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 short proof of the Cushing-Henson conjecture. (English) Zbl 1149.39300
Summary: We give a short proof of the Cushing-Henson conjecture [cf. {\it J. M. Cushing} and {\it S. M. Henson}, J. Difference Equ. Appl. 8, No. 12, 1119--1120 (2002; Zbl 1023.39013)] concerning the Beverton-Holt difference equation [cf. {\it R. J. Beverton} and {\it S. J. Holt}, On the dynamics of exploited fish populations. Fish. Invest. 19, HM50, London (1957)] which is important in theoretical ecology. The main result shows that a periodic environment is always deleterious for populations modeled by the Beverton-Holt difference equation.

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
92D25Population dynamics (general)
Full Text: DOI EuDML
[1] L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 217-250, 2005. · Zbl 1104.45001
[2] R. J. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, vol. 19 of Fish. Invest., HMSO, London, 1957.
[3] J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859-872, 2001. · Zbl 1002.39003 · doi:10.1080/10236190108808308
[4] J. M. Cushing and S. M. Henson, “A periodically forced Beverton-Holt equation,” Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1119-1120, 2002. · Zbl 1023.39013 · doi:10.1080/1023619021000053980
[5] S. Elaydi and R. J. Sacker, “Global stability of periodic orbits of non-autonomous difference equations and population biology,” Journal of Differential Equations, vol. 208, no. 1, pp. 258-273, 2005. · Zbl 1067.39003 · doi:10.1016/j.jde.2003.10.024
[6] S. Elaydi and R. J. Sacker, “Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures,” in Proceedings of the 8th International Conference on Difference Equations and Applications, pp. 113-126, Chapman & Hall/CRC, Florida, 2005. · Zbl 1087.39504
[7] S. Elaydi and R. J. Sacker, “Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures,” Journal of Difference Equations and Applications, vol. 11, no. 4-5, pp. 337-346, 2005. · Zbl 1084.39005 · doi:10.1080/10236190412331335418
[8] H.-F. Huo and W.-T. Li, “Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 135-144, 2005. · Zbl 1111.39007 · doi:10.1155/DDNS.2005.135 · eudml:127292
[9] M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Florida, 2002. · Zbl 0981.39011
[10] P. Liu and K. Gopalsamy, “On a model of competition in periodic environments,” Applied Mathematics and Computation, vol. 82, no. 2-3, pp. 207-238, 1997. · Zbl 0872.39009 · doi:10.1016/S0096-3003(96)00044-6
[11] S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence xn+1=f(xn - 1,xn),” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631-638, 2006. · Zbl 1100.39001
[12] L.-L. Wang and W.-T. Li, “Existence and global attractivity of positive periodic solution for an impulsive delay population model,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, vol. 12, no. 3-4, pp. 457-468, 2005. · Zbl 1078.34049
[13] X.-X. Yan, W.-T. Li, and Z. Zhao, “On the recursive sequence xn+1=\alpha - (xn/xn - 1),” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 269-282, 2005. · Zbl 1068.39030 · doi:10.1007/BF02936054