Delay Placement Governs Feasibility and Hopf Bifurcation in a Pollution-Based Tourism Model

Abstract

We compare eight variants of a three-variable pollution-based tourism model obtained by assigning the same discrete delay to different terms. The delay may enter three slots: the visitor factor in the deterrence term, pollution perception, or visitation-driven growth and reinvestment. We call a placement feasible when every nonnegative initial history produces a solution that exists for all future time and remains nonnegative. For every $τ>0$, feasibility holds exactly when the visitor factor in the deterrence term is current, which occurs in four of the eight placements. In three of these four feasible placements, the positive equilibrium is locally asymptotically stable for every delay. In the fourth -- the perception-lag model, with delayed pollution perception but current visitation -- a conjugate pair of characteristic roots crosses the imaginary axis at a finite critical delay for every set of positive parameters. The equilibrium then loses stability and does not restabilise as the delay increases. A center-manifold reduction and numerical parameter sweep show that both supercritical and subcritical Hopf bifurcations can occur. The visitor-delayed placement, closest to earlier delayed-tourism models, can undergo a local Hopf bifurcation under additional conditions, but it does not preserve nonnegativity for all initial histories.