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Computationally Checking if a Reaction Network is Monotone or Non-expansive

Alon Duvall

TL;DR

The paper develops a unified, algorithmic framework to determine whether a reaction network defined by $\\dot{x} = \\Gamma R(x)$ is monotone with respect to some cone or non-expansive with respect to some norm. It introduces a constructive procedure that either produces a certifying cone (or norm) or proves nonexistence, and establishes a duality: monotonicity of a network implies monotonicity of its dual with respect to the corresponding dual cone. Key ideas include cross-positivity criteria, lifting to relate monotonicity and non-expansivity, and a cone-building procedure from a starting vector using region-based operations. The work also proves a monotonicity dichotomy, provides termination and convergence guarantees for the algorithm, and demonstrates the method through several worked examples, including recoveries of known cones and a demonstration of non-monotonicity w.r.t. any cone in a particular network. Overall, the approach yields both constructive certificates and theoretical insights into the structure of monotone and non-expansive reaction networks, with implications for stability and global dynamics.

Abstract

We present a systematic procedure for testing whether reaction networks exhibit non-expansivity or monotonicity. This procedure identifies explicit norms under which a network is non-expansive or cones for which the system is monotone-or provides proof that no such structures exist. Our approach reproduces known results, generates novel findings, and demonstrates that certain reaction networks cannot exhibit monotonicity or non-expansivity with respect to any cone or norm. Additionally, we establish a duality relationship which states that if a network is monotone, so is its dual network.

Computationally Checking if a Reaction Network is Monotone or Non-expansive

TL;DR

The paper develops a unified, algorithmic framework to determine whether a reaction network defined by is monotone with respect to some cone or non-expansive with respect to some norm. It introduces a constructive procedure that either produces a certifying cone (or norm) or proves nonexistence, and establishes a duality: monotonicity of a network implies monotonicity of its dual with respect to the corresponding dual cone. Key ideas include cross-positivity criteria, lifting to relate monotonicity and non-expansivity, and a cone-building procedure from a starting vector using region-based operations. The work also proves a monotonicity dichotomy, provides termination and convergence guarantees for the algorithm, and demonstrates the method through several worked examples, including recoveries of known cones and a demonstration of non-monotonicity w.r.t. any cone in a particular network. Overall, the approach yields both constructive certificates and theoretical insights into the structure of monotone and non-expansive reaction networks, with implications for stability and global dynamics.

Abstract

We present a systematic procedure for testing whether reaction networks exhibit non-expansivity or monotonicity. This procedure identifies explicit norms under which a network is non-expansive or cones for which the system is monotone-or provides proof that no such structures exist. Our approach reproduces known results, generates novel findings, and demonstrates that certain reaction networks cannot exhibit monotonicity or non-expansivity with respect to any cone or norm. Additionally, we establish a duality relationship which states that if a network is monotone, so is its dual network.
Paper Structure (25 sections, 34 theorems, 44 equations, 4 algorithms)

This paper contains 25 sections, 34 theorems, 44 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Some preliminary results
  5. Lifted networks
  6. Building a cone from a vector
  7. Strong monotonicity and weak contractivity
  8. A monotonicity dichotomy
  9. Choosing a starting vector
  10. Existence of starting vectors
  11. Algorithm termination conditions
  12. Convergence of the procedure
  13. Procedure for monotonicity and contractivity
  14. Duality
  15. Three worked out examples
  16. ...and 10 more sections

Key Result

Theorem 1

WALCHER2001543 A system $\dot{x} = f(x)$ with phase space $\mathbb{R}_{\geq}^n$ is monotone with respect to a proper, pointed and convex cone $K$ iff for any $x \in \mathbb{R}_{\geq}^n$ and for all $k_1\in \hbox{bd}(K)$ and $k_2 \in K^*$ such that $k_2^{\top} k_1 = 0$ we have that $k_2^{\top} \math

Theorems & Definitions (69)

  • Theorem 1
  • Corollary 1
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 59 more