利用符号计算方法研究生物系统全时滞稳定性

生物系统全时滞稳定性表明系统对于时滞具有很好的可靠性,因此一直是学者们研究的热点,该研究通常采用传统的数学方法或数值计算方法.针对高维非线性含参数的生物系统,利用Hurwitz判据和多项式完全判别系统提出了带参数的非线性生物系统全时滞稳定性的一个充要代数判据.在此基础上,研究了如何利用Grbner基、三角化分解和实解分类等符号计算方法来处理得到的代数问题,并提出了一个利用符号计算方法系统化、算法化和自动化分析生物系统全时滞稳定性问题的方法.该方法使用的计算均是精确的,这为生物学家以及工程师研究某些生物系统的稳定性提供了理论基础.最后,通过对实际生物模型,比如时滞Lotka-Volterra模型和SIR传染病模型全时滞稳定性问题分析得到的有效结果,证明了符号计算方法分析生物系统全时滞稳定性的可行性及其相较于传统数学方法的优越性.

Analysis of all time-delay stability for biological systems using symbolic computation methods

All time-delay stability for biological systems shows that the time-delay system possess good reliability, so this issue has always been the highlight of the scholars research. However, researchers usually adopt the traditional mathematical methods or numerical calculation methods. Based on Hurwitz criterion and polynomial complete discriminant system, a sufficient and necessary algebraic criterion of all time-delay stability for nonlinear biological systems with parameters was introduced. By using symbolic computation methods, such as the methods of Grbner basis,triangular decomposition and real solution classification, a systematic and algorithmic approach for automatically analyzing all time-delay stability of biological systems with parameters was proposed. All the computations in our approach are all exact, which may help biologists and engineers to perform algebraic analysis for certain biological models. The successful experiments on the all time-delay stability analysis of several biological models, such as time delayed Lotka-Volterra systems and SIR epidemic models with time delay, showed the feasibility of our algebraic approach and also the superiority of symbolic computation methods compared with traditional mathematic methods.