Where You Put the Delay Matters More Than How Long It Is

Seven of eight ways to build a time lag into a simple tourism model are fundamentally broken. Only one surviving design — the one where visitors make decisions based on today's pollution, not yesterday's — produces a world that never turns negative and stays calm forever. Flip the timing of a single feedback loop, and the same equations start oscillating in endless boom-and-bust cycles. The finding isn't about tourism per se, though the backdrop is protected areas trying to reconcile conservation with crowds. It's about a subtle mathematical fact with sharp consequences: in systems that track pollution and visitors, where you put the delay can matter more than how long it is.

This is the core of a new paper by Taylan Şengül and Bünyamin Kurtkaya (Şengül & Kurtkaya, 2026), two mathematicians at Marmara University and the Turkish Naval Academy in Istanbul. They took a family of three-variable models describing visitors $V$, pollution $P$, and capital $K$ — the built infrastructure that accommodates tourists — and asked a disarmingly concrete question. When a system involves time delays, where exactly should the delay sit in the equations to keep the model physically meaningful? The answer turns out to be both elegant and unsettling.

The Science

Let's unpack what these models actually describe. A national park, a protected coastline, a fragile island: visitors arrive, they generate pollution, and they also generate the capital — hotels, boardwalks, entry fees — that supports more visitors. Meanwhile, pollution deters visitors. Too much degradation and people stop coming clicked, or at least the delicate ecosystems people came to see start to suffer promotions.

The three state variables interact in a loop. Visitors $V$ generate pollution $P$ (at a per-capita emission rate $p_1$, with pollution decaying away naturally at rate $p_2$). Visitors also build capital $K$ (conversion rate $k_1$, with capital depreciating at $k_3$). Capital in turn attracts more visitors, at rate $v_3$. And pollution deters visitors through a "deterrence term" $-v_{2}VP$: the more pollution and the more visitors, the stronger the repulsive drag.

Here's where the subtlety enters. In the real world, these interactions aren't instantaneous. Visitors respond to pollution they perceive, which may reflect a previous time's conditions. Visitor-driven growth and reinvestment might lag behind current visitor numbers — you don't build a hotel the day the tourists arrive. And the deterrence term could plausibly respond to past rather than current visitor levels cryptography checked. Each of these three "slots" — the visitor factor in the deterrence term (call it ${\tau }_1$), pollution perception (${\tau }_2$), and visitation-driven growth (${\tau }_3$) — could carry a delay $\tau$, or not.

That gives eight possible placements, the corners of a three-dimensional "placement cube" $\{0,\tau {\}}^3$. The authors' first move was to define what makes a placement acceptable. They call a model feasible if every nonnegative starting history — every way of seeding the system with nonnegative past values of $V$, $P$, and $K$ — produces a solution that exists forever and never dips below zero.

This matters because these variables have physical meaning. A negative number of visitors is nonsense. Negative pollution is nonsense. And here's the trap: in a delayed system, a "negative interaction" that uses a past value of a variable might not vanish even when that variable is currently at zero. The system can get pushed into a physically meaningless negative region. Şengül and Kurtkaya emphasize this is not a technicality — it's a real failure mode that earlier delayed-tourism models have stumbled into.

The methodology is classical dynamical systems theory applied with unusual care. The authors linearize each placement around its equilibrium, derive a characteristic equation of the form $\mathcal{P}(\lambda )+\mathcal{Q}(\lambda )e^{-\lambda \tau }=0$, and search for purely imaginary roots $\lambda =i\omega$ — the signature that stability is about to be lost. They convert this into a polynomial root problem via an auxiliary cubic $\mathcal{H}(\xi )=0$, where $\xi ={\omega }^2$. When a root crosses the imaginary axis with nonzero speed (transversality) and isn't accompanied by other resonant roots, that's a Hopf bifurcation: the equilibrium loses stability and periodic oscillations are born (Şengül & Kurtkaya, 2026).

What They Found

The headline result is a clean dichotomy. For every positive delay $\tau >0$, a placement is feasible if and only if the visitor factor in the deterrence term is current — that is, ${\tau }_1=0$. Four of the eight placements meet this condition; the other four fail.

