Nullcline geometry constrains the location of oscillatory instabilities in planar predator--prey systems

Abstract

We prove that the critical structure of the prey nullcline governs where oscillatory instabilities can emerge in planar predator--prey systems. For a broad class of Gause--type models, we establish a general geometric localization theorem: the prey coordinate of every Hopf bifurcation point is confined between consecutive critical points of the prey nullcline. The mechanism is purely geometric. Along the nullcline, the diagonal Jacobian entry $J_{11}$ is proportional to the nullcline slope $g'(x)$, independently of the bifurcation parameter, while $J_{22}\le 0$ at any coexistence equilibrium. At critical points of the nullcline ($g'=0$), the Jacobian trace becomes strictly negative, creating a spectral barrier that precludes oscillatory instability. This barrier partitions the state space into dynamically distinct regions, confining the onset of limit--cycle oscillations to the ascending branches of the nullcline. We illustrate the principle in three canonical families---Bazykin's model (quadratic nullcline), a Leslie--Holling type~IV system with harvesting (cubic nullcline), and the Crowley--Martin model with predator interference (rational nullcline)---for which we obtain closed--form Hopf bifurcation loci. The same geometric mechanism extends to discrete time: for the forward Euler map with step size $\tau$, an exact identity $\det(J^G)-1=\tau[\operatorname{tr}(J)+\tau\det(J)]$ shows that the Neimark--Sacker locus is strictly disjoint from the continuous Hopf locus yet tracks it on the same ascending branch at an $O(\tau)$ distance, crossing the spectral barrier only in the coarse--step regime. The results demonstrate that the functional responses responsible for predator saturation and interference not only bound the efficiency of interactions but also imprint a geometric architecture on the bifurcation landscape.