Spectral Rigidity and Geometric Localization of Hopf Bifurcations in Planar Predator-Prey Systems

TL;DR

This paper uncovers a geometric principle called spectral rigidity that determines where Hopf bifurcations occur in planar predator-prey models, linking nullcline critical points to bifurcation locations across continuous and discrete systems.

Contribution

It introduces the spectral rigidity principle, providing explicit geometric and algebraic criteria for bifurcation localization in various predator-prey models and their discretizations.

Findings

Bifurcations occur between consecutive nullcline critical points.

Spectral conditions are constrained by nullcline geometry.

The principle applies to both continuous and discretized models.

Abstract

We identify a geometric principle governing the location of Hopf and Bogdanov--Takens bifurcations in planar predator--prey systems. The prey coordinate of any coexistence equilibrium undergoing such a bifurcation lies between consecutive critical points of the prey nullcline. The mechanism is algebraic. At critical points of the nullcline, the vanishing of its derivative induces constraints on the Jacobian that prevent the spectral conditions required for bifurcation from being satisfied. We refer to this phenomenon as \emph{spectral rigidity}. The principle is established for three model families and one discrete counterpart with qualitatively different nullcline geometries: a quadratic case (Bazykin model), a cubic case (Holling type~IV with harvesting), and a rational case (Crowley--Martin functional response). In each case, the localization follows from explicit parametric characterizations and symbolic reduction. The analysis extends to discrete-time systems. For a map obtained by forward Euler discretization of the Crowley--Martin model, the Neimark--Sacker bifurcation occurs on the descending branch of the nullcline, providing a continuous--discrete duality governed by the same mechanism. We conjecture that this localization holds for general smooth prey nullclines, with critical points acting as spectral barriers that organise the bifurcation structure.