JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (7): 73-81.doi: 10.6040/j.issn.1671-9352.0.2020.496

Previous Articles    

Stability and Turing instability of a predator-prey reaction-diffusion system with schooling behavior

ZHOU Yan, ZHANG Cun-hua*   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Published:2021-07-19

PDF (PC)

152

Abstract: A predator-prey reaction-diffusion system of fish with schooling behavior is considered. By analyzing the distribution of the roots of the characteristic equation of the linearized system at the unique constant positive equilibrium solution in the complex plane in detail, the stability and the Turing instability of the unique positive equilibrium solution of the system are discussed. In addition,the obtained theoretical conclusions are verified numerically by MATLAB software.

Key words: schooling behavior, predator-prey reaction-diffusion system, stability, Turing instability

CLC Number: 

  • O175.2
[1] MURRAY J D. Mathematical biology Ⅱ[M]. Heidelberg: Springer-Verlag, 2002.
[2] YI Fengqi, WEI Junjie, SHI Junping. Bifurcation and spatio-temporal patterns in a homogeneous diffusive predator-prey system[J]. Journal of Differential Equations, 2009, 246:1944-1977.
[3] SONG Yongli, ZOU Xingfu. Bifurcation analysis of a diffusive ratio-dependent predator-prey model[J]. Nonlinear Dynam, 2014, 78:49-70.
[4] SONG Yongli, PENG Yahong, ZOU Xingfu. Persistence, stability and Hopf bifurcation in a diffusive ratio-dependent predator-prey model with delay[J]. Int J Bifur Chaos, 2014, 24:1450093.
[5] YUAN Sanling, XU Chaoqun, ZHANG Tonghua. Spatial dynamics in a predator-prey model with herd behavior[J]. Chaos, 2013, 23:0331023.
[6] AJRALDI Valerio, PITTAVINO Marta, VENTURINO Ezio. Modeling herd behavior in population systems[J]. Nonlinear Anal, RWA, 2011, 12:2319-2333.
[7] TANG Xiaosong, SONG Yongli. Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator-prey model with herd behavior[J]. Appl Math Comput, 2015, 254:375-391.
[8] MANNA D, MAITI A, SAMANTA G P. Analysis of a predator-prey model for exploited fish populations with schooling behavior[J]. Appl Math Comput, 2018, 317:35-48.
[9] JIANG Heping. Turing bifurcation in a diffusive predator-prey model with schooling behavior[J]. Appl Math Lette, 2019, 96:230-235.
[10] 丁亚君,张存华. Lengyel-Epstein反应扩散系统的稳定性和Turing不稳定性[J]. 高校应用数学学报,2018,33:272-280. DING Yajun, ZHANG Cunhua. Stability ang Turing instability in a Lengyel-Epstein reaction-diffusion system[J]. Appl Math J Chinese Univ, 2018, 33:272-280.
[11] YAN Xiang-ping, ZHANG Cun-hua. Stability and turing stability in a diffusive predator-prey system with Beddington-DeAngelis functional response[J]. Nonlinear Analysis: Real World Applications, 2014, 20(20):1-13.
[1] ZHU Yan-lan, ZHOU Wei, CHU Tong, LI Wen-na. Complex dynamic analysis of the duopoly game under management delegation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(7): 32-45.
[2] GUO Hui-ying, YANG Fu-xia, ZHANG Cui-ping. Stability of modules with finite Ext-strongly Ding projective dimensions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(4): 31-38.
[3] Jie TANG,Ling WEI,Rui-si REN,Si-yu ZHAO. Granule description using possible attribute analysis [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(1): 75-82.
[4] YANG Zhong-liang, GUO Gai-hui. Bifurcation analysis of positive solutions for a predator-prey model with B-D functional response [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(7): 9-15.
[5] YANG Yang, WU Bao-wei, WANG Yue-e. Input-output finite time stability of asynchronous switched systems with event-triggered [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(2): 118-126.
[6] WEN Xiao, LIU Qi, GAO Zhen, DON Wai-sun, LYU Xian-qing. Application of local non-intrusive reduced basis method in Rayleigh-Taylor instability [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(2): 109-117.
[7] CHEN Lu, ZHANG Xiao-guang. Research of an epidemic model on adaptive networks [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(9): 76-82.
[8] WANG Zhan-ping, YUAN Kai-ying. Strongly Gorenstein injective modules with respect to a cotorsion pair [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 102-107.
[9] FENG Na-na, WU Bao-wei. Input-output finite time stability for event-triggered control of switched singular systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(3): 75-84.
[10] ZHANG Yu, ZHAO Ren-yu. Gorenstein FP-projective modules and its stability [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 79-85.
[11] DAI Li-hua, HUI Yuan-xian. Almost automorphic solutions for shunting inhibitory cellular neural networks with leakage delays on time scales [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 97-108.
[12] LIU Hua, YE Yong, WEI Yu-mei, YANG Peng, MA Ming, YE Jian-hua, MA Ya-lei. Study of dynamic of a discrete host-parasitoid model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(7): 30-38.
[13] LI Cui-ping, GAO Xing-bao. A neural network for solving l1-norm problems with constraints [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 90-98.
[14] FENG Xiao-zhou, XU Min, WANG Guo-hui. Coexistence solution of a predator-prey system with B-D functional response and toxin effects [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 53-61.
[15] SONG Liang, FENG Jin-shun, CHENG Zheng-xing. Existence and stability for multiple gabor frames [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 17-24.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd