带恐惧因子和强Allee效应的捕食者-食饵扩散模型的Hopf分支
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摘要
研究一类带恐惧因子和强Allee效应的捕食者-食饵扩散模型的Hopf分支问题.首先分析非负平衡点的局部渐近稳定性,然后以捕获者死亡率作为Hopf分支参数,给出了扩散模型Hopf分支存在的条件;利用中心流形定理和规范型理论,讨论了扩散系统Hopf分支的方向及分支周期解的稳定性.最后利用数值模拟验证了所得结论.
The Hopf bifurcation of a diffusive predator-prey model with fear factors and strong Allee effects is considered.Firstly,the local asymptotic stability of the non-negative equilibrium points is given.Secondly,by choosing the predator's natural growth rate as a bifurcation parameter,the existence conditions of Hopf bifurcation for the model are obtained.Next,the Hopf branch direction of diffusive system and the conditions for the stability of periodic solutions are discussed by using the center manifold theory and the normal form method.Finally,some numerical simulations are presented to verify these theoretical results.
The Hopf bifurcation of a diffusive predator-prey model with fear factors and strong Allee effects is considered.Firstly,the local asymptotic stability of the non-negative equilibrium points is given.Secondly,by choosing the predator's natural growth rate as a bifurcation parameter,the existence conditions of Hopf bifurcation for the model are obtained.Next,the Hopf branch direction of diffusive system and the conditions for the stability of periodic solutions are discussed by using the center manifold theory and the normal form method.Finally,some numerical simulations are presented to verify these theoretical results.
引文
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[2] LIAN Fu-yun,XU Yuan-tong.Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay[J].Applied Mathematics and Computation,2009,215(4):148.
[3] WANG Ming-xin,WU Qiang.Positive solutions of a prey-predator model with predator saturation and competition[J].Journal of Mathematical Analysis and Applications,2008,345(2):708.
[4] YAN Xiang-ping,ZHANG Cun-hua.Stability and Turing instability in a diffusive predator-prey system with Beddington-DeAngelis functional response[J].Nonlinear Analysis:Real World Applications,2014,20(20):1.
[5] WANG Jin-feng,SHI Jun-ping,WEI Jun-jie.Predator-prey system with strong Allee effect in prey[J].Journal of Mathematical Biology,2011,62(3):291.
[6] SASMAL S K.Population dynamics with multiple Allee effects induced by fear factors-A mathematical study on prey-predator interactions[J].Applied Mathematical Modelling,2018(64):10.
[7] LI Zi-zhen,GAO Meng,HUI Cang,et al.Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation[J].Ecological Modelling,2005,185(2):245.
[8] WANG Xiao-ying,ZANETTE L,ZOU Xing-fu.Modelling the fear effect in predator-prey interactions[J].Journal of Mathematical Biology,2016,73(5):1179.
[9] CRANDALL M G,RABINOWITZ P H.Bifurcation from simple eigenvalues[J].Journal of Functional Analysis,1971,8(2):321.
[10] HASSARD B D,KAZARINOFF N D,WAN Y H.Theory and Applications of Hopf Bifurcation[M].Cambridge:Cambridge University Press,1981.
[11] YI Feng-qi,WEI Jun-jie,SHI Junping.Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system[J].Journal of Differential Equations,2009,246(5):1944.