The proof is a thing of beauty in its simplicity (Lemma 2.4). The system's vector field is "quasi-positive": on the faces where $P=0$ or $K=0$, the state always flows back into the nonnegative region. The only face that can fail is where $V=0$. There, the growth and deterrence terms compete. If ${\tau }_1=0$, then at the moment visitors hit zero the deterrence cross-term vanishes — there are no current visitors to be deterred. The cone of nonnegative histories is forward-invariant. But if ${\tau }_1=\tau >0$, you can construct a history with zero current visitors but positive visitors one delay in the past, plus high enough past pollution. The deterrence term, drawing on that past value, yanks $V$ negative.

Four of eight delay placements are physically feasible

Label	Value
Feasible (τ1=0) placements	4
Infeasible (τ1=τ) placements	4

Data from Lemma 2.4 (Feasibility dichotomy): a placement leaves the cone of nonnegative histories forward invariant if and only if τ1=0.

.

The second surprise is what happens among the feasible placements. Three of the four — the delay-free, the instantaneous-deterrence, and the pollution-delayed models — have a positive equilibrium that is locally asymptotically stable for every delay, no matter how large. No Hopf bifurcation, ever (Lemma 2.8). But the fourth feasible placement, which the authors call the perception-lag model $(0,\tau ,0)$, behaves differently. Here pollution perception is delayed, but visitors respond to current deterrence. For every set of positive parameters — not just special tuned values — a conjugate pair of characteristic roots crosses the imaginary axis at a finite critical delay ${\tau }_0$. The equilibrium loses stability and, crucially, never restabilizes as the delay grows.

Among feasible placements, only the perception-lag model destabilizes

Label	Value
Perception-lag (τ1=0, τ2=τ, τ3=0)	1
Delay-independently stable feasible placements	3
Infeasible placements	4

Data from Table 2 and Theorem 2.14: for (0,τ,0), a conjugate pair of characteristic roots crosses the imaginary axis at a finite critical delay for every positive parameter set.

shows the eight placements and their fates: four infeasible, three delay-independently stable, one doomed to oscillate.

The perception-lag model's instability is remarkable because it's universal. Most delay models become unstable only for parameter values in some favorable region. Here, every positive parameter set produces the collapse of stability. A center-manifold reduction and numerical parameter sweep reveal that the ensuing Hopf bifurcation can be supercritical or subcritical — meaning the emerging periodic orbits can be stable, small-amplitude oscillations, or unstable large-amplitude ones (Şengül & Kurtkaya, 2026).

Perception-lag Hopf bifurcation can be supercritical or subcritical

Label	Value
Supercritical (stable oscillations)	1
Subcritical (unstable oscillations)	1

Data from Şengül & Kurtkaya (2026), Section 3: the bifurcation direction depends on parameters in the perception-lag model.

maps how the bifurcation direction flips as parameters drift.

The authors also examine one infeasible placement closely — the visitor-delayed model $(\tau ,0,\tau )$. It most closely resembles the delayed-tourism models in the earlier literature, which is precisely why it's a cautionary tale. It can undergo a local Hopf bifurcation under additional simplicity, transversality, and nonresonance conditions. But it doesn't preserve nonnegativity for all initial histories. So a model of this shape can oscillate — while also producing nonsensical negative visitors from perfectly reasonable starting conditions. An earlier paper by Caraballo and colleagues proposed a fix that preserved nonnegativity; Şengül and Kurtkaya show, in their Appendix B, that this fix, while saving positivity, simultaneously destroys the Hopf bifurcation. You cannot have both, in that construction.

Why This Changes Things

This is a paper about model design, not about any specific park. But that's exactly why it matters for the world. Delayed dynamical models are everywhere: epidemiology, economic policy, climate, fisheries, conservation. Whenever the present state of a system depends on its past, someone writes a delay differential equation. The received wisdom has been that the magnitude of a delay drives oscillations — that longer lags destabilize. Şengül and Kurtkaya's result complicates that picture sharply: the placement of the delay can be the decisive feature.

