Jacobi stability of dynamical systems with applications to biology Ileana Rodica Nicola and Vladimir Balan Abstract The paper investigates the structural stability of hepatocyte physiology in the case of bursting (explosive) behavior, based on the five KCC-invariants of the second-order canonic extension of the characterizing SODE. Mathematics Subject Classification: 37N25, 92C45. Key words: dynamical system, KCC-invariants, Jacobi stability. 1 Introduction. It is well known that the SODE (system of ordinary diferential equations) which describes the intra-cell calcium variation in time exhibit a very rich and complex dynamical behavior. In the present work we investigate the robustness and the fragility of a mathe- matical biological model which describes the Calcium variations in time in the living cell, by means of the deviation curvature tensor of the attached SODE (KCC or Ja- cobi stability). By robustness we mean both the relative insensitivity to alteration of their internal parameters and the ability to adapt to changes in their environment. From the mathematical point of view, the differential geometrical theory of varia- tional equations studying the deviation of nearby trajectories allows us to estimate the admissible perturbation arround the steady states of the SODE. By admissible mean perturbations we have in view the ones which do not change the stability ranges of the system. The applicative biological aspects of our model represent an important open ques- tion in the field, and are subject of further research. The calcium variations in time model is based on the mechanism of calcium in- duced calcium release (CICR). This model takes into account calcium-stimulated degradation of inositol 1,4,5- triphosphate (InsP 3 ) by a 3-kinase. Complex calcium (Ca 2+ ) variations in time have been observed in certain cell types, particularly in hepatocytes, as a response to stimulation by certain substances. * Proceedings of The 3-rd International Colloquium ”Mathematics in Engineering and Numerical Physics” October 7-9 , 2004, Bucharest, Romania, pp. 195-201. c  Balkan Society of Geometers, Geometry Balkan Press 2005.