Evolution of autocatalytic sets in a competitive percolation model

The evolution of autocatalytic sets (ACSs) is a widespread process in biological, chemical, and ecological systems and is of great significance in many applications such as the evolution of new species or complex chemical organization. In this paper, we propose a competitive model with an m-selection rule in which an abrupt emergence of a macroscopic independent ACS is observed. By numerical simulations, we find that the maximal increase of the size grows linearly with the system size. We analytically derive the threshold t_α where the explosive transition occurs and verify it by simulations. Moreover, our analysis explains how this giant independent ACS grows and reveals that, as the selection rule becomes stricter, the phase transition is dramatically postponed, and the number of the largest independent ACSs coexisting in the system increases accordingly. Our result indicates that suppression during evolution could lead to the abrupt appearance of giant ACSs.