### Multistationarity in mass action networks by linear inequalities

4 and 12 November 2010

Carsten Conradi

Max-Planck Institute for Dynamics of Complex Technical Systems

Magdeburg, Germany

#### Abstract

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, there exist necessary and sufficient conditions for multistationarity that take the form linear inequality systems.

In the first part of this seminar an approach for establishing multistationarity is presented that is based on ideas from Chemical Reaction Network Theory. This approach yields necessary and sufficient conditions for multistationarity that consist of linear inequality systems and, in general, a small number of nonlinear equations.

In the second part of the seminar these nonlinear equations are further examined: if the reaction network is of a certain structure, then the nonlinear equations can be replaced by linear inequality systems, hence yielding necessary and sufficient conditions for multistationarity involving only linear inequality systems.

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