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)
Bifurcations for a predator-prey system with two delays. (English) Zbl 1132.34053
Authors’ abstract: In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global bifurcation results of J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], we may show the global existence of periodic solutions.
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K60Qualitative investigation and simulation of models
92D25Population dynamics (general)
[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 · doi:10.1006/jmaa.1996.0471
[2]Chow, S. -N.; Hale, J. K.: Methods of bifurcation theory, (1982)
[3]Cushing, J. M.: Periodic time-dependent predator – prey systems, SIAM J. Appl. math. 32, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[4]Faria, T.; Magalháes, 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 · doi:10.1006/jdeq.1995.1145
[5]Faria, T.; Magalháes, L. T.: Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations 122, 181-200 (1995) · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[6]Faria, T.: Stability and bifurcation for a delay predator – prey model and the effect of diffusion, J. math. Anal. appl. 254, 433-463 (2001) · Zbl 0973.35034 · doi:10.1006/jmaa.2000.7182
[7]Giannakopoulos, F.; Zapp, A.: Local and Hopf bifurcation in a scalar delay differential equation, J. math. Anal. appl. 237, 425-450 (1999) · Zbl 1126.34371 · doi:10.1006/jmaa.1999.6431
[8]Hadelen, K. P.; Tomiuk, J.: Periodic solutions of differential-difference equations, Arch. ration. Mech. anal. 65, 87-95 (1977) · Zbl 0426.34058 · doi:10.1007/BF00289359
[9]Hale, J.; Lunel, S. V.: Introduction to functional differential equations, (1993)
[10]He, X.: Stability and delays in a predator – prey system, J. math. Anal. appl. 198, 355-370 (1996)
[11]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993)
[12]Leung, A.: Periodic solutions for a prey – predator differential delay equation, J. differential equations 26, 391-403 (1977) · Zbl 0365.34078 · doi:10.1016/0022-0396(77)90087-0
[13]Mallet-Paret, J.; Nussbaum, R. D.: Global continuation and asymptotic behavior for periodic solutions of a differential – delay equation, Ann. math. Pura appl. 145, 33-128 (1986) · Zbl 0617.34071 · doi:10.1007/BF01790539
[14]Mallet-Paret, J.; Nussbaum, R. D.: A differential – delay equation arising in optics and physiology, SIAM J. Math. anal. 20, 249-292 (1989) · Zbl 0676.34043 · doi:10.1137/0520019
[15]Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays, Proc. roy. Soc. Edinburgh sect. A 129, 1017-1032 (1999) · Zbl 0946.34062 · doi:10.1017/S0308210500031061
[16]Song, Y.; Wei, J.: Local and global Hopf bifurcation in a delayed hematopoiesis model, Internat. J. Bifur. chaos appl. Sci. engrg. 14, 3909-3919 (2004) · Zbl 1090.37547 · doi:10.1142/S0218127404011697
[17]Song, Y.; Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator – prey system, J. math. Anal. appl. 301, 1-21 (2005) · Zbl 1067.34076 · doi:10.1016/j.jmaa.2004.06.056
[18]Taboas, P.: Periodic solutions of a planar delay equation, Proc. roy. Soc. Edinburgh sect. A 116, 85-101 (1990) · Zbl 0719.34125 · doi:10.1017/S0308210500031395
[19]Wang, W.; Ma, Z.: Harmless delays for uniform persistence, J. math. Anal. appl. 158, 256-268 (1991) · Zbl 0731.34085 · doi:10.1016/0022-247X(91)90281-4
[20]Wei, J.; Li, M.: Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear anal. 60, 1351-1367 (2005) · Zbl 1144.34373 · doi:10.1016/j.na.2003.04.002
[21]Wei, J.; Li, Y.: Global existence of periodic solutions in a tri-neuron network model with delays, Phys. D 198, 106-119 (2004) · Zbl 1062.34077 · doi:10.1016/j.physd.2004.08.023
[22]Wu, J.: Theory and applications of partial functional differential equations, (1996)
[23]Wu, J.: Symmetric functional differential equations and neural networks with memory, Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2
[24]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 · doi:10.1016/0362-546X(95)00230-S