The role of cooperation and parasites in non-linear replicator delayed extinctions

In the present work we study the role of cooperation and parasites on extinction delayed transitions for self-replicating species with catalytic activity. We first use a one-dimensional continuous equation to study the dynamics of both single autocatalytic replicator and symmetric two-member hypercycles, where two well-defined phases involving survival and extinction of replicators are shown to exist. Extinction dynamics is analyzed numerically and analytically and under both deterministic and stochastic scenarios. A ghost is also found for the single autocatalytic replicator and for the asymmetric hypercycle, with an extinction time delay following the square-root scaling law near bifurcation threshold. We find that the extinction delay is longer for the two-member hypercycle than for the single autocatalytic species, indicating that cooperation among replicators might involve to spend a longer time in the bottle-neck region of the ghost. The asymmetry of the network is shown to prolong the extinction time. We also show that an attached parasite decreases the time spent in the bottle-neck region of the ghost, thus accelerating extinction in these systems of replicators. Nevertheless the effect of the parasite is not so important when replicators catalytically cooperate, being the two-member hypercycle less sensitive to the parasite than the autocatalytic species. Here the hypercycle asymmetry can also significantly increase the delaying capacity. These features make the hypercycle to undergo a longer extinction delay, thus increasing the memory effect of the ghost. We finally explore the role of the ghost in fluctuating media, where the extinction delayed transition is shown to increase the survival probability of cooperating catalytic species.