一类变消耗率微生物模型的极限环

Limit Cycle of a Class of Changing Consuming Rate on Microorganism Models

研究营养基和微生物之间的平衡是非线性生物动力系统研究的主要问题之一 .我们在讨论微生物连续培养的数学模型时 ,往往假设微生物的增长对营养基的消耗具有一个常数消耗率 1δ,但实际上不完全是这样 .一般来说 δ不是常数 ,而是营养基浓度 x的函数 ,即为 s( x) .营养基浓度越大 ,则消耗率就越低 ,因此函数 s( x)是 x的非减函数 .本文对一类微生物模型当消耗率为 s( x) ( s( x) >0 ,s′( x) >0 )时进行了讨论 ,得到的结论是 :当该模型在第一象限内唯一正平衡点是不稳定时 ,围绕该平衡点至少存在一个稳定的极限环 .

Studying the balance between Nutrition and Microorganism is one of the important problems of non linear biodynamical system.When we discuss the mathematical model of continuous culivation for microorganism,we usually assume that the nutrition consuming rate for microorganism be a constant 1δ.It is not true actually.δ is generally not a constant.It is a function of nutrient density x ,that is s(x) .The greater the nutrient density is,the lower the consuming rate.So the function s(x) is a nondecreasing function of x .We discuss a class of microorganism models when the consuming rate is s(x) (s(x) >0, s′(x) >0)and obtain that there is a stable limit cycle round the equilibrium point at least,when the model is unstable at the only equilibrium point in the first quadrant.