Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes

被引：62

作者：

Yafia, Radouane ^{[1
]}

El Adnani, Fatiha ^{[2
]}

Alaoui, Hamad Talibi ^{[2
]}

机构：

[1] Univ Ibn Zohr, Fac Polydisciplinaire, Ouarzazate, Morocco

[2] Univ Chouaib Doukkali, Dept Math, Fac Sci, El Jadida, Morocco

来源：

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS | 2008年 / 9卷 / 05期

关键词：

predator-prey system; delay differential equations; stability/unstability; Hopf bifurcation; periodic solutions;

D O I：

10.1016/j.nonrwa.2006.12.017

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The model analyzed in this paper is based on the model set forth by [M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069-1075, 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. (c) 2007 Elsevier Ltd. All rights reserved.

引用

页码：2055 / 2067

页数：13

共 12 条

[1] Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes [J].

Aziz-Alaoui, MA ;

Okiye, MD .

APPLIED MATHEMATICS LETTERS, 2003, 16 (07) :1069-1075

[2] Global analyses in some delayed ratio-dependent predator-prey systems [J].

Beretta, E ;

Kuang, Y .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1998, 32 (03) :381-408

[3] DISCRETE DELAY, DISTRIBUTED DELAY AND STABILITY SWITCHES [J].

COOKE, KL ;

GROSSMAN, Z .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1982, 86 (02) :592-627

[4]

CUSHING J. M, 2013, Integrodifferential Equations and Delay Models in Population Dynamics, DOI DOI 10.1007/978-3-642-93073-7

[5]

Gopalsamy K., 2013, Stability and Oscillations in Delay Differential Equations of Population Dynamics, V74

[6]

Hale JK, 1997, THEORY FUNCTIONAL DI

[7]

Kuang Y., 1993, DELAY DIFFERENTIAL E

[8]

Macdonald N., 1978, TIME LAGS BIOL MODEL, DOI DOI 10.1007/978-3-642-93107-9

[9] Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay [J].

Nindjin, A. F. ;

Aziz-Alaoui, M. A. ;

Cadivel, M. .

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 2006, 7 (05) :1104-1118

[10] Effect of seasonality on the dynamics of 2 and 3 species prey-predator systems [J].

Upadhyay, RK ;

Iyengar, SRK .

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 2005, 6 (03) :509-530

← 1 2 →