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)
Periodicity and asymptotic stability of a predator-prey system with infinite delays. (English) Zbl 1201.34111
Summary: By using a fixed point theorem and Lyapunov functional, an especially easily checked criterion is obtained for the global existence and global asymptotic stability of positive periodic solutions of a periodic predator-prey system with infinite delays. Moreover, the global existence theorem is also sufficient and necessary. This result improves and generalizes noticeably some known results.
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K20Stability theory of functional-differential equations
[1]Ackland, G. J.; Gallagher, I. D.: Stabilization of large generalized Lotka–Volterra foodwebs by evolutionary feedback, Phys. rev. Lett. 93, No. 15, 158701-1-158701-4 (2004)
[2]Brauer, F.; Castillo-Chávez, C.: Mathematical models in population biology and epidemiology, (2001)
[3]Gopalsamy, K.: Stability and oscillation in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[4]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[5]Bereketoglu, H.; Gyori, I.: Global asymptotic stability in a nonautonomous Lotka–Volterra type system with infinite delay, J. math. Anal. appl. 210, 279-291 (1997) · Zbl 0880.34072 · doi:10.1006/jmaa.1997.5403
[6]Chen, F.; Sun, D.: Periodic solutions of scalar neutral Volterra integro-differential equations with infinite delay, Ann. differential equations. 19, 250-255 (2003) · Zbl 1049.45010
[7]Fan, M.; Wang, K.: Global existence of positive periodic solutions of periodic predator–prey system with infinite delays, J. math. Anal. appl. 262, 1-11 (2001) · Zbl 0995.34063 · doi:10.1006/jmaa.2000.7181
[8]Fan, M.; Wang, K.: Existence and global attractivity of positive periodic solution of multi-species ecological competition system, Acta math. Sinica 43, 77-82 (2000) · Zbl 1005.92036
[9]Gopalsamy, K.; Aggarwala, B. D.: Limit cycles in two species competition with time delay, J. aust. Math. soc. Ser. B 22, 148-160 (1992) · Zbl 0458.92014 · doi:10.1017/S033427000000223X
[10]Kuang, Y.: Global stability of gause-type predator–prey systems, J. math. Biol. 28, 463-474 (1990) · Zbl 0742.92022 · doi:10.1007/BF00178329
[11]Kuang, Y.; Smith, H. L.: Global stability of infinite delay Lotka–Volterra type systems, J. differential equations 103, 221-246 (1993) · Zbl 0786.34077 · doi:10.1006/jdeq.1993.1048
[12]May, R. M.: Stability and complexity in model ecosystems, (1973)
[13]Xia, Y.; Cao, J.: Almost-periodic solution for an ecological model with infinite delays, Proc. edinb. Math. soc. 50, 229-249 (2007) · Zbl 1130.34044 · doi:10.1017/S0013091504001233
[14]Zhao, Y.; Hutson, V.: Permanence in kolomogorov periodic predator–prey models with diffusion, Nonlinear anal. 23, 651-668 (1994) · Zbl 0823.92031 · doi:10.1016/0362-546X(94)90244-5
[15]Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[16]Krasnoselskii, M. A.: Positive solutions of operator equations, (1964) · Zbl 0121.10604
[17]Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)