### Persistence and permanence in biological interaction networks

14 July 2010

Gheorghe Craciun

Department of Mathematics

University of Wisconsin at Madison

#### Abstract

Dynamical system models are very commonly used to analyze biological interaction networks, such as the dynamics of concentrations in biochemical reaction networks, the spread of infectious diseases within a population, and the dynamics of species in an ecosystem. Persistence and permanence are properties of dynamical systems that provide information about the long-term behavior of the system. For example, the persistence property is relevant in deciding if, in the long term, a chemical species will be completely consumed by a reaction network, an infection will die off, or a species in an ecosystem will become extinct. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to persistent systems, even if we allow the reaction rate parameters to vary in time. These results also apply to power-law systems and other nonlinear dynamical systems. This is joint work with Fedor Nazarov and Casian Pantea.

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