一类混合比率依赖食物链扩散模型的Hopf分支
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摘要
研究了一类基于混合比率依赖的三种群食物链扩散模型.利用Hurwitz判据讨论了非负常数平衡解的稳定性,并通过理论分析研究了该系统空间齐次和空间非齐次的Hopf分支,同时利用规范型理论和中心流形定理给出Hopf分支方向和分支周期解稳定性的判据.最后借助Matlab软件进行数值模拟,验证补充理论分析结果.
A class of hybrid ratio-dependent three species food chain model with diffusion is investigated.Hurwitz criterion is used to discuss the stability of the non-negative constant equilibrium solution about this system. Then, the Hopf bifurcation is considered. At the same time, the Hopf bifurcation direction and stability of bifurcation periodic solutions are discussed making use of the normal form method and the center manifold theorem. Finally with the help of matlab software, some numerical simulations are shown to support and supply the results of theoretical analysis.
A class of hybrid ratio-dependent three species food chain model with diffusion is investigated.Hurwitz criterion is used to discuss the stability of the non-negative constant equilibrium solution about this system. Then, the Hopf bifurcation is considered. At the same time, the Hopf bifurcation direction and stability of bifurcation periodic solutions are discussed making use of the normal form method and the center manifold theorem. Finally with the help of matlab software, some numerical simulations are shown to support and supply the results of theoretical analysis.
引文
[1]李艳玲.应用偏微分方程.西安:西安交通大学出版社,2009(Li Yanling. The Application of Partial Differential Equation. Xi'an:Xi'an Jiaotong University, 2009)
[2] Hastings A, Powell T. Chaos in a three-species food chain. Ecology, 1991, 72(3):896-903
[3] Gakkhar S, Naji R K. Chaos in a three species ratio dependent food chain. Chaos, Solitons and Fractals, 2002, 14(1):771-778
[4] Wang F Y, Pang G P. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons and Fractals, 2008, 36(5):1366-1376
[5] Yi F Q, Wei J J, Shi J P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predatorprey system. Journal of Differential Equations, 2009, 246(5):1944-1977
[6] Zuo W J, Song Y L. Stability and bifurcation analysis of a reaction-diffusion equation with distributed delay. Nonlinear Dynamics, 2015, 79(1):437-454
[7] Shi Q Y, Shi J P, Song Y L. Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition. J. Differential Equations, 2017, 263(10):6537-6575
[8] Guo G H, Li B F, Wei M H, Wu J H. Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model. J. Math. Anal. Appl., 2012, 391(1):265-277
[9] Guo G H, Liu L, Li B F, Li J. Qualitative analysis on positive steady-state solutions for an autocatalysis model with high order. Nonlinear Anal. RWA,2018, 41:665-691
[10] Liu W M. Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl.,1994,182(1):250-256
[11] Hassard B D, Kazarinoff N D, Wan Y H. Theory and application of Hopf bifurcation. Cambridge University Press, 1981
[2] Hastings A, Powell T. Chaos in a three-species food chain. Ecology, 1991, 72(3):896-903
[3] Gakkhar S, Naji R K. Chaos in a three species ratio dependent food chain. Chaos, Solitons and Fractals, 2002, 14(1):771-778
[4] Wang F Y, Pang G P. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons and Fractals, 2008, 36(5):1366-1376
[5] Yi F Q, Wei J J, Shi J P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predatorprey system. Journal of Differential Equations, 2009, 246(5):1944-1977
[6] Zuo W J, Song Y L. Stability and bifurcation analysis of a reaction-diffusion equation with distributed delay. Nonlinear Dynamics, 2015, 79(1):437-454
[7] Shi Q Y, Shi J P, Song Y L. Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition. J. Differential Equations, 2017, 263(10):6537-6575
[8] Guo G H, Li B F, Wei M H, Wu J H. Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model. J. Math. Anal. Appl., 2012, 391(1):265-277
[9] Guo G H, Liu L, Li B F, Li J. Qualitative analysis on positive steady-state solutions for an autocatalysis model with high order. Nonlinear Anal. RWA,2018, 41:665-691
[10] Liu W M. Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl.,1994,182(1):250-256
[11] Hassard B D, Kazarinoff N D, Wan Y H. Theory and application of Hopf bifurcation. Cambridge University Press, 1981