Multi-valued characteristics and Morse decompositions

A rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area.We combine a theory of monotone input–output systems with a classical theory of Morse decompositions in the context of ordinary differential equations models of biochemical reactions. We show that a multi-valued input–output characteristic can be used to define non-trivial Morse decompositions which provide information about a global structure of the attractor. The previous work on input–output characteristics is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory.We apply our tools to a model of cell cycle maintenance. We show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions.