非自治浮游植物—浮游动物模型的建模与分析（英文）
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摘要
基于非线性常微分方程模型,研究一类具有非自治密度依赖的有毒浮游植物―浮游动物系统的动力学性质。为使模型更具有一般形式,模型中对消耗功能反应函数及毒素释放功能反应函数引进了Beddington-DeAngelis型功能反应函数。通过引进特定的集合Γ,得到系统持久的弱充分条件;根据这个集合Γ和Brouwer不动点定理,得到系统正周期解存在的充分条件。由于由正周期解的全局吸引性可得到周期解的唯一性,通过建立适当的Lyapunov函数,得到周期解全局吸引的充分条件。数值模拟验证了理论分析的正确性。
The dynamical behaviors of a non-autonomous density-dependent toxic-phytoplankton-zooplankton system are investigated based upon nonlinear ordinary differential equation model. Beddington-DeAngelis functional response is induced to both of the consume response function and distribution of toxic substance term such that the model is in a more general form. Firstly, a weaker sufficient condition is established for the permanence of the system by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, the existence conditions of positive periodic solution are obtained. Secondly, since the uniqueness of positive periodic solution can be ensured by global attractiveness, a sufficient condition for global attractiveness of the periodic solutions is obtained by constructing a suitable Lyapunov function. Finally, several groups of illustrations are performed to justify analytical findings.
The dynamical behaviors of a non-autonomous density-dependent toxic-phytoplankton-zooplankton system are investigated based upon nonlinear ordinary differential equation model. Beddington-DeAngelis functional response is induced to both of the consume response function and distribution of toxic substance term such that the model is in a more general form. Firstly, a weaker sufficient condition is established for the permanence of the system by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, the existence conditions of positive periodic solution are obtained. Secondly, since the uniqueness of positive periodic solution can be ensured by global attractiveness, a sufficient condition for global attractiveness of the periodic solutions is obtained by constructing a suitable Lyapunov function. Finally, several groups of illustrations are performed to justify analytical findings.
引文
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[5] ZHANG T,WANG W.Hopf bifurcation and bistability of a nutrient phytoplankton-zooplankton model [J].Applied Mathematical Modelling,2012,36(12):6225-6235.
[6] FAN M,KUANG Y.Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response [J].Journal of Mathematical Analysis and Applications,2004,295(1):15-39.
[7] FAN M,WANG Q,ZOU X.Dynamics of a non-autonomous ratio-dependent predator-prey system [J].Proceedings of the Royal Society of Edinburgh,2003,133A:97-118.
[8] KAKUTANI S.A generalization of Brouwer fixed-point theorem [J].Duke Mathematical Journal,1941,8(3):457-459.
[9] KUANG Y,BERETTA E.Global qualitative analysis of a ratio-dependence predator-prey system [J].Journal of Mathematical Biology,1998,36:389-406.
[10] LI H,YASUHIROT.Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response [J].Journal of Mathematical Biology,2011,374(2):644-656.
[11] LI H Y,SHE Z K.Uniqueness of periodic solutions of a nonautomous density-dependent predator-prey system [J].Journal of Mathematical Analysis and Applications,2015,422(1):886-905.
[12] TENG Z D,MEHBUBA R.Persistence in nonautonomous predator-prey systems with infinite delay [J].Journal of Computational and Applied Mathematics,2006,197(2):302-321.
[13] ZHAO J,JIANG J.Permanence in nonautomous Lotka-Volterra system with predator-prey [J].Applied Mathematics and Computation,2004,152(1):99-109.
[2] ODUM E P.Fundamentals of Ecology [M].Philadelphia:Elsevier,1971.
[3] GAO M,SHI H,LI Z.Chaos in a seasonnally and periodically forced phytoplankton-zooplankton system [J].Nonlinear Analysis:Real World Applications,2009,10(3):1643-1650.
[4] JANG S R J,BAGLAMA J,RICK J.Nutrient-phytoplankton-zooplankton models with a toxin [J].Mathematical and Computer Modelling,2006,43(1-2):105-118.
[5] ZHANG T,WANG W.Hopf bifurcation and bistability of a nutrient phytoplankton-zooplankton model [J].Applied Mathematical Modelling,2012,36(12):6225-6235.
[6] FAN M,KUANG Y.Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response [J].Journal of Mathematical Analysis and Applications,2004,295(1):15-39.
[7] FAN M,WANG Q,ZOU X.Dynamics of a non-autonomous ratio-dependent predator-prey system [J].Proceedings of the Royal Society of Edinburgh,2003,133A:97-118.
[8] KAKUTANI S.A generalization of Brouwer fixed-point theorem [J].Duke Mathematical Journal,1941,8(3):457-459.
[9] KUANG Y,BERETTA E.Global qualitative analysis of a ratio-dependence predator-prey system [J].Journal of Mathematical Biology,1998,36:389-406.
[10] LI H,YASUHIROT.Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response [J].Journal of Mathematical Biology,2011,374(2):644-656.
[11] LI H Y,SHE Z K.Uniqueness of periodic solutions of a nonautomous density-dependent predator-prey system [J].Journal of Mathematical Analysis and Applications,2015,422(1):886-905.
[12] TENG Z D,MEHBUBA R.Persistence in nonautonomous predator-prey systems with infinite delay [J].Journal of Computational and Applied Mathematics,2006,197(2):302-321.
[13] ZHAO J,JIANG J.Permanence in nonautomous Lotka-Volterra system with predator-prey [J].Applied Mathematics and Computation,2004,152(1):99-109.