zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Effects of seasonal growth on ratio dependent delayed prey predator system. (English) Zbl 1221.34187
Summary: The Beddington--DeAngelis ratio dependent prey predator model with time delay is discussed. The existence of Hopf bifurcation is established. The numerical simulations have shown that seasonal growth and delay can give rise to variety of attractors including periodic, quasi-periodic as well as chaotic oscillations. The degree of complexity in the system increases with increasing magnitude of delay, or frequency of seasonal variation. The model parameters involved in functional response can also affect the complexity of the system.

34K18Bifurcation theory of functional differential equations
92D25Population dynamics (general)
Full Text: DOI
[1] Freedman, H. I.: Deterministic mathematical models in population ecology, (1980) · Zbl 0448.92023
[2] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[3] May, R. M.: Stability and complexity in model ecosystems, (1973)
[4] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[5] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator -- prey systems, Nonlinear anal TMA 32, 381-408 (1998) · Zbl 0946.34061 · doi:10.1016/S0362-546X(97)00491-4
[6] Cushing, J. M.: Periodic time-dependent predator -- prey system, SIAM J appl math 32, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[7] Beddington, J. R.: Mutual interference between parasites or predators and its effect on searching efficiency, J animal ecol 44, 331-340 (1975)
[8] Inoue, M.; Kamifukumoto, H.: Scenarios leading to chaos in forced Lotka -- Volterra model, Prog theor phys 71, 930-937 (1984) · Zbl 1074.37522 · doi:10.1143/PTP.71.930 · http://ptp.ipap.jp/link?PTP/71/930/
[9] Macdonald, N.: Time delay in prey -- predator models, Math biosci 28, 321-330 (1976) · Zbl 0324.92016 · doi:10.1016/0025-5564(76)90130-9
[10] May, R. M.: Time delay versus stability in population models with two and three trophic levels, Ecology 4, 315-325 (1973)
[11] Gakkhar, S.; Singh, B.: Prey -- predator models with seasonal variations, Chaos solitons fract 28, 229-239 (2004)
[12] Rinaldi, S.; Muratori, S.; Kuznetsov, Y.: Multiple attractors catastrophes and chaos in seasonally perturbed predator -- prey communities, Bull math biol 55, 15-35 (1993) · Zbl 0756.92026
[13] 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
[14] Wang, W.; Ma, Z.: Harmless delays for uniform persistence, J math anal appl 158, 256-268 (1991) · Zbl 0731.34085 · doi:10.1016/0022-247X(91)90281-4