Summary
A method for studying biological control loops has been developed, which suffices to prove global stability for the Goodwin equations when the Hill coefficient θ is equal to 1. This holds for arbitrary reaction constants, even if time delays are included in the system. For a generalized class of repfessible systems, including the Goodwin Equations for θ>1, the method gives a sufficient condition for global stability, in terms of solutions of an algebraic equation in a single variable. When the criterion is not satisfied, the same equation gives bounds on any possible limit cycles. The method also shows that inducible systems with a unique equilibrium are globally stable. The system of equations studied allows each reaction rate equation to be non-linear, and to include a time delay.
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Allwright, D.J. A global stability criterion for simple control loops. J. Math. Biol. 4, 363–373 (1977). https://doi.org/10.1007/BF00275084
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DOI: https://doi.org/10.1007/BF00275084