Dynamics of small autocatalytic reaction networks. I: Bifurcations, permanence and exclusion
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.
catalysed multiplication of autocatalysis
game dynamics of Maynard- Smith models of altruism
scientific article; zbMATH DE number 4162704