Functional shift-induced degenerate transcritical Neimark-Sacker bifurcation in a discrete hypercycle

In this article 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 degradader. 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 manner. All points of a line containing $P$ become fixed points at the bifurcation and only at the bifurcation. Hofbauer and Iooss~\cite{HofbauerIooss1984} presented and proved a result that provides sufficient conditions for a Neimark-Sacker bifurcation (the authors called it"Hopf") to occur in a special degenerate situation. They use it to prove the existence of an invariant curve for the model when a parameter related to the time discreteness of the system goes to infinity becoming a continuous-time system. Here we study the bifurcation that governs the functional shift and demonstrate the existence of an invariant curve when the cooperation parameter approaches zero and thus approaches the switch to degrading species. This invariant curve lives in a different domain and exists for a different set of values of the parameters described by these authors. In order to apply the mentioned result we uncouple the Neimark-Sacker and the transcritical bifurcations. This is accomplished by a preliminary singular change of coordinates that puts the involved fixed points at a fixed position, so that they stay at a fixed distance among them. Finally, going back to the original variables, we can describe mathematically the details of this bifurcation.