Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III

The present note is an addendum to the paper of Etoua–Rousseau (2010) [1] which presented a study of a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . Complete bifurcation diagrams were proposed, but some parts were conjectural. An organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3. This point was of codimension 3 when b≠0, and was conjectured to be of infinite codimension when b=0. This conjecture was in line with a second conjecture that the Hopf bifurcation of order 2 degenerates to a Hopf bifurcation of infinite codimension when b=0. In this note we prove these two conjectures.