Reachability of nonnegative and symbiotic states for linear differential systems

Let A $\in$ $R\sp{n\times n }$ be essentially nonnegative and consider the linear differential system $\dot x$ = Ax. An initial point $x\sb0$ $\in$ $R\sp{n}$ lies in the reachability cone of the nonnegative orthant, $X\sb{A}(R\sbsp{+}{n})$, if the trajectory emanating from $x\sb0$ reaches $R\sbsp{+}{n}$ at some finite time $t\sb0$ = $t(x\sb0)$ $\geq$ 0 and remains in $R\sbsp{+}{n}$ thereafter. In the present work we characterize numerically $X\sb{A}(R\sbsp{+}{n}$). We show that there exists a constant h(A) $>$ 0, not necessarily small, such that $x\sb0$ $\in$ $X\sb{A}(R\sbsp{+}{n}$) if and only if there exists a nonnegative integer $k\sb0$ = $k(x\sb0)$ for which (I + $hA)\sp{k\sb0}x\sb0$ $\geq$ 0, where h $\in$ (0, h(A)) and det(I + hA) $\not=$ 0.

An initial point in $X\sb{A}(R\sbsp{+}{n}$) is called a symbiosis point if also the initial velocity vector lies in $X\sb{A}(R\sbsp{+}{n}$). As a consequence, each component of the trajectory emanating from a symbiosis point becomes and remains a nonnegative and nondecreasing function of time. Utilizing our results on $X\sb{A}(R\sbsp{+}{n}$), we characterize the symbiosis points numerically.

We also give a descriptive characterization of the symbiosis points, based on the eigenspace decomposition of trajectories, and when A is weakly stable we describe them from a matrix-combinatorial point of view. In the latter case we also characterize trajectories whose higher order derivatives lie in $X\sb{A}(R\sbsp{+}{n}$).