zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Stability and bifurcation analysis for a delayed Lotka--Volterra predator--prey system. (English) Zbl 1095.92071
Summary: The present paper deals with a delayed Lotka-Volterra predator-prey system. By linearizing the equations and by analyzing the locations on the complex plane of the roots of the characteristic equation, we find necessary conditions that the parameters should verify in order to have oscillations in the system. In addition, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. To verify the obtained conditions, a special numerical example is also included.

MSC:
92D40Ecology
34K18Bifurcation theory of functional differential equations
37L10Normal forms, center manifold theory, bifurcation theory
37L15Stability problems of infinite-dimensional dissipative systems
37N25Dynamical systems in biology
34K13Periodic solutions of functional differential equations
WorldCat.org
Full Text: DOI
References:
[1] Beretta, E.; Kuang, Y.: Convergence results in a well-known delayed predator -- prey system. J. math. Anal. appl. 204, 840-853 (1996) · Zbl 0876.92021
[2] Chow, S. -N.; Hale, J. K.: Methods of bifurcation theory. (1982) · Zbl 0487.47039
[3] Faria, T.: Stability and bifurcation for a delayed predator -- prey model and the effect of diffusion. J. math. Anal. appl. 254, 433-463 (2001) · Zbl 0973.35034
[4] Faria, T.; Magalhães, L. T.: Normal form for retarded functional differential equations and applications to bogdanov -- Takens singularity. J. differential equations 122, 201-224 (1995) · Zbl 0836.34069
[5] Freedman, H. I.; Rao, V. S. H.: Stability for a system involving two time delays. SIAM J. Appl. math. 46, 552-560 (1986) · Zbl 0624.34066
[6] Freedman, H. I.; Ruan, S.: Uniform persistence in functional differential equations. J. differential equations 115, 173-192 (1995) · Zbl 0814.34064
[7] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[8] Hale, J. K.; Infante, E. F.; Tsen, F. -S.P.: Stability in linear delay equations. J. math. Anal. appl. 105, 533-555 (1985) · Zbl 0569.34061
[9] He, X.: Stability and delays in a predator -- prey system. J. math. Anal. appl. 198, 355-370 (1996) · Zbl 0873.34062
[10] Kuang, Y.: Delay differential equations with application in population dynamics. (1993) · Zbl 0777.34002
[11] Liu, Z.; Yuan, R.: Stability and bifurcation in a harmonic oscillator with delays. Choas, solitons fractals 23, 551-562 (2005) · Zbl 1078.34050
[12] Ma, W.; Takeuchi, Y.: Stability analysis on a predator -- prey system with distributed delays. J. comput. Appl. math. 88, 79-94 (1998) · Zbl 0897.34062
[13] Wang, W.; Ma, Z.: Harmless delays for uniform persistence. J. math. Anal. appl. 158, 256-268 (1991) · Zbl 0731.34085
[14] Wei, J.; Li, M.: Global existence of periodic solutions in a tri-neuron network model with delays. Physica D 198, 106-119 (2004) · Zbl 1062.34077
[15] Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physica D 130, 255-272 (1999) · Zbl 1066.34511
[16] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator -- prey systems. Nonlinear anal. 28, 1373-1394 (1997) · Zbl 0872.34047