JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (9): 83-90.doi: 10.6040/j.issn.1671-9352.0.2021.500

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Stability and Turing instability in the diffusive Beddington-DeAngelis-Tanner predator-prey model

SUN Chun-jie, ZHANG Cun-hua*   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Published:2022-09-15

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Abstract: The diffusive Beddington-DeAngelis-Tanner predator-prey model with Neumann boundary condition is considered. By using the linearization method and analyzing the distribution of the eigenvalues of the associated eigenvalue problem on the complex plane, the local asymptotic stability and Turing instability of the unique constant positive equilibrium solution are obtained. The numerical simulation using MATLAB software package is carried out to verify the theoretical results obtained.

Key words: diffusive predator-prey model, local asymptotic stability, Turing instability

CLC Number: 

  • O175.26
[1] XU Xiaofeng, LIU Ming. Global Hopf bifurcation of a general predator-prey system with diffusion and stage structures[J]. Journal of Differential Equations, 2020, 269(10):8370-8386.
[2] 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.
[3] 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.
[4] JIANG Heping. Turing bifurcation in a diffusive predator-prey model with schooling behavior[J]. Appl Math Lette, 2019, 96:230-235.
[5] XIA Lei, SUN Jiaojiao, YIN Zuguang, et al. Dynamics and response reshaping of nonlinear predator-prey system undergoing random abrupt disturbances[J]. Applied Mathematics and Mechanics, 2021, 42(8):1123-1134.
[6] QIU Shuyuan, MU Chunlai, TU Xinyu. Dynamics for a three-species predator-prey model with density-dependent motilities[J]. Journal of Dynamics and Differential Equations, 2021, 33:1-25.
[7] MAY R M. Stability and complexity in model ecosystems[M]. New Jersey: Princeton University Press, 1973: 89-96.
[8] TANNER J T. The stability and intrinsic growth rates of prey and predator populations[J]. Ecology, 1975, 56:855-867.
[9] YAN Xiangping, ZHANG Cunhua. Stability and turing instability in a diffusive predator-prey system with Beddington-DeAngelis function response[J]. Nonlinear Analysis: Real World Applications, 2014, 20(20):1-13.
[10] ARANCIBIA I C, BODE M, FLORES J, et al. Turing patterns in a diffusive Holling-Tanner predator-prey model with an alternative food source for the predator[J]. Communications in Nonlinear Science and Numerical Simulation, 2021, 99:1-15.
[11] 程惠东, 侯晓雨.具有Holling-Tanner功能反应的捕食模型研究[J].数学建模及其应用, 2021,10(2):32-43. CHENG Huidong, HOU Xiaoyu. Study on the predator-prey model with Holling-Tanner functional response[J]. Mathematical Modeling and Its Applications, 2021, 10(2):32-43.
[1] ZHOU Yan, ZHANG Cun-hua. Stability and Turing instability of a predator-prey reaction-diffusion system with schooling behavior [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(7): 73-81.
[2] LI Hai-xia. Qualitative analysis of a diffusive predator-prey model with density dependence [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(9): 54-61.
[3] ZHANG Dao-xiang, ZHAO Li-xian, HU Wei. Turing instability induced by cross-diffusion in a three-species food chain model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(1): 88-97.
[4] ZHANG Li-na, WU Shou-yan. Global behavior of solutions for a modified LeslieGower #br# predator-prey system with diffusion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(1): 86-91.
[5] HOU Qiang， JIN Zhen. Analysis of a HollingTanner model with ratiodependence
incorporating a prey refuge [J]. J4, 2009, 44(3): 56-60 .
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