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)
The effect of dispersal on permanence in a predator-prey population growth model. (English) Zbl 1032.92032
Summary: We consider a periodic predator-prey system where the prey can disperse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation. We assume Volterra within-patch dynamics. Under the assumption that the average of the dispersal rate from Patch 1 to Patch 2 is less than that of the intrinsic growth rate of prey in Patch 1, we provide a sufficient and necessary condition to guarantee the prey and predator species to be permanent by using the main techniques of {\it Z. Teng}, Appl. Anal. 72, 339-352 (1999; Zbl 1031.34045).

34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
34D05Asymptotic stability of ODE
37N25Dynamical systems in biology
Full Text: DOI
[1] Teng, Z.: Uniform persistence of the periodic predator-prey Lotka-Volterra systems. Appl. anal. 72, 339-352 (1999) · Zbl 1031.34045
[2] Miliski, M.; Heller, R.: Influence of a predator on the optimal foraging behaviour of stickbacks (Gasteropodus aculeatus L.). Nature (London) 275, 642-644 (1978)
[3] Cerri, R. D.; Fraser, D. F.: Predation and risk in foraging minnows: balancing conflicting demands. Amer. nat. 121, 552-561 (1983)
[4] Miliski, M.: The patch choice model: no alternative to balancing. Amer. nat. 125, 317-320 (1985)
[5] Beretta, E.; Solimano, F.; Takeuchi, Y.: Global stability and periodic orbits for two patch predator-prey diffusion delay models. Math. biosci. 85, 153-183 (1987) · Zbl 0634.92017
[6] Song, X.; Chen, L.: Persistence and periodic orbits for two species predator prey system with diffusion. Canad. appl. Math. quart. 6, No. 3, 233-244 (1998) · Zbl 0941.92032
[7] Luo, M.; Ma, Z.: The persistence of two species Lotka-Volterra model with diffusion. J. biomath. 12, 52-59 (1997) · Zbl 0898.92026
[8] Allen, L. J. S.: Persistence and extinction in Lotka-Volterra reaction-diffusion equations. Math. biosci. 65, 1-12 (1983) · Zbl 0522.92021
[9] 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
[10] Chewning, W. C.: Migratory effect in predation prey systems. Math. biosci. 23, 253-262 (1975) · Zbl 0301.92010
[11] Freedman, H. I.; Waltman, P.: Mathematical models of population interaction with dispersal. I. stability of two habitats with and without a predator. SIAM J. Math. 32, 631-648 (1977) · Zbl 0362.92006
[12] 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
[13] Freedman, H. I.; Takeuchi, Y.: Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear anal., TMA 13, 993-1002 (1989) · Zbl 0685.92018
[14] Hastings, A.: Spatial heterogeneity and the stability of predator prey systems. Theor. pop. Biol. 12, 37-48 (1977) · Zbl 0371.92016
[15] Hassell, M. P.: The dynamics of arthropod predator-prey systems. (1978) · Zbl 0429.92018
[16] Holt, R. D.: Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution. Theor. pop. Biol. 28, 181-208 (1985) · Zbl 0584.92022
[17] 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
[18] Levin, S. A.: Dispersion and population interactions. Amer. natur. 108, 207-228 (1974)
[19] Takeuchi, Y.: Global stability in generalized Lotka-Volterra diffusion systems. J. math. Anal. appl. 116, 209-221 (1986) · Zbl 0595.92013
[20] Takeuchi, Y.: Diffusion effect on stability of Lotka-Volterra model. Bull. math. Biol. 46, 585-601 (1986) · Zbl 0613.92025
[21] Song, X.; Chen, L.: Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay. Computers math. Applic. 35, No. 6, 33-40 (1998) · Zbl 0903.92029
[22] Cui, J.; Chen, L.: The effect of diffusion on the time varying logistic population growth. Computers math. Applic. 36, No. 3, 1-9 (1998) · Zbl 0934.92025
[23] Cui, J.; Chen, L.: The effects of habitat fragmentation and ecological invasion on population sizes. Computers math. Applic. 38, No. 1, 1-11 (1999) · Zbl 0939.92033
[24] Smith, H. L.: Cooperative systems of differential equation with concave nonlinearities. Nonlinear analysis 10, 1037-1052 (1986) · Zbl 0612.34035
[25] Tineo, A.: An iterative scheme for the N-competing species problem. J. diff. Equ. 116, 1-15 (1995) · Zbl 0823.34048