Stability analysis and bifurcations in a diffusive predator-prey system

Previous Article
Nonlocal problems for parabolic inclusions
PROC Home
This Issue
Next Article
A viscoelastic model for avascular tumor growth

2009, 2009(Special): 92-100. doi: 10.3934/proc.2009.2009.92

Stability analysis and bifurcations in a diffusive predator-prey system

Leonid Braverman ^1, and Elena Braverman ^2,

1.	Athabasca University, 1 University Drive, Athabasca, AB T9S 3A3, Canada
2.	Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

Received July 2008 Revised August 2009 Published September 2009

Abstract
Related Papers

We consider a predator-prey system with logistic-type growth and linear diffusion for the prey, Holling type II functional response and the nonlinear diffusion $\nabla \left( \sigma n b \nabla b)$ for the predator, where $n$ is the prey (nutrient) and $b$ is the predator (bacteria) density, respectively. This corresponds to a collective-type behavior for predators: they spread faster when numerous enough at a front line. We present the complete linear stability analysis for this case, discuss some results of numerical simulations: the asymptotic behavior of the model (with the zero Neumann boundary conditions in a 2-D domain) was similar to the relevant Lotka-Volterra system of ordinary differential equations.

Keywords: linear stability analysis, nonlinear diffusion term, Diffusive predator-prey system, Holling type II functional response, Hopf bifurcation.

Mathematics Subject Classification: Primary: 92D25 Secondary: 35K57, 35Q51, 35Q80.

Citation: Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92

[1]	Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure & Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481
[2]	Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[3]	Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
[4]	Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[5]	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
[6]	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
[7]	Jian Zu, Wendi Wang, Bo Zu. Evolutionary dynamics of prey-predator systems with Holling type II functional response. Mathematical Biosciences & Engineering, 2007, 4 (2) : 221-237. doi: 10.3934/mbe.2007.4.221
[8]	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
[9]	Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653
[10]	Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
[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]	Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621
[13]	Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295
[14]	Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
[15]	Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857
[16]	Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807
[17]	Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117
[18]	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
[19]	Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607
[20]	Wan-Tong Li, Yong-Hong Fan. Periodic solutions in a delayed predator-prey models with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 175-185. doi: 10.3934/dcdsb.2007.8.175

Impact Factor:

American Institute of Mathematical Sciences