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

Abstract

Along the prey nullcline $y=g(x)$ of a planar system, the Jacobian entry satisfies $J_{11}=P{g}^{\prime }$, where $P=-f_{1y}{|}_{\gamma }$. Under $P>0$ and predator self-damping, the Hopf trace condition forces $g^{\prime }(x^{\ast })>0$: 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 $\tau =D{|}_{\gamma }$. 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 $\tau$ and $\nu =f_2{|}_{\gamma }$, 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 $\phi \mapsto g$ removes $p^{\prime \prime \prime }$. They have different scopes. For predator-dependent responses, comparison with the $y$-affine truncation of the $3$-jet at fixed linear part gives ${\ell }_1={\ell }_1^{\mathrm{aff}}+\Delta$, where $\Delta =\Lambda (f_{1xyy}+{\omega }_0^{-2}⟨{\partial }_{x}f,\nabla D⟩f_{1yy})$ and $\Lambda >0$. On the Hopf locus, $⟨{\partial }_{x}f,\nabla D⟩=J_{11}{\tau }^{\prime }+({\omega }_0^2/P)D_y$. Thus the same quantity $J_{11}=P{g}^{\prime }$ 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 ${\ell }_1,{\ell }_1^{\mathrm{aff}},\Delta ,\Lambda$ scale by $|c{|}^2$ under $q\mapsto cq$; their signs and ratios are unchanged.

22 pages, 1 figure. Includes a supplementary Wolfram Language notebook