Geometric localization conjecture for Hopf, Bogdanov–Takens, and Neimark–Sacker bifurcations

Consider a smooth ($C^2$ at least) predator–prey system of the form given in the paper in the first quadrant of $\mathbb{R}^2$, whose prey nullcline $y=g(x)$ has exactly two critical points in $(0,\infty)$: a local minimum at $x_{\min}$ and a local maximum at $x_{\max}$, with $0<x_{\min}<x_{\max}$. Suppose the system admits exactly three coexistence equilibria.

Geometric localization conjecture. Every coexistence equilibrium at which a Hopf or Bogdanov–Takens bifurcation occurs has prey coordinate in $(x_{\min},x_{\max})$, where $g'(x^*)>0$. In the discrete-time map version, the Neimark–Sacker bifurcation occurs at equilibria satisfying $g'(x^*)<0$, on descending branches of the nullcline. At the critical points, $g'(x)=0$, an algebraic dependence among the Jacobian entries precludes the spectral conditions for bifurcation: trace zero in flows and unit determinant in maps.

The continuous-time assertion is established for quadratic, cubic, and rational nullclines, and the discrete-time assertion is established for the studied forward-Euler model. A general proof for arbitrary smooth nullclines without case-by-case computation remains open.