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)
Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects. (English) Zbl 1181.37119
Summary: This paper is concerned with a delayed predator-prey diffusive system with Neumann boundary conditions. The bifurcation analysis of the model shows that Hopf bifurcation can occur by regarding the delay as the bifurcation parameter. In addition, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are also discussed by employing the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, the effect of the diffusion on bifurcated periodic solution is considered.
MSC:
37N25Dynamical systems in biology
92D25Population dynamics (general)
37G40Symmetries, equivariant bifurcation theory
References:
[1]Fan, Y. H.; Li, W. T.: Permanence in delayed ratio-dependent predator–prey models with monotonic functional responses, Nonlinear anal. RWA 8, 424-434 (2007) · Zbl 1152.34368 · doi:10.1016/j.nonrwa.2005.12.003
[2]Fan, Y. H.; Li, W. T.; Wang, L. L.: Periodic solutions of delayed ratio-dependent predator–prey models with monotonic or nonmonotonic functional responses, Nonlinear anal. RWA 5, 247-263 (2004) · Zbl 1069.34098 · doi:10.1016/S1468-1218(03)00036-1
[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 · doi:10.1006/jmaa.2000.7182
[4]Faria, T.; Magalhães, 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
[5]Li, W. T.; Yan, X. P.; Zhang, C. H.: Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos solitons fractals 38, 227-237 (2008) · Zbl 1142.35471 · doi:10.1016/j.chaos.2006.11.015
[6]Yan, X. P.; Li, W. T.: Bifurcation and global periodic solutions in a delayed facultative mutualism system, Physica D 227, 51-69 (2007) · Zbl 1123.34055 · doi:10.1016/j.physd.2006.12.007
[7]Zhou, L.; Tang, Y.; Hussein, S.: Stability and Hopf bifurcation for a delay competitive diffusion system, Chaos solitons fractals 14, 1201-1225 (2002) · Zbl 1038.35147 · doi:10.1016/S0960-0779(02)00068-1
[8]Celik, C.: The stability and Hopf bifurcation for a predator–prey system with time delay, Chaos solitons fractals 37, 87-99 (2008) · Zbl 1152.34059 · doi:10.1016/j.chaos.2007.10.045
[9]Fowler, M. S.; Ruxton, G. D.: Population dynamic consequences of allee effects, J. theor. Biol. 42, 728-737 (1980)
[10]Hadjiavgousti, D.; Ichtiaroglou, S.: Allee effect in a predator–prey system, Chaos solitons fractals 36, 334-342 (2008) · Zbl 1128.92045 · doi:10.1016/j.chaos.2006.06.053
[11]Jang, S. R. J.: Allee effects in a discrete-time host–parasitoid model, J. difference equ. Appl. 12, 165-181 (2006) · Zbl 1088.92058 · doi:10.1080/10236190500539238
[12]Scheuring, I.: Allee effect increases the dynamical stability of populations, J. theor. Biol. 199, 407-414 (1999)
[13]Liu, Z.; Yuan, R.: The effect of diffusion for a predator–prey system with nonmonotonic functional response, Internat. J. Bifur. chaos 12, 4309-4316 (2004) · Zbl 1074.35055 · doi:10.1142/S0218127404011867
[14]Yan, X. P.; Li, W. T.: Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. chaos 18, 441-453 (2008) · Zbl 1162.35319 · doi:10.1142/S0218127408020434
[15]Wu, J.: Theory and applications of partial functional differential equations, (1989)
[16]Faria, T.: Normal form and Hopf bifurcation for partial differential equations with delays, Trans. amer. Math. soc. 352, 2217-2238 (2000) · Zbl 0955.35008 · doi:10.1090/S0002-9947-00-02280-7