Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III
This note is an addendum to the paper [\textit{R. M. Etoua} and \textit{C. Rousseau}, J. Differ. Equations 249, No. 9, 2316--2356 (2010; Zbl 1217.34080)], where a generalized Gause model with prey harvesting and a generalized Holling response function of type III: \(p(x)={m x^2 \over a x^2 + b x + 1}\) was studied and complete bifurcation diagrams were proposed. A conjecture was proposed for a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3. It was conjectured that the nilpotent point is of infinite codimension for \(b=0\). This conjecture was in line with a second conjecture about the codimension of Hopf bifurcation for \(b=0\). These two conjectures are proved in this note. The method of proof is quite original. It consists in finding analytic transformation and time scaling of the system bringing it to a time-reversible system.
generalized Gause model with prey harvesting
scientific article; zbMATH DE number 5957032