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Global stability of deficiency zero chemical networks. (English) Zbl 0823.92011

We study deficiency zero chemical networks which are weakly reversible and “weakly repelling”. In this situation it will be shown that each positive solution tends to the unique positive equilibrium in its compatibility class. This result will be shown to be applicable in the following situations:
(1) There exists no critical species in the network, (2) there is one critical species which appears in at most one complex in each linkage class. It will also be shown to be applicable to two types of enzyme systems not satisfying (1) or (2).
The “weakly repelling” condition is that any solution in a positive compatibility class which is on the boundary at some time must be in the interior for all later time. This concept is somewhat related to “persistence” or “permanence” where solutions are kept away from the boundary. In our work, the weakly repelling condition arises from a study of conservation laws and the Strict Positive Theorem of A. Vol’pert [see A. I. Vol’pert and S. I. Khudyaev, Analysis in classes of discontinuous functions and equations of mathematical physics. (1985; Zbl 0564.46025)].

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
05C90 Applications of graph theory
92E20 Classical flows, reactions, etc. in chemistry
34A99 General theory for ordinary differential equations
34D99 Stability theory for ordinary differential equations

Citations:

Zbl 0564.46025
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