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 in a two-competitor, one-prey system with time delay. (English) Zbl 1181.34090
The author considers a delayed two-competitor/one-prey system in which both two competitors exhibit Holling II functional response. By choosing the time delay as a bifurcation parameter, it is found that Hopf bifurcation occurs when the delay passes through a certain critical value. Numerical simulations are performed to illustrate the analytical results.
MSC:
34K60Qualitative investigation and simulation of models
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
References:
[1]Ayala, F. J.: Experimental invalidation of the principle of competitive exclusion, Nature 224, 1076-1079 (1969)
[2]Butler, G. J.; Waltman, P.: Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat, J. math. Biol. 12, 295-310 (1981) · Zbl 0475.92017 · doi:10.1007/BF00276918
[3]Cushing, J. M.: Periodic two-predator one-prey interactions and the time sharing of a resource niche, SIAM J. Appl. math. 44, 392-410 (1984) · Zbl 0554.92016 · doi:10.1137/0144026
[4]Gause, G. F.: The struggle for existence, (1934)
[5]Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981)
[6]Hsu, S. B.; Hubbell, S. P.; Waltman, P.: Competing predators, SIAM J. Appl. math. 35, 617-625 (1978) · Zbl 0394.92025 · doi:10.1137/0135051
[7]Li, X. L.; Wei, J. J.: On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos solitons fract. 26, 519-526 (2005) · Zbl 1098.37070 · doi:10.1016/j.chaos.2005.01.019
[8]Martin, A.; Ruan, S. G.: Predator – prey models with delay and prey harvesting, J. math. Biol. 43, 247-267 (2001) · Zbl 1008.34066 · doi:10.1007/s002850100095
[9]Muratori, S.; Rinaldi, S.: Remarks on competitive coexistence, SIAM J. Appl. math. 49, 1462-1472 (1989) · Zbl 0681.92020 · doi:10.1137/0149088
[10]Kareiva, P.: Renewing the dialogue between theory and experiments in population ecology, Perspectives in ecological theory, 68-88 (1989)
[11]Ruan, S. G.; Ardito, A.; Ricciardi, P.; Deangelis, D. L.: Coexistence in competition models with density-dependent mortality, Comptes rend. Biol. 330, 845-854 (2007)
[12]Ruan, S. G.; Wei, J. J.: On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. appl. Medic. biol. 18, 41-52 (2001) · Zbl 0982.92008
[13]Ruan, S. G.; Wei, J. J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. cont. Dis. 10, 863-874 (2003) · Zbl 1068.34072
[14]Smith, H. L.: The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model, SIAM J. Appl. math. 42, 27-43 (1982) · Zbl 0489.92019 · doi:10.1137/0142003
[15]Song, Y. L.; Wei, J. J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos, Chaos solitons fract. 22, 75-91 (2004) · Zbl 1112.37303 · doi:10.1016/j.chaos.2003.12.075
[16]Wang, A. Y.; Wei, J. J.: Bifurcation analysis in an approachable haematopoietic stem cells model, J. math. Anal. appl. 345, 276-285 (2008) · Zbl 1152.34061 · doi:10.1016/j.jmaa.2008.04.014
[17]Wei, J. J.: Bifurcation analysis in a scalar delay differential equation, Nonlinearity 20, 2483-2498 (2007) · Zbl 1141.34045 · doi:10.1088/0951-7715/20/11/002
[18]Wei, J. J.; Ruan, S. G.: Stability and bifurcation in a neural network model with two delays, Physica D 130, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[19]Xiao, D. M.; Li, W. X.: Stability and bifurcation in a delayed ratio-dependent predator – prey system, Proc. ed. Math. soc. 45, 205-220 (2002) · Zbl 1041.92028 · doi:10.1017/S0013091500001140
[20]Yan, X. P.; Zhang, C. H.: Hopf bifurcation in a delayed lokta – Volterra predator – prey system, Nonlinear anal.: real world appl. 9, 114-127 (2008) · Zbl 1149.34048 · doi:10.1016/j.nonrwa.2006.09.007
[21]Zhang, J. F.; Li, W. T.; Yan, X. P.: Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system, Appl. math. Comput. 198, 865-876 (2008) · Zbl 1134.92034 · doi:10.1016/j.amc.2007.09.045