一个非线性连续培养模型的极限环及其周长

Limit Cycles in a Nonlinear Continuous Culture Model

用微分动力系统的定性理论研究了一个非线性连续微生物培养动力系统的极限环问题,探讨和估计了存在极限环的条件和极限环的周长。本文的模型是对本文作者之一在多年前所提出和研究过的发酵模型的推广[5],这里我们对发酵产物与基质浓度的非线性函数关系做了进一步的深入研究。这一模型最近又被Pilyugin和Waltman在生化培养的研究中再次导出的[14]。工程界对于连续发酵的非线性振荡方面的研究大多是采用从实验数据归纳出来的经验模型,很难反映过程的本质;生物数学界有关这方面的模型,其中的函数大多数是文[5,14]的特例。本文的工作无论从生化的角度还是从数学的角度考虑都是有意义的。

In this paper, the limit cycles in a nonlinear deterministic model in the continuous culture vessel with variable yield is studied. The conditions of the existence of limit cycles and the perimeter of the limit cycles are discussed and estimated. The model is a generalization of the one previously proposed and studied by one of the authors years ago[5], and derived again recently by Pilyugin ＆ Waltman[14]. In the literature in China, most of the related work in biochemical engineering are experimental models based on the lab data, which are hard to explain the nonlinear oscillatory phenomena, and most of the work in biomathematics consider the yield functions as constant, linear functions, or some special cases of our general function[5,14]. Obviously, this generalization is useful for the further study of the microbial growth.