987) · Zbl 0627.92021
[10] Freedman, H. I.; Takeuchi, Y.: Predator survival versus extinction as a function of dispersal in a predator-prey model with patchy environment. Appl. anal. 31, 247-266 (1989) · Zbl 0641.92016
[11] Cushing, J. M.: Integro-differential equations and delay models in population dynamics. Lecture notes in biomathematics 20 (1977) · Zbl 0363.92014
[12] Freedman, H. I.; Rao, V. S. H.: The tradeoff between mutual interference and time lags in predator-prey systems. Bull. math. Biol. 45, 991-1004 (1983) · Zbl 0535.92024
[13] Hofbauer, J.; Sigmund, K.: Dynamical systems and the theory of evolution. (1988) · Zbl 0678.92010
[14] Selgrade, J. F.: Asymptotic behavior of solutions to a single loop positive feedback system. J. differential equations 38, 80-103 (1980) · Zbl 0419.34054
[15] Selgrade, J. F.: On the existence and uniqueness of connecting orbits. Nonlinear anal. 7, 1123-1125 (1983) · Zbl 0524.58039
[16] Hale, J. K.; Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. anal. 20, 388-395 (1989) · Zbl 0692.34053
[17] Lakshmikantham, V.; Leela, S.: Differential and integral inequalities. (1969) · Zbl 0177.12403
[18] Wendi, Wang; Mazhien: Harmless delays for uniform persistence. J. math. Anal. appl. 158, 256-268 (1991) · Zbl 0731.34085
[19] Wendi, Wang: Uniform persistence in competition models. J. biomath. 2, 164-169 (1991) · Zbl 0825.92108
[20] Yulin, C.; Gard, T. C.: Uniform persistence for population models with delay using multiple Lyapunov functions. Differential equations 6, 883-898 (1993) · Zbl 0780.92019
[21] Yulin, C.; Gard, T. C.: Extinction in predator-prey models with time delay. Math. biosci. 118, 197-210 (1993) · Zbl 0806.92015
[22] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[23] Y. Kuang, Periodic solutions in a class of delayed predator-prey system
[24] Freedman, H. I.; So, J.; Waltman, P.: Coexistence in a model of competition in the chemostat incorporating discrete time delays. SIAM J. Appl. math. 49, 859-870 (1989) · Zbl 0676.92013