Equilibria, stability and reachability of Leslie systems with nonnegative inputs

Leslie systems, a particular class of positive systems used in population dynamics and control, are defined and analyzed. A simple property relating sign and stability of their equilibria, and some geometric characteristics of their reachability set (in cases of bounded and unbounded inputs) are proved. All the properties do not hold, in general, in other classes of positive systems. It is shown that Leslie systems have strictly positive equilibria if and only if they are asymptotically stable. Their reachability set in the case of unbounded inputs is the positive cone generated by the reachability vectors, while in the case of bounded inputs, it is a polyhedron whose vertices can be easily computed. All of these properties follow from the nonpositiveness of the coefficients of the characteristic polynomial of the system