非自治浮游植物—浮游动物模型的建模与分析

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.