Bulletin of Mathematical Biology

, Volume 40, Issue 6, pp 839–851 | Cite as

Interactions among biological systems: An analysis of asymptotic stability

  • Francesco Andrietti


An analysis of the interactions among asymptotically stable dynamical systems is formulated by making use of the dynamical system theory. Some results coming from previous mathematical analyses have been slightly modified to take into account some typical biological constraints as the boundedness properties of the solutions. In particular it has been shown that when the “coupling” among the subsystems is “loose” enough (in a sense that has to be made mathematically precise) the asymptotic behaviour of a complex system is the same of that of its individual components. The mathematical theory has been used to analyze two systems of biological significance: the coupling among chemical reactions and the stability properties of a 4-dimensional system describing the kinetics of a chemical transmitter.


Asymptotic Stability Composite System Negative Real Part Dynamical System Theory Boundedness Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. De Groot, S. R. 1963.Thermodynamics of Irreversible Processes. Amsterdam: North-Holland Publishing Co.Google Scholar
  2. De Sobrino, L. 1975. “The Glansdorff-Prigogine Thermodynamic Stability Criterion in the Light of Liapunov's Theory.”J. Theor. Biol.,54, 323–333.CrossRefGoogle Scholar
  3. Hahn, W. 1967.Stability of Motion. New York: Springer.Google Scholar
  4. Krasovskii, N. N. 1963.Stability of Motion. Stanford: Stanford University Press.Google Scholar
  5. Leibovic, K. N. L. 1975. “Analysis of Light Absorption and Response of Vertebrate Photoreceptor.”5th International Biophysics Congress. Copenhagen, Denmark.Google Scholar
  6. — and F. Andrietti. 1977. “Analysis of a model for transmitter kinetics.”Biol. Cybernetics,27, 165–173.MATHCrossRefGoogle Scholar
  7. Michel, A. N. 1974. “Stability Analysis of Interconnected Systems.”SIAM J. Control.,12 (3), 554–580.MATHMathSciNetCrossRefGoogle Scholar
  8. Rosen, R. 1968. “Some Comments on the Physico-Chemical Desciption of Biological Activity.”J. Theor. Biol.,18, 380–386.CrossRefGoogle Scholar
  9. Ŝiljak, D. D. 1972. “Stability of Large-Scale Systems.”Proc. Fifth World Congress of IFAC (Session 9, Nonlinear Systems): Paris.Google Scholar
  10. — 1975. “When is a Complex Ecosystem Stable?.”Math. Biosci.,25, 25–50.MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1978

Authors and Affiliations

  • Francesco Andrietti
    • 1
  1. 1.Istituto di Fisiologia Generale e di ChimicaBiologica dell'Università degli Studi di MilanoMilanoItaly

Personalised recommendations