Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all time $t$ if it starts from a positive vector $x(0)$. Let the unique largest eigenvalue (which is always purely real) of $A$ be negative. Then the vector $x$ will converge surely to 0. Usual linear dynamical systems theory tells us that $||x(t)||\le ||x(0)|| e^{\lambda_{max} t}$.

So far so good. But there is another property to Metzler matrices: the eigenvector corresponding to the maximum eigenvalue is positive. I am wondering if there is any special interpretation to this normalised eigenvector. Positive eigenvectors in the context of PageRank algorithm stand for relative importance of different pages. What does it stand for in the context of postive linear systems?