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A proof of bistability for the dual futile cycle. (English) Zbl 1331.34089
The paper investigates a mathematical model of the dual futile cycle in cell biology that originally consists of a system of nine autonomous ordinary differential equations. By rescaling the time variable, this system adopts the caracter of fast-slow dynamics, thus representing a singularly perturbed system of six differential equations, which in the limit leads to a planar vector field (MM) of Michaelis-Menten type. Under some simplifying assumptions, the authors first apply standard tools of bifurcation theory, such as centre manifold theory, to show the existence of cusp bifurcation for (MM), with two stable equilibria and an unstable one. Then, using results from geometric singular perturbation theory, it is proved that the perturbed system inherits the properties of system (MM) and thus exhibits two asymptotically stable equilibria and one saddle. In final remarks, the authors refer to the MAPK cascade, whose mathematical treatment leads to a three-dimensional unperturbed system of a structure similar to (MM). The question of periodic solutions for (MM) is also raised.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 34C23 Bifurcation theory for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 92C37 Cell biology 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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