Map dynamics of autocatalytic networks and the replicator equations

Abstract

Processes of replication and mutation pivotal to molecular evolution may be modelled by a set of coupled nonlinear differential equations descriptive of autocatalytic networks. Solutions of the four dimensional system reveal aperiodic behaviours and chaos, punctuated by regions of periodic oscillations of the population variables. This complicated dynamics is encapsulated in terms of polynomial mappings which cast the relevant features of these behaviours in compact form and reproduces many of the fine details of the sequences of bifurcations. The equations descriptive of replication are topologically equivalent to generalized Lotka-Volterra equations, and thus the present map dynamics analysis finds a corresponding broader range of potential future application.