The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch

Abstract

We work with the trace field restricted to the prey nullcline of a planar predator--prey system and reorganize the first Lyapunov coefficient around this local geometry. For a graph first nullcline, we show that $J_{11}=Pg'$, where $P=-f_{1y}$ is the marginal predation rate. Under predator self-damping, this identity localizes Hopf bifurcations to ascending branches of the prey nullcline. Using nullcline-adapted coordinates and Hadamard's lemma, we derive a five-term closed expression for the first Lyapunov coefficient $\ell_1$ that remains valid for predator-dependent functional responses. We establish three distinct structural reduction mechanisms: two are general consequences of the cubic pairing with the critical eigenvectors and nullcline straightening, while the third is specific to the Gause class. Comparing the full coefficient with a $y$-affine third-order truncation yields an exact decomposition $\ell_1=\ell_1^{\mathrm{aff}}+Δ$, with $Δ$ governed by two predator-dependent cubic channels and a positive structural factor. On the Hopf locus, we derive a bridge identity showing that the same quantity $J_{11}=Pg'$ that localizes the bifurcation also weights the tangential derivative of the trace field in the criticality correction. Exact and high-precision validations for representative predator--prey models reproduce the analytical identities and numerical values.