Dynamics of small autocatalytic reaction networks. I: Bifurcations, permanence and exclusion
type: Item
instance of: scholarly article
title: Dynamics of small autocatalytic reaction networks. I: Bifurcations, permanence and exclusion
zbMATH Open document ID: 0707.92009
author: Peter F. Stadler
author: Peter Schuster
published in: Bulletin of Mathematical Biology
publication date: 1990-01-01T00:00:00Z
review text: The ordinary differential equation: \[ (*)\quad \dot x_ k=x_ k([Ax]_ k-x'Ax),\quad k=1,2,...,n \] (where \(x=(x_ 1,...,x_ n)\) stands for a column vector) describes the kinetics of catalysed multiplication of autocatalysis under the constraint of constant population size. Important issues in biophysics and biology are modelled by this equation: replication assays involving catalytic RNA, idiotypic networks in theoretical immunology, game dynamics of Maynard-Smith models of altruism in animal behaviour. The present paper makes a systematic study of low-dimensional cases of (*). The nine rate constants of the general replicator equation (*) for three species \((n=3)\) are reduced to three relevant angular parameters. The qualitative behaviour of (*) for \(n=3\) and all generic bifurcations are readily derived from the knowledge of the three angular coordinates. At non-generic transitions two or more critical surfaces are crossed in the parameter space simultaneously. Then bifurcations are more complicated, but the paper analyses the major classes of non-generic bifurcations in non-degenerate networks. The phase portraits of degenerate networks depend on two or one angular coordinates if one or two variables do not enter the r.h.s. of (*). In these cases, the complete classification of the dynamics in terms of phase portraits is listed and described.
Mathematics Subject Classification ID: 92C45
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 92C05
Mathematics Subject Classification ID: 80A30
zbMATH DE Number: 4162704
zbMATH Keywords: autocatalytic reaction networks
zbMATH Keywords: differential equations of game dynamics
zbMATH Keywords: catalysed multiplication of autocatalysis
zbMATH Keywords: constant population size
zbMATH Keywords: catalytic RNA
zbMATH Keywords: idiotypic networks
zbMATH Keywords: immunology
zbMATH Keywords: game dynamics of Maynard- Smith models of altruism
zbMATH Keywords: low-dimensional cases
zbMATH Keywords: replicator equation
zbMATH Keywords: angular parameters
zbMATH Keywords: generic bifurcations
zbMATH Keywords: non-generic transitions
zbMATH Keywords: critical surfaces
zbMATH Keywords: non-generic bifurcations
zbMATH Keywords: non-degenerate networks
zbMATH Keywords: phase portraits
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4162704