BIRS Workshop Lecture Videos
Mass-action systems: From linear to non-linear inequalities Yu, Polly
For mass-action kinetics, a common model for biochemistry, much work has gone into relating network structure to the possible dynamics of the resulting systems of polynomial ODEs. A family of mass-action systems, complex-balancing, is defined by having a positive equilibrium that balances monomials across vertices. Surprisingly, every positive equilibrium of such a system similarly balance monomials across vertices. These systems enjoy a variety of algebraic and stability properties: toricity in the steady state variety and in parameter space; Lyapunov and conjectured global stability. Unfortunately, most systems are vertex-balanced if and only if the parameters come from a toric ideal. By searching for different graphs representing the same ODEs, we can expand the parameter region for which the system is dynamically equivalent to a complex-balanced system. The expanded region is defined in the space of states and parameters, and the challenge is to eliminate the state variables to obtain explicit conditions on parameters (that is, to perform quantifier elimination over the reals). In this talk, I will introduce and set up the problem via examples.
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