Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

Finding invariant sets for biological systems using monomial domination

“…In [1] have been presented general conditions that imply non-vanishing of polynomial functions on the positive orthant. They are based on the observation that a monomial can be dominated by a linear combination of other monomials if the convex hull of their exponents contains the exponent of the monomial to be dominated.…”

“…In Section II, we summarise the findings presented in [1]. Importantly, Section II provides the basis for our novel analysis framework presented in Section III, where we provide a methodology based on Lyapunov theory to find an invariant set of a dynamical system; importantly, this can be used to guarantee that the population of a particular system will not explode on the one hand, or go extinct on the other hand.…”

“…It is then shown that, for relevant classes of n-dimensional polynomials, global and local injectivity are equivalent [2], and local injectivity can be checked by verifying that the determinant of a Jacobian matrix is a non-vanishing polynomial function. In this paper, we further develop the monomial domination approach presented in [1] to analyse and characterise biological systems. The main contribution of this paper is to describe how we can use this approach to address various qualitative properties of biological models, different from the multistability problem discussed in [1].…”

“…In this paper, we further develop the monomial domination approach presented in [1] to analyse and characterise biological systems. The main contribution of this paper is to describe how we can use this approach to address various qualitative properties of biological models, different from the multistability problem discussed in [1]. We show that this approach has the advantage that no a priori information is required about the values of parameters present in the model.…”

Tensor Approximation of Stationary Distributions of Chemical Reaction Networks

“…Specifically, we show here that for Chemical Reaction Networks (CRNs for short) with deficiency zero in the sense of Feinberg [15,16,17] and for a certain class of propensity functions that the stationary PDF of such systems allows for approximation of accuracy ε > 0 with tensor ranks r which scale logarithmically in the accuracy as well as in the maximum copy numbers, and which are linearly scaling in the number of chemical species present in the system. The outline of this paper is as follows.…”