The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch
Along the prey nullcline $y=g(x)$ of a planar system, the Jacobian entry satisfies $J_{11}=P\,g'$, where $P=-f_{1y}|_γ$. Under $P>0$ and predator self-damping, the Hopf trace condition forces $g'(x^\ast)>0$: critical points of $g$ are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let $D=f_{1x}+f_{2y}$ and $τ=D|_γ$. Using nullcline coordinates $(u=x,v=y-g(x))$ and Hadamard's lemma gives $\dot u=-vH$, $\dot v=W$, with $H(u,0)=P$. The mixed jet of $W$ is determined by the jets of $τ$ and $ν=f_2|_γ$, yielding a closed five-term formula for the first Lyapunov coefficient $\ell_1$ in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives $\partial\ell_1/\partial f_{1yyy}=\partial\ell_1/\partial f_{2xxx}=0$. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization $φ\mapsto g$ removes $p'''$. They have different scopes. For predator-dependent responses, comparison with the $y$-affine truncation of the $3$-jet at fixed linear part gives $\ell_1=\ell_1^{\mathrm{aff}}+Δ$, where $Δ=Λ\bigl(f_{1xyy}+ω_0^{-2}\langle\partial_x f,\nabla D\rangle f_{1yy}\bigr)$ and $Λ>0$. On the Hopf locus, $\langle\partial_x f,\nabla D\rangle=J_{11}τ'+(ω_0^2/P)D_y$. Thus the same quantity $J_{11}=Pg'$ that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while $\ell_1,\ell_1^{\mathrm{aff}},Δ,Λ$ scale by $|c|^2$ under $q\mapsto cq$; their signs and ratios are unchanged.