Necessary conditions for multistationarity and stable periodicity. (English) Zbl 0982.92001

Summary: We show that, for a differential system defined by a quasi-monotonous function \(f\) (with constant sign partial derivatives) the existence of a positive loop in the interaction graph associated to the Jacobian matrix of \(f\) is a necessary condition for multistationarity, and the existence of a negative loop comprising at least two elements is a necessary condition for stable periodicity. This gives a formal proof of R. Thomas’ conjectures [Springer Ser. Synerg. 9, 180-193 (1981)].


92B05 General biology and biomathematics
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37N25 Dynamical systems in biology
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