  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.
