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 of a predator-prey system with stage structure and harvesting. (English) Zbl 1207.34106
Summary: A two-species predator-prey system with stage structure and harvesting is investigated. The existence of Hopf bifurcations for the system is proven. Also, the stability and directions of the Hopf bifurcations are determined by applying normal form theory and the center manifold theorem.
MSC:
34K60Qualitative investigation and simulation of models
92D25Population dynamics (general)
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K19Invariant manifolds (functional-differential equations)
References:
[1]Hale, J. K.: Theory of functional differential equations, (1977)
[2]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
[3]Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays, Physica D 130, 225-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[4]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
[5]Freedman, H. I.: Deterministic mathematical models in population ecology, (1980)
[6]Takeuchi, Y.: Global dynamical properties of Lotka–Volterra systems, (1996) · Zbl 0844.34006
[7]Aiello, W. G.; Freedman, H. I.: A time-delay model of single-species growth with stage structure, Math. biol. 101, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[8]Aiello, W. G.; Freedman, H. I.; Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. math. 52, 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[9]Freedman, H. I.; Wu, J.: Persistence and global asymptotical stability of single species dispersal model with stage structure, Quart. appl. Math. 49, 351-371 (1991) · Zbl 0732.92021
[10]Gourley, S. A.; Kuang, Y.: A stage structured predator–prey model and its dependence on maturation delay and death rate, J. math. Biol. 49, 188-200 (2004) · Zbl 1055.92043 · doi:10.1007/s00285-004-0278-2
[11]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[12]Song, X.; Chen, L.: Optimal harvesting and stability for a predator–prey system with stage structure, Acta math. Appl. sin. 18, 307-314 (2002) · Zbl 1054.34125 · doi:10.1007/s102550200042
[13]Diekmann, O.; Nisbet, R.; Gurney, W.; Bosch, F.: Simple mathematical models for cannibalism: a critique and a new approach, Math. biol. 78, 21-46 (1986) · Zbl 0587.92020 · doi:10.1016/0025-5564(86)90029-5
[14]Murray, J. D.: Mathematical biology, (1989)
[15]Hassard, B.; Kazarinoff, D.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[16]Yosida, K.: Functional analysis, (1996)