Functional Shift-Induced Degenerate Transcritical Neimark–Sacker Bifurcation in a Discrete Hypercycle

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point $P$ while, simultaneously, two fixed points collide with $P$ in a transcritical bifurcation. Moreover, all points of a line containing $P$ become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation.