J Electr Eng Technol Vol. 8, No. 6: 1530-1541, 2013 http://dx.doi.org/10.5370/JEET.2013.8.6.1530 1530 On the Stability of Critical Point for Positive Systems and Its Applications to Biological Systems Joo-Won Lee*, Nam Hoon Jo † , Hyungbo Shim** and Young Ik Son*** Abstract – The coexistence and extinction of species are important concepts for biological systems and can be distinguished by an investigation of stability. When determining local stability of nonlinear systems, Lyapunov indirect method based on the Jacobian linearization has been widely employed due to its simplicity. Despite such popularity, it is not applicable to singular systems whose Jacobian has at least one eigenvalue that is equal to zero. In such singular cases, an appropriate Lyapunov function should be sought to determine the stability of systems, which is rather difficult and quite involved. In this paper, we seek for a simple criterion to determine stability of the equilibrium that is located at the boundary of the positive orthant, when one of eigenvalues of the Jacobian is zero. The goal of the paper is to present a generalized condition for the equilibrium to attract all trajectories that starting from initial condition in the positive orthant and near the equilibrium. Unlike the Lyapunov direct method, the proposed method requires just a simple algebraic computation for checking the stability of the critical point. Our approach is applied to various biological systems to show the effectiveness of the proposed method. Keywords: Prey-predator model, Positive system, Center manifold, Jacobian linearization, Lya-punov function 1. Introduction Stability is an important concept that can provide powerful insight into qualitative behavior of various biological systems [1, 2] such as predator-prey systems, viral and immune systems, epidemic systems and so on. Stability analysis tells us whether a solution trajectory near an equilibrium will converge toward or move away from it. For predator-prey system [3-15], an investigation of stability distinguishes the coexistence of predators and prey from the extinction of some species. In addition, stability analysis may determine the basin of attraction of each equilibrium, that is, the region of coexistence and the region of extinction. Aside from predator-prey systems, some research works on virus infection of CD4 + cells have been performed in [16-20]. In their works, mathematical models describing infection dynamics have infection-free equilibrium and chronic-infection equilibrium. The infection will die out if the former is stable, while the infection will persist if the latter is stable. For the epidemic model with some vaccination strategy [21-23], an examination of stability can predict whether the vaccination successfully leads to disease eradication. An epidemic outbreak is produced if an endemic equilibrium is stable, whereas the disease will die out if a disease-free equilibrium is stable. In order to check the local stability of a nonlinear system at its certain equilibrium, Lyapunov indirect method based on the Jacobian linearization has been widely employed due to its simplicity. Despite such popularity, the method cannot be applied to determine stability when one eigenvalue of the Jacobian matrix is equal to zero. In fact, although the research works in [2-12, 15-18] have used the Jacobian matrix to determine stability, most of them have not provided any conclusion on stability when the Jacobian matrix has an eigenvalue at the origin of the complex plane. While the extinction of some species can be predicted by investigating the stability of equilibria with some zero component, stability of such equilibria has not been determined in [2-12, 15-18] since the Jacobian has an eigenvalue at the origin. In order to analyze stability for such cases, an appropriate Lyapunov function should be constructed as in [24], or some other nonlinear stability theory needs to be employed [25]. Even though Lyapunov stability theory and LaSalle’s invariance principle can be employed to determine stability [26], it is rather difficult and quite involved to construct an appropriate Lyapunov function. An alternative method may be to study the stability of reduced system by invoking the center manifold theory [26, 27]. However, the verification procedure is not convenient in general, because a solution to the partial differential equation for the center manifold needs to be obtained. To avoid the difficulty in solving the partial differential equation, the authors have presented in [28] an alternative technique under a certain assumption of the Jacobian † Corresponding Author: Dept. of Electrical Engineering, Soongsil University, Korea. (nhjo@ssu.ac.kr) * Dept. of Electrical Engineering, Soongsil University, Korea. ** School of Electrical Engineering, Seoul National University, Korea. *** Dept. of Electrical Engineering, Myongji University, Korea. Received: November 8, 2012; Accepted: April 5, 2013 ISSN(Print) 1975-0102 ISSN(Online) 2093-7423