On the effect of time lags on a saddle-node remnant in hyperbolic replicators

Saddle-node (s-n) bifurcations can be responsible for abrupt changes between alternative states in nonlinear dynamical systems. It is known that once a s-n bifurcation takes place, a s-n remnant (also named ghost or delayed transition) can continue attracting the flows in the phase space before they achieve another attractor. The time needed to pass through the saddle remnant, which causes an extremely long transient after the bifurcation, is known to follow an inverse square-root law. In this manuscript we investigate the effect of time lags in the transient dynamics near a s-n bifurcation by means of delay differential equations. To do so we use a one-variable dynamical system describing the dynamics of an autocatalytic replicator, introducing a time lag, tau, in the process of hyperbolic replication, becoming an infinity-dimensional dynamical system. We show that the delayed transitions found in the lagged system become much longer than those found in the system without time lags, although the inverse square-root law is preserved. The time the flows spend crossing the ghost is shown to increase linearly with tau. The implications of these transients' enlargement are discussed in the framework of prebiotic evolution.