Lotka-Volterra representation of general nonlinear systems

TL;DR

This paper explores the structure and properties of the Generalized Lotka-Volterra form for nonlinear differential equations, showing how various biological models can be embedded into this framework and providing methods for their transformation.

Contribution

It introduces a formal framework for representing diverse nonlinear systems in the GLV form, including procedures and theorems ensuring consistent transformations.

Findings

GLV form invariance under quasimonomial transformations

Embedding of S-systems and mass-action systems into GLV

Procedures for transforming general nonlinear systems into GLV

Abstract

In this paper we elaborate on the structure of the Generalized Lotka-Volterra (GLV) form for nonlinear differential equations. We discuss here the algebraic properties of the GLV family, such as the invariance under quasimonomial transformations and the underlying structure of classes of equivalence. Each class possesses a unique representative under the classical quadratic Lotka-Volterra form. We show how other standard modelling forms of biological interest, such as S-systems or mass-action systems are naturally embedded into the GLV form, which thus provides a formal framework for their comparison, and for the establishment of transformation rules. We also focus on the issue of recasting of general nonlinear systems into the GLV format. We present a procedure for doing so, and point at possible sources of ambiguity which could make the resulting Lotka-Volterra system dependent on the path followed. We then provide some general theorems that define the operational and algorithmic framework in which this is not the case.