Mathematical models for resource depletion under progressively aggressive extraction regimes, culminating in the Gompertz decline curve, derived from the oil-shock model framework in Mathematical Geoenergy (GEM), Chapter 8 (Alternative Characterization and Models, Sections 8.10–8.12).
Chapter 8 examines what happens to oil (and general resource) production when the extraction rate is no longer constant. Starting from the simplest proportional-depletion ODE, the chapter builds up three increasingly aggressive extraction scenarios:
Constant extraction rate — the textbook case, giving pure exponential reserve decline and a monotonically-falling production curve.
Linearly-increasing extraction rate — an accelerating pump-rate prolongs the production plateau but inevitably leads to a sharper subsequent collapse, shifting the peak by a finite, asymptotically-limited delay (≈ 8 years, as discussed in § 8.10).
Exponentially-increasing extraction rate — the most aggressive scenario, in which demand-driven extraction accelerates exponentially. This produces the Gompertz decline curve (§ 8.11): a slow build-up followed by an abrupt collapse that is far steeper than the symmetric Hubbert (logistic) curve.
The Gompertz equation arose historically in population biology (Gompertz, 1825) and survival analysis, but it also describes any system in which a resource is exploited at an exponentially growing rate from a finite stock. Concrete examples discussed in § 8.12 include:
Nauru phosphate — discovered as a singular event around 1900; worldwide fertiliser demand then grew at ≈ 9–10 % per year, producing a Gompertz-like production-and-collapse curve that tracks actual data better than the logistic Hubbert model for this constrained, single-discovery resource.
Passenger pigeon — the colonial-era hunting of an effectively non-dispersive population, whose culling accelerated until the species became extinct in the late 1800s, mirrors the Gompertz collapse.
The key difference from the Hubbert model is the absence of significant discovery dispersion: oil has many discoveries spread over decades (which mitigates the collapse), whereas a single island’s phosphate ore or a single population has no such spread.
Eq. 8-40 — Proportional depletion ODE
\[\frac{dU}{dt} = -k\\;\cdot U(t)\]The extraction rate is proportional to remaining reserves: every unit of reserve has the same probability k of being extracted per unit time. This is the founding hypothesis of Hubbert’s shock model.
Eq. 8-41 — Exponential reserve decline
\[U(t) = U_0\\;\cdot e^{-kt}\]Eq. 8-42 — Exponential production curve
\[P(t) = -\frac{dU}{dt} = k\\;\cdot U_0\\;\cdot e^{-kt} = k\\;\cdot U(t)\]Production falls monotonically from $P(0)=kU_0$. The total resource produced is conserved: $\int_0^\infty P(t)\,dt = U_0$.
Eq. 8-43 — Linear extraction ramp ODE
\[\frac{dU}{dt} = -(k + c\\,t)\\;\cdot U(t)\]Eq. 8-44 — Separated form
\[\frac{dU}{U} = -(k + c\\,t)\\,dt\]Eq. 8-45 — Integrated log-reserve
\[\ln U - \ln U_0 = -\\!\left(k t + \frac{c\\,t^2}{2}\right)\]Eq. 8-46 — Gaussian-modulated reserve decline
\[U(t) = U_0\\;\cdot\exp\\!\left(-k t - \frac{c\\,t^2}{2}\right)\]Eq. 8-47 — Plateauing production curve
\[P(t) = (k + c\\,t)\\;\cdot U_0\\;\cdot\exp\\!\left(-k t - \frac{c\\,t^2}{2}\right)\]When $c > 0$ the extraction rate grows linearly, creating a momentary production plateau. The peak occurs at:
\[t^* = \frac{\sqrt{c} - k}{c}\]This is positive only when $\sqrt{c} > k$, i.e. the ramp coefficient is large enough to offset the initial constant rate. In practice the peak delay approaches an asymptote: no matter how fast the ramp grows, the exponential damping in reserves limits the achievable delay to ≈ 8 years (as demonstrated numerically in the book).
Eq. 8-48 — Exponential extraction ramp (separated)
\[\frac{dU}{U} = -\\!\left(k + a\\,e^{bt}\right)dt\]The extraction rate $k(t) = k + a\,e^{bt}$ grows exponentially with growth rate $b$ and initial amplitude $a$.
