T HE fields of population biology and ecology deal with the analysis of growth and decline of populations in nature and the struggle of species to predominate over one another. Many mathematical population models were proposed over the last few decades, with the most significant contributions coming from the work of Lotka [1] and Volterra [2]. The predator–prey models of Lotka andVolterra, studied extensively by ecologists and population biologists, consists of a set of nonlinear ordinary differential equations, and stability of the equilibrium solutions of these models has been a subject of intense study for students of life sciences [3–6]. For example, many standard textbooks on mathematical models in biology, such as [7], cover these issues. These small perturbations from equilibrium can be modeled as linear state-space systems in which the state-space plant matrix is the Jacobian, and it is important to analyze the stability of these state-space (Jacobian) matrices. For communities of five or more species, the order of these matrices is high enough to cause difficulties in assessing the stability. For this reason, to circumvent these difficulties, alternative concepts of reduced computation have been proposed, and one such important concept is that of qualitative (or sign) stability. The technique of qualitative stability applies ideally to large-scale systems in which there is no quantitative information about the interrelationship of species or subsystems. Themotivation for this method actually came from economics. The paper by economists Quirk and Ruppert [8] was later followed by further research and application to ecology by May [9] and Jeffries [10]. Note that in a complex community composed of many species, numerous interactions take place. The magnitudes of the mutual effects of species on each other are seldom accurately known, but one can establish with greater certainty whether predation, competition, or other influences are present. This means that in the Jacobian matrix, one does not technically know the actualmagnitudes of the partial derivatives, but their signs are known with certainty. Thus, the qualitative information about the species is represented by the signs , , or 0. Thus, the (i,j)th entry of the statespace (Jacobian) matrix simply consists of signs , , or 0, with the positive sign indicating that species j has a positive influence on species i, the negative sign indicating a negative influence, and zero indicating no influence. An alternative visual representation of this situation can also be given by a directed graph (or, simply, a digraph), as shown in Fig. 1. For example, with respect to the digraph of Fig. 1a, the corresponding sign-pattern matrix is given by