Abstract：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 Grbner 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.
柳君, 牛薇. 利用符号计算方法研究生物系统全时滞稳定性[J]. 北京航空航天大学学报, 2015, 41(12): 2363-2369.
LIU Jun, NIU Wei. Analysis of all time-delay stability for biological systems using symbolic computation methods. JOURNAL OF BEIJING UNIVERSITY OF AERONAUTICS AND A, 2015, 41(12): 2363-2369.
秦元勋.有时滞的系统的无条件稳定性[J].数学学报,1960,10(1):125-142. Chin Y S.Unconditional stability of systems with time lags[J].Acta Mathematica Sinica,1960,10(1):125-142(in Chinese).
[2]
秦元勋,俞元洪.一类时滞微分系统无条件稳定的条件[J].控制理论与应用,1984,1(1):23-35. Chin Y S,Yu Y H.Unconditional stability conditions for a class of differential systems with time delay[J].Journal of Control Theory and Applications,1984,1(1):23-35(in Chinese).
[3]
周超顺,邓聚龙.线性定常时滞系统全时滞渐近稳定的充分代数判据[J].自动化学报,1990,16(1):62-65. Zhou C S,Deng J L.A sufficient algebra criteria for stability of linear constant time-delay system[J].Acta Automatica Sinica,1990,16(1):62-65(in Chinese).
[4]
Cao D Q,He P,Ge Y M.Simple algebraic criteria for stability of neutral delay-differential systems[J].Journal of the Franklin Institute,2005,342(3):311-320.
[5]
Hu G D,Hu G D,Cahlon B.Algebraic criteria for stability of linear neutral systems with a single delay[J].Journal of Computational and Applied Mathematics,2001,135(1):125-133.
[6]
He P,Cao D Q.Algebraic stability criteria of linear neutral systems with multiple time delays[J].Applied Mathematics and Computation,2004,155(3):643-653.
[7]
Niu W,Wang D.Algebraic analysis of bifurcation and limit cycles for biological systems[M].Berlin,Heidelberg:Springer,2008:156-171.
[8]
Niu W,Wang D.Algebraic approaches to stability analysis of biological systems[J].Mathematics in Computer Science,2008,1(3):507-539.
[9]
Wang D.Elimination methods[M].Berlin,Heidelberg:Springer,2001:193-224.
[10]
Buchberger B.Gröbner bases:An algorithmic method in polynomial ideal theory[J].Multidimensional Systems Theory,1985:184-232.
[11]
Faugère J C.A new efficient algorithm for computing Gröbner bases (F4)[J].Journal of Pure and Applied Algebra,1999,139(1):61-88.
[12]
Yang L,Xia B.Real solution classification for parametric semi-algebraic systems[C]//Algorithmic Algebra and Logic[S.l.:s.n.],2005:281-289.
[13]
宋永利,韩茂安,魏俊杰.多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支[J].数学年刊:A辑,2005,25(6):783-790. Song Y L,Han M A,Wei J J.Stability and global Hopf bifurcation for a predator-prey model with two delays[J].Chinese Annals of Mathematics:Series A,2005,25(6):783-790(in Chinese).
[14]
Bhattacharyya S P,Chapellat H,Keel L H.Robust control-the parametric approach[M].New York:Prentice Hall PTR,1995:446-472.
[15]
Lancaster P,Tismenetsky M.The theory of matrices:With applications[M].Pittsburgh:Academic Press,1985:89-103.
[16]
Xia B.DISCOVERER:A tool for solving semi-algebraic systems[J].ACM Communications in Computer Algebra,2007,41(3):102-103.
[17]
Kar T K,Pahari U K.Non-selective harvesting in prey-predator models with delay[J].Communications in Nonlinear Science and Numerical Simulation,2006,11(4):499-509.
[18]
Cooke K L.Stability analysis for a vector disease model[J].Journal of Mathmatics,1979,9(1):31-42.
[19]
Meng X,Chen L,Wu B.A delay SIR epidemic model with pulse vaccination and incubation times[J].Nonlinear Analysis:Real World Applications,2010,11(1):88-98.
2015, Vol. 41 