The feasibility dichotomy has a plain-language translation. If a model lets the deterrence term consult past visitors rather than current ones, then the model can trap itself in a state where visitors are already zero but the system keeps "remembering" that there used to be plenty of them — and the polluted memory drags the solution below zero, into a region where visits and pollution no longer mean anything. This is not an exotic edge case. The construction in the proof (Eq. 2.3) uses strictly positive initial histories, so even perfectly reasonable, all-positive starting data can drive the visit count negative in finite time.

The deeper lesson is about how feedback loops in social-ecological systems should be modeled. Consider what "deterrence" means in tourism. Visitors don't just react to the pollution level they perceive; they react in proportion to how many of them are present — congestion and crowding effects scale with the crowd. The model encodes this with the product $-v_{2}VP$. Whether the $V$ in that product is current or delayed turns out to be the single most consequential modeling choice in the entire family. It determines physical feasibility. It determines whether the system oscillates. It determines whether the model can even be believed.

This connects to a broader theme in the recent literature that the authors cite: in delayed logistic-type equations, whether you get oscillations depends on how the delay enters the feedback, not just on how big it is (Şengül & Kurtkaya, 2026, citing references on delayed logistic equations). The present paper is a systematic demonstration of that principle in a realistic three-variable system.

There's also a sobering check on earlier work. The visitor-delayed placement is "closest to earlier delayed-tourism models," and those models found Hopf bifurcations. Şengül and Kurtkaya's analysis shows that such bifurcations can be real — but they occur in a model that simultaneously fails to preserve nonnegativity. In other words, some of the celebrated oscillation results in delayed-tourism theory may have been describing dynamics in a regime where the model had already become physically incoherent. That's not to say the oscillations are artifacts — the visitor-delayed model genuinely bifurcates under the right conditions — but it means the model class needs to be re-examined before its predictions about real parks and real tourist economies are taken at face value.

What's Next

The paper is explicitly a first step, not a conclusion. The authors restricted themselves to common-delay placements, where a single $\tau$ is either zero or present in each slot. The full problem with three independent delays ${\tau }_1,{\tau }_2,{\tau }_3$ remains open — "the complete specification" of sixteen distinct multi-delay placements with possibly different lags. The restriction was a deliberate choice to isolate placement effects from magnitude effects inherent: the lesson that placement dominates is cleanest when the magnitude is held fixed. Real systems presumably have three different lags, each with its own physical origin — and the interaction between those lags is exactly the kind of thing this framework is built to probe next.

Several open threads jump out. First, the delay-independently stable feasible placements are stable for every finite $\tau$, but the intuition is worth interrogating: why does adding pollution perception delay alone to a stable system trigger universal instability, while adding growth-and-reinvestment delay alone, or both together, leaves it rock-solid? The answer must live in which feedback loop is being desynchronized. Teasing out that mechanism in biological or economic terms would be valuable.

Second, the subcritical Hopf bifurcations deserve scrutiny. A subcritical bifurcation means the stable equilibrium coexists with unstable large-amplitude oscillations just before the critical delay. In practical terms, a system that looks stable might be sitting next to violent periodic swings it can switch into under perturbation. For tourism management, that's the difference between "the park will settle down" and "the park might suddenly lurch into a boom-bust cycle." Getting the bifurcation direction right is not a theoretical footnote; it's the difference between a reassuring and a cautionary model.

Third and most important, the feasibility criterion should become a standard screening tool. Before any delayed social-ecological model is used to draw conclusions about real places, it should pass the nonnegativity test that Şengül and Kurtkaya formalize. Their machinery — the quasi-positivity face-by-face check, the Gronwall-boundedness argument for global existence — is general enough to apply beyond tourism, to any positive-variable delay system. That's a genuinely useful contribution to how the field builds models, not just what one model says.

The broader significance is almost philosophical. This work is a reminder that the structure of a model — which variables talk to which, and with what lag — can be a more powerful determinant of its behavior than its parameters. Two models with identical numbers but different delay placements can disagree about whether a tourist economy is stable or perpetually oscillating, whether it respects the physical limits of nonnegativity, or whether it spirals into nonsense. For anyone trying to predict the future of a real place — a fragile island, a beloved national park, a coastline buckling under visitation — the choice of where to put the delay is not a technical detail. It's the whole argument.