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)
Permanence and global attractivity of a discrete semi-ratio dependent predator-prey system with Holling II type functional response. (English) Zbl 1213.49046
Summary: We propose a discrete semi-ratio dependent predator-prey system with Holling II type functional response. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

49N75Pursuit and evasion games in calculus of variations
49M25Discrete approximations in calculus of variations
34C25Periodic solutions of ODE
34D23Global stability of ODE
Full Text: DOI
[1] Chen, F.D.: Permance and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems. Appl. Math. Comput. 59, 804--814 (1991)
[2] Saito, Y., Ma, W., Hara, T.: A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays. J. Math. Anal. Appl. 256, 162--174 (2001) · Zbl 0976.92031 · doi:10.1006/jmaa.2000.7303
[3] Fan, M., Wang, K.: Periodic solution of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Model. 35, 951--961 (2002) · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
[4] Berryman, A.A.: The origins and evolution of predator-prey theory. Ecology 75, 1530--1535 (1992) · doi:10.2307/1940005
[5] Huo, H.F., Li, W.T.: Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model. Math. Comput. Model. 34, 261--269 (2004) · Zbl 1067.39008 · doi:10.1016/j.mcm.2004.02.026
[6] Yang, X.T.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316, 161--177 (2006) · Zbl 1107.39017 · doi:10.1016/j.jmaa.2005.04.036
[7] Fang, Y.H., Li, W.T.: Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response. J. Math. Anal. Appl. 299, 357--374 (2004) · Zbl 1063.39013 · doi:10.1016/j.jmaa.2004.02.061
[8] Fan, M., Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete Contin. Dyn. Syst. Ser. B 4, 563--574 (2004) · Zbl 1100.92064 · doi:10.3934/dcdsb.2004.4.563
[9] Wang, Q., Fan, M., Wang, K.: Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses. J. Math. Anal. Appl. 278, 443--471 (2003) · Zbl 1029.34042 · doi:10.1016/S0022-247X(02)00718-7
[10] Arrowsmith, D.K., Place, C.M.: Dynamical Systems. Chapman and Hall, London (1992)
[11] Beltrami, E.: Mathematics for Dynamical Modelling. Academic Press, San Diego (1987) · Zbl 0625.58012
[12] Huo, H.F., Li, W.T.: Permanence and global stability for nonautonomous discrete model of plankton allelopathy. Appl. Math. Lett. 17, 1007--1013 (2004) · Zbl 1067.39009 · doi:10.1016/j.aml.2004.07.002
[13] Chen, Y.M., Zhou, Z.: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 277, 358--366 (2003) · Zbl 1019.39004 · doi:10.1016/S0022-247X(02)00611-X