Replicator Networks with Intermediates

The behaviour of a class of autocatalytic systems with intermediates has been studied. A general chemical model has been derived, and the corresponding kinetic differential equations have been studied by both analytical and numerical methods. Two special cases were investigated in more detail: a competitive model, closely related to the second order Schlögl model, and a mutualistic model, derived from the hypercycle model due to Eigen and Schuster. The reaction systems have been investigated under three different kinds of boundary conditions: the continuously stirred tank reactor, a closed system where reaction products decay to the substrate, consuming energy from outside the system, and the evolution reactor. For a special choice of the rate constants, stabilities of all fixed points could be calculated both in the competitive and the mutualistic model. While the competitive model admits only saddle node bifurcations under these conditions, sequences of Hopf bifurcations can be found in the mutualistic model. Analytical formulae for the critical parameters where the bifurcations occur have been derived. Finally, the behaviour of the systems under diffusion has been studied. In both models, Turing instabilities can be found under appropriate conditions when starting near a fixed point that is stable in the absence of diffusion. For stable limit cycles in the mutualistic model an interesting behaviour was found: they can form either stationary but spatially inhomogeneous patterns or oscillating patches. Limit cycles on a two dimensional domain can lead to the formation of spirals.