  The evolution of autocatalytic sets (ACS) is a widespread process in
biological, chemical and ecological systems which is of great significance in
many applications, such as the evolution of new species or complex chemical
organizations. In this paper, we propose a competitive model with a 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{\alpha}
where the abrupt jump happens and verify it by simulations. Moreover, our
analysis explains how this giant independent ACS grows and reveals that, as the
selection rule becomes more strict, the phase transition is dramatically
postponed, and the number of the largest independent ACSs coexisting in the
system increases accordingly. Our research work deepens the understanding of
the evolution of ACS and should provide useful information for designing
strategies to control the emergence of ACS in corresponding applications.
