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)
Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes. (English) Zbl 1156.34342
Summary: The model analyzed in this paper is based on the model set forth by {\it M.A. Aziz-Alaoui} and {\it M. Daher Okiye} [Appl. Math. Lett. 16, No. 7, 1069--1075 (2003; Zbl 1063.34044)]; {\it A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel}, Analysis of a a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., in Press.] with time delay, which describes the competition between predator and prey. This model incorporates a modified version of Leslie-Gower functional response as well as that of the Holling-type II. In this paper, we consider the model with one delay and a unique non-trivial equilibrium $E^{*}$ and the three others are trivial. Their dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation at $E^{*}$ for small and large delays and at the third trivial equilibrium that is proven to exist as the delay (taken as a parameter of bifurcation) crosses some critical values. We illustrate these results by numerical simulations.

34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K60Qualitative investigation and simulation of models
Full Text: DOI
[1] Aziz-Alaoui, M. A.; Okiye, M. Daher: Boundedness and global stability for a predator -- prey model with modified Leslie -- gower and Holling-type II schemes, Appl. math. Lett. 16, 1069-1075 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[2] Beretta, E.; Kuang, Y.: Global analyses in some delayed ratio-depended predator -- prey systems, Nonlinear anal. Theory methods appl. 32, No. 3, 381-408 (1998) · Zbl 0946.34061 · doi:10.1016/S0362-546X(97)00491-4
[3] Cooke, K. L.; Grossman, Z.: Discrete delay, distributed delay and stability switches, J. math. Anal. appl. 86, 592-627 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[4] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics, (1977) · Zbl 0363.92014
[5] Gopalsamy, K.: Stability on the oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[6] Hale, J. K.: Theory of functional differential equations, (1997) · Zbl 1098.34552
[7] Kuang, Y.: Delay differential equations, with applications in population dynamics, (1993) · Zbl 0777.34002
[8] Macdonald, N.: Time lags in biological models, (1978) · Zbl 0403.92020
[9] Nindjin, A. F.; Aziz-Alaoui, M. A.; Cadivel, M.: Analysis of a predator -- prey model with modified Leslie -- gower and Holling-type II schemes with time delay, Nonlinear anal. Real world appl. 7, No. 5, 1104-1118 (2006) · Zbl 1104.92065 · doi:10.1016/j.nonrwa.2005.10.003
[10] Upadhyay, R. K.; Rai, V.: Crisis-limited chaotic dynamics in ecological systems, Chaos solitons fractals 12, No. 2, 205-218 (2001) · Zbl 0977.92033 · doi:10.1016/S0960-0779(00)00141-7
[11] Upadhyay, R. K.; Iyengar, S. R. K.: Effect of seasonality on the dynamics of 2 and 3 species prey -- predator system, Nonlinear anal.: realworld appl. 6, 509-530 (2005) · Zbl 1072.92058 · doi:10.1016/j.nonrwa.2004.11.001
[12] Xu, R.; Chaplain, M. A. J.: Persistence and global stability in a delayed predator -- prey system with michaelis -- menten type functional response, Appl. math. Comput. 130, 441-455 (2002) · Zbl 1030.34069 · doi:10.1016/S0096-3003(01)00111-4