Dynamics of Autocatalytic Replicator Networks Based on Higher Order Ligation Reactions


Bärbel M.R. Stadler, Peter F. Stadler, Peter Schuster

A class of autocatalytic reaction networks based on template dependent ligation and higher order catalysis is analyzed. Apart from an irreversible ligation reaction we consider only reversible aggregation steps that provide a realistic description of molecular recognition. The over-all dynamics can be understood by means of replicator equations with highly non-linear interaction functions. The dynamics depends crucially on the total concentration c0 of replicating material.
For small c0 in the hyperbolic growth regime, we recover the familiar dynamics of second order replicator equations with its wealth of complex dynamics ranging from multi-stability to periodic and strange attractors as well as to heteroclinic orbits. For large c0, in the parabolic growth regime, product inhibition becomes dominating and we observe a single globally stable equilibrium tantamount to permanent coexistence. In an intermediate parameter range we sometimes observe a behavior that is reminiscent of ``survival of the fittest''. Independently replicating species (Schlögel's model) and the hypercycle are discussed in detail.

Submitted to Bull.Math.Biol..

Keywords: Autocatalytic Networks, Replicator Equation, Ligation, Hypercycle

Return to 1999 working papers list.