Eq. 8-49 — Integrated log-reserve
\[\ln U - \ln U_0 = -\\!\left(k t + \frac{a\\,e^{bt}}{b}\right)\]Eq. 8-50 — Gompertz reserve decline
\[U(t) = U_0\\;\cdot\exp\\!\left(-k t - \frac{a\\,e^{bt}}{b}\right)\]At $t=0$ the reserves are already reduced by the initial exponential term: $U(0) = U_0\,e^{-a/b}$. As $t\to\infty$, $U\to 0$ (complete depletion).
Eq. 8-51 — Gompertz production curve
\[P(t) = \left(k + a\\,e^{bt}\right)\\;\cdot U_0\\;\cdot \exp\\!\left(-k t - \frac{a\\,e^{bt}}{b}\right)\]For the pure Gompertz ($k=0$) the peak time and peak value are:
\[t^* = \frac{\ln(b/a)}{b}, \qquad P(t^*) = U_0\\,b\\,e^{-1} \approx 0.368\\;U_0\\,b\]The right-flank collapse is faster than the left-flank rise, giving the characteristic asymmetric shape with a narrow, sharp peak and an abrupt post-peak collapse — in contrast to the symmetric Hubbert (logistic) curve.
For the pure Gompertz ($k=0$), the logarithmic derivative of production is:
\[\frac{d\ln P}{dt} = b\bigl(1 - e^{bt}\cdot(a/b)\bigr) = b\\!\left(1 - e^{b(t-t^*)}\right)\]The Hubbert logistic by contrast has symmetric half-widths on both flanks. Numerically, for representative parameters ($a=0.01,\; b=0.15$) the Gompertz left half-width is ≈ 9.7 yr while the right half-width is only ≈ 6.6 yr (asymmetry ratio ≈ 0.67), confirming the steeper right-side collapse.
The cumulative production equals $U(0) = U_0\,e^{-a/b}$, which is strictly less than $U_0$: not all reserves are extracted within finite time under the Gompertz regime when reset to $t=0$. Conservation is guaranteed by the fundamental theorem of calculus applied to the ODE: $\int_0^\infty P(t)\,dt = U(0) - U(\infty) = U_0\,e^{-a/b}$.
| File | Purpose |
|---|---|
gompertz_symbolic.py |
Symbolic derivation of all Chapter 8 Gompertz equations using SymPy |
gompertz_numerical.py |
Numerical implementation, validation, and composite figure |
gompertz_model_output.png |
Output figure (6 panels) generated by gompertz_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python gompertz_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python gompertz_numerical.py
Proportional depletion is the baseline: the ODE $dU/dt = -kU$ encodes the physical constraint that extraction rate is proportional to the remaining reserve. Every increase in extraction rate must be paid for by a correspondingly sharper future decline — there is no free lunch.
Linear ramp delays the peak only finitely: increasing $k(t) = k + ct$ does shift the production peak, but the delay asymptotes to ≈ 8 years regardless of how large $c$ becomes. The exponential damping in reserves swamps any polynomial increase in extraction, making a prolonged plateau impossible in a finite-resource system.
Exponential extraction drives the Gompertz collapse: once the extraction rate itself grows exponentially ($a\,e^{bt}$), the Gompertz function replaces the logistic. The right-side collapse is exponential-in-exponential — far steeper than the symmetric logistic decline — because reserves fall just as the demand for them peaks.
Gompertz applies to single-discovery, unconstrained resources: the Nauru phosphate case demonstrates Gompertz dynamics because (a) all reserves were discovered essentially at once (no dispersive discovery curve), and (b) production was not supply-limited (phosphate can be extracted as fast as demand requires, unlike oil in a low-permeability reservoir). The oil case is mitigated by dispersed multi-field discovery.
The Gompertz peak is shallower but the collapse is steeper: compared to the Hubbert (logistic) model with the same total resource and same peak time, the Gompertz curve has a similar front but a narrower, sharper peak and a much faster post-peak decline. This is consistent with the phosphate data where the Hubbert Linearization underestimates the decline rate (Fig. 8-30 in the book).
Minimal parameterisation: the entire Gompertz model needs only three parameters — $k$, $a$, and $b$ (or just $a$ and $b$ in the pure case) — plus the total initial resource $U_0$. The MaxEnt principle selects the exponential extraction prior as the distribution that maximises entropy given only a mean growth rate $b$, providing the simplest explanation for the observed accelerating extraction dynamics.