Abstract
Abstract A class of nonautonomous linear ordinary differential equations is considered. The coefficient matrices are Metzler matrices with zero column sums such that their directed graphs have a common directed spanning tree. It is shown that if the off-diagonal elements of the coefficients are uniformly positive along the common directed spanning tree, then under mild additional assumptions the convergence of the Perron vectors of the coefficient matrices implies that all solutions tend to a finite limit at infinity. The value of the limit can be expressed in terms of the initial data.
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