一类合作系统的渐近性态的代数判别

AN ALGEBRAIC CRITERION FOR ASYMPTOTIC BEHAVIOR OF A CLASS OF COOPERATIVE SYSTEMS

A.Lajmanovich 和 J.Yorke 用如下的不可约的合作向量场来模拟淋病的传播:)=Ax+N(X),(1)其中 A=diag(c_1,c_2,…,c_n)B—diag(α_1,α_2,…,α_n),N=diag(x_1,x_2,…,x_n)Bx,B 为一个非负的不可约阵.在[1]中,Lajmanovich 和 Yorke 证明了(1)具有相当有趣的性质:当 A 的特征值的最大实部 S(A)为非正时,所有轨线收敛于原点;当 S(A)>0时,所有轨线收敛于一个正奇点.

In this paper an algebraic criterion for asymptotic behavior of cooperative systems withconcave nonlinearities is given.The global behaviol of such systems is determined by thesign of all principal minors of the Jacobian matrix of the vector field at the origin.