Bifurcations of a discrete prey-predator model with Holling type II functional response

Previous Article
Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets
DCDS-B Home
This Issue
Next Article
The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise

July 2010, 14(1): 159-176. doi: 10.3934/dcdsb.2010.14.159

Bifurcations of a discrete prey-predator model with Holling type II functional response

1.	Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.	Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received August 2009 Revised November 2009 Published April 2010

Abstract
Related Papers

Recently a discrete-time prey-predator model with Holling type II was discussed for its bifurcations so as to show its complicated dynamical properties. Simulation illustrated the occurrence of invariant cycles. In this paper we first clarify the parametric conditions of non-hyperbolicity, correcting a known result. Then we apply the center manifold reduction and the method of normal forms to completely discuss bifurcations of codimension 1. We give bifurcation curves analytically for transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation separately, showing bifurcation phenomena not indicated in the previous work for the system.

Keywords: discrete prey-predator model; invariant cycle; transcritical bifurcation; flip bifurcation; Neimark-Sacker bifurcation..

Mathematics Subject Classification: 37G10, 39A05, 34C23, 92D3.

Citation: Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159

[1]	Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009
[2]	Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[3]	Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[4]	Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019
[5]	Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[6]	Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101
[7]	Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141
[8]	Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002
[9]	Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
[10]	Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[11]	Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[12]	Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209
[13]	Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[14]	Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737
[15]	Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[16]	Hui Cao, Yicang Zhou, Zhien Ma. Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1399-1417. doi: 10.3934/mbe.2013.10.1399
[17]	R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423
[18]	Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[19]	M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133
[20]	Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences