恒化器中具有不同移动速率的环状模型解的定性分析

Qualitative analysis of solution in chemostat with different removing rate

为模拟自然界中既包含竞争关系又包含捕食-被捕食关系的生态系统,建立用常微分方程组表示的环状模型.模型中的微生物种群具有不同的死亡率,由此导致系统能量不守恒,降维法失效.通过直接求解三个方程组成的方程组,得到半平凡平衡点的存在性.运用常微分方程的定性理论讨论平衡点的局部渐进稳定性并证明系统的一致持续生存性质.用Matlab软件对相应平衡点的存在性和稳定性进行数值模拟.结果表明:适当调整系统参数,系统会出现振荡,从而产生分歧现象.

An annular model for chemostat was depicted by a group of ODE （ordinary differential equation） to imitate the ecosystem which included competition and predator-prey relations in nature. The different death rates of the microorganism led to the failure of conservation law and the method of dimensionality reduction. The existence of semi-trivial equilibria was obtained by directly solving the group which consisted of three ODE. The local asymptotic stability of the equilibria together with the uniform persistence of the system were discussed with help of qualitative theory of ODE. Numerical simulation of corresponding equilibria＇ s existence and stability were presented. The results show that the system will oscillate so as to cause bifurcations if the parameters of the system are regulated properly.