Please wait a minute...
高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2014, Vol. 29 Issue (4): 431-442    DOI:
    
The stability and bifurcation behavior of pest and natural enemy models with piecewise constant arguments and refuge
WANG Lie
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China
Download:   PDF(0KB)
Export: BibTeX | EndNote (RIS)      

Abstract  In this paper, the stability and bifurcation behavior of the pest and natural enemy model with piecewise constant arguments and refuge are investigated. First the discrete solution determined the dynamical behavior of the model is achieved by calculation. Thus, the linearized stability theorem is applied to find some sufficient conditions for the local asymptotic stability of equilibria. Secondly by choosing the intrinsic rate of increase or the refuge ratio as the bifurcation parameter, it is shown that the discrete solution of the model undergoes Flip bifurcation and NeimarkSacker bifurcation by using the bifurcation theory. Furthermore, the explicit formulaes determining the stability of bifurcating periodic solution are derived by applying the normal form and center manifold theorems. Finally, numerical simulations are performed to illustrate the analytic results and exhibit the complex dynamical behaviors.

Key words： natural enemy      pest      piecewise constant arguments      refuge      stability      bifurcation     
Received: 16 April 2014      Published: 08 June 2018
CLC:  O175.1  
Service
E-mail this article
Add to my bookshelf
Add to citation manager
E-mail Alert
RSS
Articles by authors
WANG Lie
Cite this article:

WANG Lie. The stability and bifurcation behavior of pest and natural enemy models with piecewise constant arguments and refuge. Applied Mathematics A Journal of Chinese Universities, 2014, 29(4): 431-442.

URL:

http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2014/V29/I4/431


带有分段常数变量和避难所的天敌-害虫模型的稳定性和分支行为

研究一类带有分段常数变量和避难所的天敌-害虫模型的稳定性和分支行为. 首先通过计算转化得到天敌-害虫模型对应的差分模型, 利用线性稳定性理论讨论了正平衡态局部渐近稳定的充分条件. 其次以害虫种群的内禀增长率或逃脱率为分支参数, 利用分支理论研究了模型正平衡态处产生翻转分支周期解和Neimark-Sacker分支周期解的充分条件; 并且使用正规形理论和中心流形定理构造了判断分支周期解稳定性的阈值. 最后数值模拟验证了理论分析的正确性, 并展示了该模型复杂的动力学行为.

关键词： 天敌,  害虫,  分段常数变量,  避难所,  稳定性,  分支 
[1] SHI Lei , XIAO Qing-kun , LIU Bao-qing. Existence and stability of front solutions of the amplitude equations for rotating magnetoconvection[J]. Applied Mathematics A Journal of Chinese Universities, 2017, 32(2): 142-148.
[2] KONG Xiang-shan, LI Hai-tao, ZHAO Hong-xin, LV Xun-jing. Bifurcation of positive solutions for a class of integral boundary value problems of fractional differential equations[J]. Applied Mathematics A Journal of Chinese Universities, 2017, 32(1): 13-22.
[3] GAO Ping. Strong stability of $(\alpha,\beta)$-mixing sequences[J]. Applied Mathematics A Journal of Chinese Universities, 2016, 31(4): 405-412.
[4] TIAN Xiao-hong, XU Rui, WANG Zhi-li. Global exponential stability and Hopf bifurcation of inertial Cohen-Grossberg neural networks with time delays in leakage terms[J]. Applied Mathematics A Journal of Chinese Universities, 2016, 31(4): 428-440.
[5] LI Shao-e, FENG Wei-zhen. Practical stability of impulsive switched systems with time delay by Lyapunov-Razumikhin method[J]. Applied Mathematics A Journal of Chinese Universities, 2016, 31(3): 327-337.
[6] YANG Yu, ZHOU Jin-ling. Global stability of a diffusive virus dynamics model with Beddington-DeAngelis incidence function[J]. Applied Mathematics A Journal of Chinese Universities, 2016, 31(2): 161-166.
[7] FU Jin-bo, CHEN Lan-sun, CHENG Rong-fu. Global stability of a delayed viral infection model with latent period and immune response[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(4): 379-388.
[8] XU Chao-qun, LI Hong-en, YUAN San-ling. Study on a stochastic phage-bacteria model in which the death rate of phage is influenced by noise[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(3): 253-261.
[9] ZHANG Dao-xiang, CAO Lei. Analysis of stability and bifurcation of a SEIR epidemic model with nonlinear incidence[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(2): 157-164.
[10] YANG Xiao-zhong, ZHANG Xue, WU Li-fei. A kind of efficient difference method for time-fractional option pricing model[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(2): 234-244.
[11] YANG Hong, ZHU Huan. Dynamical behavior of SIR epidemical model with time delay[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(2): 165-170.