Dynamical Behaviors in a Discrete Predator-prey Model with a Prey Refuge
By incorporating a prey refuge, this paper proposes new discrete Leslie–Gower predator–prey systems with and without Allee effect. The existence of fixed points are established and the stability of fixed points are discussed by analyzing the modulus of characteristic roots.
[1] Yongli Song, Sanling Yuan, Jianming Zhang. Bifurcation analysis in the
delayed Leslie-Gower predator-prey system. Appl. Math. Modelling 33
(2009), 4049-4061.
[2] Eduardo Gonz'alez-Olivares, Rodrigo Ramos-Jiliberto. Dynamic consequences
of prey refuges in a simple model system: more prey, fewer
predators and enhanced stability. Ecological Modelling 166 (2003), 135-
146.
[3] Liujuan Chen, Fengde Chen, Lijuan Chen. Qualitative analysis of a
predator-prey model with Holling type II functional response incorporating
a constant prey refuge. Nonlinear Analysis: RWA 11 (2010), 246-252.
[4] Vlastimil Kˇrivan. Effects of Optimal Antipredator Behavior of Prey on
PredatorCPrey Dynamics: The Role of Refuges. Theoretical Population
Biology 53 (1998), 131-142.
[5] J.D. Murray. Mathematical Biology. New York: Springer-Verlag, 1989.
[6] R.P. Agarwal, P.J.Y. Wong. Advance Topics in Difference Equations.
Dordrecht: Kluwer, 1997.
[7] Canan Celik, Oktay Duman. Allee effect in a discrete-time predator-prey
system. Chaos, Solitons and Fractals 40 (2009), 1956-1962.
[8] P.H. Leslie. Some further notes on the use of matrices in population
mathematics. Biometrika 35 (1948), 213-245.
[9] P.H. Leslie. A stochastic model for studying the properties of certain
biological systems by numerical methods. Biometrika 45 (1958), 16-31.
[10] W. Ko, K. Ryu. Qualitative analysis of a predator-prey model with
Holling type II functional response incorporating a prey refuge. Journal
of Differential Equations 231 (2006), 534-550.
[11] T. Kumar Kar. Stability analysis of a prey-predator model incorporating
a prey refuge. Communications in Nonlinear Science and Numeric
Simulation 10 (2005), 681-691.
[12] Y. Huang, F. Chen, Z. Li. Stability analysis of a prey-predator model with
Holling type III response function incorporating a prey refuge. Applied
Mathematics and Computation 182 (2006), 672-683.
[13] Xiaoli Liu, Dongmei Xiao. Complex dynamic behaviors of a discrete-
time predator-prey system. Chaos, Solitons and Fractals 32 (2007), 80-
94.
[1] Yongli Song, Sanling Yuan, Jianming Zhang. Bifurcation analysis in the
delayed Leslie-Gower predator-prey system. Appl. Math. Modelling 33
(2009), 4049-4061.
[2] Eduardo Gonz'alez-Olivares, Rodrigo Ramos-Jiliberto. Dynamic consequences
of prey refuges in a simple model system: more prey, fewer
predators and enhanced stability. Ecological Modelling 166 (2003), 135-
146.
[3] Liujuan Chen, Fengde Chen, Lijuan Chen. Qualitative analysis of a
predator-prey model with Holling type II functional response incorporating
a constant prey refuge. Nonlinear Analysis: RWA 11 (2010), 246-252.
[4] Vlastimil Kˇrivan. Effects of Optimal Antipredator Behavior of Prey on
PredatorCPrey Dynamics: The Role of Refuges. Theoretical Population
Biology 53 (1998), 131-142.
[5] J.D. Murray. Mathematical Biology. New York: Springer-Verlag, 1989.
[6] R.P. Agarwal, P.J.Y. Wong. Advance Topics in Difference Equations.
Dordrecht: Kluwer, 1997.
[7] Canan Celik, Oktay Duman. Allee effect in a discrete-time predator-prey
system. Chaos, Solitons and Fractals 40 (2009), 1956-1962.
[8] P.H. Leslie. Some further notes on the use of matrices in population
mathematics. Biometrika 35 (1948), 213-245.
[9] P.H. Leslie. A stochastic model for studying the properties of certain
biological systems by numerical methods. Biometrika 45 (1958), 16-31.
[10] W. Ko, K. Ryu. Qualitative analysis of a predator-prey model with
Holling type II functional response incorporating a prey refuge. Journal
of Differential Equations 231 (2006), 534-550.
[11] T. Kumar Kar. Stability analysis of a prey-predator model incorporating
a prey refuge. Communications in Nonlinear Science and Numeric
Simulation 10 (2005), 681-691.
[12] Y. Huang, F. Chen, Z. Li. Stability analysis of a prey-predator model with
Holling type III response function incorporating a prey refuge. Applied
Mathematics and Computation 182 (2006), 672-683.
[13] Xiaoli Liu, Dongmei Xiao. Complex dynamic behaviors of a discrete-
time predator-prey system. Chaos, Solitons and Fractals 32 (2007), 80-
94.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57381", author = "Kejun Zhuang and Zhaohui Wen", title = "Dynamical Behaviors in a Discrete Predator-prey Model with a Prey Refuge", abstract = "By incorporating a prey refuge, this paper proposes new discrete Leslie–Gower predator–prey systems with and without Allee effect. The existence of fixed points are established and the stability of fixed points are discussed by analyzing the modulus of characteristic roots.
", keywords = "Leslie-Gower, predator–prey model, prey refuge, allee effect.", volume = "5", number = "8", pages = "1296-3", }
