赤潮模型中代数-微分方程组的分岔理论和新型数值算法研究

In this project, we will derive the existence conditions of singularity-induced bifurcation of the differential-algebraic equations in Nutrient-Phytoplankton-Zooplankton-Detritus(N-P-Z-D) type models, obtain the critical value sets of parameters which can be used to predict occurrences of red tides, and construct the new effective numerical method for the models. As follows is idiographic research: Firstly, since the singularity-induced bifurcation often leads to the pulse of the system which indicates the sudden outbreak of the red tide, we will improve the known models for some ocean areas (Jiaozhou Bay), investigate the structure of parameters of the singularity-induced bifurcation in the improved models, especially the existence conditions of singularity-induced bifurcation, and obtain the critical value sets of parameters about the occurrences of red tides. Secondly, according to the characteristics of differential-algebraic equations in red tides, we will structure the effective new methods. The red models maybe has some singular points, the models are large-scale, and the monitoring data are often un-uniformly distributed. For solving the three different problems, we will introduce Sinc methods, waveform iterative methods and meshless methods, respectively. Meanwhile, taking advantage of preconditioning techniques and matrix splittings, we will explore the fast numerical methods for solving the large-scale singular linear equations which are obtained during the above process.

本课题主要研究N-P-Z-D（营养盐(Nutrient)一浮游植物(Phytoplankton)一浮游动物(Zooplankton)一碎屑(Detritus)）模式的赤潮模型中微分-代数方程组的奇异诱导分岔存在的条件，从而给出赤潮发生的参数临界值，构造求解此类模型的新型数值算法。具体如下：1. 针对具体海域（胶州湾海域），对现有的模型进行局部改进，研究该系统出现奇异诱导分岔的参数结构，特别是出现奇异诱导分岔的条件，给出赤潮发生的参数临界值。因为奇异诱导分岔常常会导致系统产生脉冲，而脉冲现象表现为赤潮的突然爆发； 2. 根据赤潮模型的特点构造相应的新型算法。针对赤潮模型的大规模性、监测数据的不均匀分布和模型存在奇异点，分别引入波形迭代法、无网格方法和Sinc方法。同时，利用预处理子和矩阵分裂，为求解过程中产生的具有特殊结构的大规模奇异线性方程组构造快速的数值算法。

在实际的海洋生态系统中,藻类的生长率是随着季节及环境等各方面因素的变化而不断变化的。首先，该课题将藻类的生长率R变为g=R-∆Sinωt，并在单种群P-Z赤潮生态模型基础上，建立了基于两类浮游动物、一种浮游植物的Beddington-DeAngelis型赤潮生态模型，研究了参数变化对系统的影响。其次，我们针对代数-微分方程组构造了新型算法并分析了算法的收敛性。在利用sinc方法求解代数-微分系统时出现了大规模线性方程组。我们根据方程组的结构特征，构造了一类块三对角矩阵作为预处理子，并估计了预优后矩阵的特征值边界。进而，我们得到了求解上述大规模方程组的快速算法。数值实验表明该类预处理子是有效的。