Shaking Verhulst: From Oscillation to Aperiodicity in Quasi-Periodically Forced Populations

We investigate the global dynamics of the Verhulst logistic model driven by time-dependent carrying capacity. For single-frequency forcing ("wobbling"), we characterize the system as a "hunting" dynamic driven by inevitable topological crossings of the nullcline manifold. We derive the exact closed-form solution and prove the global asymptotic stability of a unique limit cycle. When the forcing becomes quasi-periodic ("shaking")—specifically using the Golden Ratio φ to break resonance—the system transitions from a smooth torus attractor to a Strange Non-Chaotic Attractor (SNA). We derive the exact integral condition for the critical transition parameter, marking the onset of fractal "bumping" dynamics via a grazing bifurcation. Finally, we derive the analytic solution for the stochastic variance of the system, demonstrating how noise acts as a "smearing kernel" that converts the fractal grazing boundary into a probabilistic extinction tunnel.