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 on general Gause-type predator-prey models with delay. (English) Zbl 1246.37095
A class of three-dimensional Gause-type predator-prey models with delay is considered. Firstly, some sufficient conditions for the existence of a Hopf bifurcation are obtained by employing the polynomial theorem and analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

MSC:
37N25Dynamical systems in biology
92B05General biology and biomathematics
35B32Bifurcation (PDE)
37G15Bifurcations of limit cycles and periodic orbits
WorldCat.org
Full Text: DOI
References:
[1] J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707-723, 1968. · doi:10.1002/bit.260100602
[2] M. S. Bartlett, “On theoretical models for competitive and predatory biological systems,” Biometrika, vol. 44, pp. 27-42, 1957. · Zbl 0080.36301 · doi:10.1093/biomet/44.1-2.27
[3] F. Brauer and A. C. Soudack, “Stability regions and transition phenomena for harvested predator-prey systems,” Journal of Mathematical Biology, vol. 7, no. 4, pp. 319-337, 1979. · Zbl 0397.92019 · doi:10.1007/BF00275152
[4] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, HIFR Consulting, Edmonton, Canada, 1987. · Zbl 0448.92023
[5] K. S. Cheng, S. B. Hsu, and S. S. Lin, “Some results on global stability of a predator-prey system,” Journal of Mathematical Biology, vol. 12, no. 1, pp. 115-126, 1981. · Zbl 0464.92021 · doi:10.1007/BF00275207
[6] A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427-451, 1994. · Zbl 0823.92030 · doi:10.1007/BF00160167
[7] S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763-783, 1995. · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[8] M. C. Varriale and A. A. Gomes, “A study of a three species food chain,” Ecological Modelling, vol. 110, no. 2, pp. 119-133, 1998. · doi:10.1016/S0304-3800(98)00062-3
[9] S. W. Zhang and L. S. Chen, “Chaos in three species food chain system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 73-83, 2005. · Zbl 1066.92060 · doi:10.1016/j.chaos.2004.07.014
[10] K. McCann and P. Yodzis, “Bifurcation structure of a 3-species food chain model,” Theoretical Population Biology, vol. 48, no. 2, pp. 93-125, 1995. · Zbl 0854.92022 · doi:10.1006/tpbi.1995.1023
[11] S. B. Hsu and T. W. Hwang, “Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type,” Taiwanese Journal of Mathematics, vol. 3, no. 1, pp. 35-53, 1999. · Zbl 0935.34035
[12] H. I. Freedman and P. Waltman, “Mathematical analysis of some three-species food-chain models,” Mathematical Biosciences, vol. 33, no. 3-4, pp. 257-276, 1977. · Zbl 0363.92022 · doi:10.1016/0025-5564(77)90142-0
[13] J. M. Ginoux, B. Rossetto, and J. L. Jamet, “Chaos in a three-dimensional Volterra-Gause model of predator-prey type,” International Journal of Bifurcation and Chaos, vol. 15, no. 5, pp. 1689-1708, 2005. · Zbl 1092.37541 · doi:10.1142/S0218127405012934
[14] A. Hastings and T. Powell, “Chaos in three-species food chain,” Ecology, vol. 72, no. 3, pp. 896-903, 1991. · doi:10.2307/1940591
[15] W. Jiang and J. Wei, “Bifurcation analysis in a limit cycle oscillator with delayed feedback,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 817-831, 2005. · Zbl 1080.34054 · doi:10.1016/j.chaos.2004.05.028
[16] H. B. Wang and W. H. Jiang, “Hopf-pitchfork bifurcation in van der Pol’s oscillator with nonlinear delayed feedback,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 9-18, 2010. · Zbl 05710064 · doi:10.1016/j.jmaa.2010.03.012
[17] J. J. Wei and W. H. Jiang, “Bifurcation analysis in van der Pol’s oscillator with delayed feedback,” Journal of Computational and Applied Mathematics, vol. 213, no. 2, pp. 604-615, 2008. · Zbl 05254086 · doi:10.1016/j.cam.2007.01.041
[18] R. Xu and M. A. J. Chaplain, “Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks,” Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 148-162, 2002. · Zbl 1013.34074 · doi:10.1006/jmaa.2001.7701
[19] G. Liu, W. Yan, and J. Yan, “Positive periodic solutions for a class of neutral delay Gause-type predator-prey system,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4438-4447, 2009. · Zbl 1181.34089 · doi:10.1016/j.na.2009.03.002
[20] Y. K. Li and Y. Kuang, “Periodic solutions in periodic delayed Gause-type predator-prey systems,” Preceedings of Dynamical Systems and Applications, vol. 3, pp. 375-382, 2001. · Zbl 1009.34063
[21] T. Zhao, Y. Kuang, and H. L. Smith, “Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems,” Nonlinear Analysis. Theory, Methods & Applications. Series A, vol. 28, no. 8, pp. 1373-1394, 1997. · Zbl 0872.34047 · doi:10.1016/0362-546X(95)00230-S
[22] X. Ding and J. Jiang, “Positive periodic solutions in delayed Gause-type predator-prey systems,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1220-1230, 2008. · Zbl 1137.34033 · doi:10.1016/j.jmaa.2007.07.079
[23] S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159-173, 2001. · Zbl 1035.34084
[24] S. G. Ruan and J. J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” Mathematical Medicine and Biology, vol. 18, no. 1, pp. 41-52, 2001. · Zbl 0982.92008
[25] X. Li and J. Wei, “On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 519-526, 2005. · Zbl 1098.37070 · doi:10.1016/j.chaos.2005.01.019
[26] B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridgem, UK, 1981. · Zbl 0474.34002