Determine the long-time behavior after subcritical Hopf destabilization

Determine the long-time behavior of trajectories of the perception-lag pollution-based tourism model after the subcritical Hopf bifurcation, including whether trajectories that depart from the positive equilibrium converge to a large-amplitude periodic orbit or another attractor.

Background

For the second numerical scenario of the perception-lag model, the Hopf bifurcation is subcritical. Above the critical delay, simulated trajectories depart from the equilibrium and exhibit rapidly growing oscillations over the finite numerical horizons considered.

The paper proves that finite-time blow-up cannot occur for nonnegative solutions, but this result does not establish boundedness as time tends to infinity or identify the eventual attractor. In particular, the computations do not determine whether the departing trajectories ultimately approach a large-amplitude periodic orbit, so their asymptotic behavior remains unresolved.

References

The long-time behaviour, including possible convergence to a large-amplitude periodic orbit, therefore remains unresolved.

— Delay Placement Governs Feasibility and Hopf Bifurcation in a Pollution-Based Tourism Model  (2609.10452 - Şengül et al., 9 Sep 2026) in Section 3, subsection “Perception-Lag Scenario 2: Subcritical Hopf,” paragraph “Departure above onset”