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)
Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters. (English) Zbl 1190.34106
The authors consider a delayed population model with delay-dependent parameters. They prove analytically that the positive equilibrium switches from being stable to unstable and then back to stable as the delay τ increases, and Hopf bifurcations occur between the two critical values of stability changes. The critical values for the stability switches and Hopf bifurcations can be analytically determined. Using the perturbation approach and Floquet technique, they also obtain an approximation to the bifurcating periodic solution and derive formulas for determining the direction and stability of the Hopf bifurcations. Finally, they illustrate their results by some numerical examples.
MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
References:
[1]Hale, J.: Theory of functional differential equations, (1977)
[2]Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[3]Ding, X.; Li, W.: Local Hopf bifurcation and global existence of periodic solutions in a kind of physiological system, Nonlinear anal. RWA 8, 1459-1471 (2007) · Zbl 1163.34053 · doi:10.1016/j.nonrwa.2006.07.013
[4]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[5]Aiello, W. G.; Freedman, H. I.: A time-delay model of single-species growth with stage structure, Math. biosci. 101, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[6]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
[7]Cooke, K. L.; Den Driesche, P. Van: Analysis of an SEIRS epidemic model with two delays, J. math. Biol. 35, 240-260 (1996) · Zbl 0865.92019 · doi:10.1007/s002850050051
[8]Cooke, K. L.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194
[9]Bence, J. R.; Nisbet, R. M.: Space-limited recruitment in open systems: the importance of time delays, Ecology 70, 1434-1441 (1989)
[10]Nisbet, R. M.; Gurney, W. S. C.: Modelling fluctuating populations, (1982) · Zbl 0593.92013
[11]Nisbet, R. M.; Gurney, W. S. C.; Metz, J. A. J.: Stage structure models applied in evolutionary ecology, Biomathematics 18, 428-449 (1989)
[12]Mackey, M. C.; Glass, L.: Oscillations and chaos in physiological control systems, Science 197, 287-289 (1977)
[13]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[14]Mackey, M. C.: Periodic auto-immune hemolytic anemia: an induced dynamical disease, Bull. math. Biol. 41, 829-834 (1979) · Zbl 0414.92012
[15]Mackey, M. C.: Some models in hemopoiesis: predictions and problems in biomathematics and cell kinetics, (1981)
[16]Hale, J.; Sternberg, N.: Onset of chaos in differential delay equations, J. comput. Phys. 77, 221-239 (1988) · Zbl 0644.65050 · doi:10.1016/0021-9991(88)90164-7
[17]Barclay, H. J.; Den Driessche, P. Van: A model for a species with two life history stages and added mortality, Ecol. model. 11, 157-166 (1980)
[18]Fisher, M. E.; Goh, B. S.: Stability results for delayed recruitment models in population dynamics, J. math. Biol. 19, 147-156 (1984) · Zbl 0533.92017 · doi:10.1007/BF00275937
[19]Tognetti, K.: The two stage stochastic model, Math. biosci. 25, 195-204 (1975)
[20]Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. anal. 33, 1144-1165 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086
[21]Stokes, A.: On the stability of a limit cycle of autonomous functional differential equations, Contr. diff. Eqns. 3, 121-140 (1964) · Zbl 0135.30903
[22]Der Heiden, U. An; Walther, H. O.: Existence of chaos in control systems with delayed feedback, J. differential equations 47, 273-295 (1983) · Zbl 0477.93040 · doi:10.1016/0022-0396(83)90037-2