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\section{Depletion Theory}

L. David Roper (roperld@vt.edu) 1976, revised 2000

\subsubsection{Introduction\protect\bigskip }

An outline of a theory of depletion of nonrenewable resources is presented
here. Start with two basic irrefutable facts:

\begin{enumerate}
\item  The earth (or any portion of the earth) is a finite source of any
mineral.

\item  As a mineral is extracted from the earth it becomes steadily more
difficult to extract the remainder. By ``more difficult'' is meant that more
materials and energy are required and more environmental degradation occurs.
\end{enumerate}

These two facts define complicated nonlinear interactions among all
minerals: The increasing scarcity with time of one mineral (say crude oil)
makes it more difficult to obtain another mineral (say iron ore) which may
be crucially important in the extraction of the first mineral. This is only
one of scores of nonlinear interactions that are simultaneously at work.

Common sense tells one the kind of long-term ``average'' production-rate
behavior to expect for any mineral. There are components of both technology
and sociology that interplay in the behavior:

\begin{enumerate}
\item  In the \textit{earliest times} the mineral is relatively readily
available, but the technology for its extraction and society's need for it
are undeveloped. Therefore, the production rate will increase slowly at
first. However, as the extracted mineral enters into the mainstream of the
society its presence will generate more need for it and thereby generate
more advanced extraction technology. Thus, it is reasonable to assume that
the production rate at earliest times will be some increasing function of
the amount already extracted at that time. Let $Q_{\infty }\equiv $ amount
that will be eventually extracted and $Q(t)\equiv $ amount left to be
extracted at time $t$; then the production rate $P(t)=-dQ/dt$ at time $t$ is
some function of [$Q_{\infty }-Q(t)$]. Since any smooth function can be
expanded as a power series in an independent variable, at the very earliest
times $P(t)$ should be proportional to some power of [$Q_{\infty }-Q(t)$].
The simplest assumption and one that often works in other similar situations
is that $P(t)\varpropto \lbrack Q_{\infty }-Q(t)]$ at the earliest times.
However, more complicated possibilities will be considered.

\item  At the \textit{latest times} when the mineral is almost completely
depleted, the principal limitation on the production rate $P$ will be the
amount left to be extracted $Q(t)$ at that time. Again, at the very latest
times $P$ should be proportional to some power of $Q(t)$. The simplest
assumption and one that often works in other similar situations is that $%
P(t)\varpropto Q(t)$ at the latest times. However, more complicated
possibilities will be considered.

\item  3. At \textit{intermediate times} there are no rational arguments
that we can muster for any particular functional form for $P(t)$ as a
function of $Q(t)$. So we shall consider several possibilities and let the
production data for a given mineral ``choose'' which of the possibilities
works best by performing least-squares fits to the data. Some obvious
statements can be made, however: After rising slowly at earliest times, the
production rate should begin to accelerate, then later (at an inflection
point) decelerate until the production rate peaks at some time. Then the
rate will begin to decline in a similar, but not necessarily symmetrical,
fashion. Finally, $P(t)$ will asymptotically approach zero. The simplest
assumption that one could make which yields this kind of behavior is that $%
P(t)$ is strictly proportional to the first power of both [$Q_{\infty }-Q(t)$%
] and $Q(t)$ at all times; i.e. 
\begin{equation}
P(t)=-\frac{dQ}{dt}=kQ(Q_{\infty }-Q)/Q_{\infty }  \tag{Eq. (1)}
\end{equation}
where $k$ is a rate constant that is a measure of the usefulness of the
mineral and the long-term economic conditions of the society. (One can
define a time constant $\tau =1/k$.) This equation, as well as more
complicated cases, are discussed below.
\end{enumerate}

Of course, in reality for a particular mineral the long-term average
behavior described above will not precisely describe the production-rate
behavior. There are short-term social phenomena, such as wars and economic
depressions, that can and sometimes do cause rather large fluctuations in
the production rates. (A detailed study of correlations of these
mineral-production fluctuations with specific social phenomena would be
interesting. We do not attempt it here.) These short-term fluctuations
exhibit behavior similar to that described above for the long-term average
behavior except that the rate constant k is greatly increased (time constant
t is greatly reduced). (We shall often refer to the long-term average
behavior as the ``background'' behavior.) There are two situations that
could exist:

\begin{enumerate}
\item  The short-term fluctuations have little or no effect on the long-term
background behavior. That is, the rate constant for the background behavior
is unchanged as short-term fluctuations occur.

\item  The short-term fluctuations are evidence of changes in the long-term
use of that mineral either because of the onset of new long-term social
phenomena or new mineral technology (e.g., substitution of another mineral
for it in its major use). That is, the rate constant for the background
behavior is changed as short-term fluctuations occur.
\end{enumerate}

Of course, it is possible that the long-term background rate ``constant'' $k$
is not really a constant in time even in the absence of fluctuations. In
fact, one would think that after a mineral has become an integral part of a
society's \textit{modus operandi} that the society will make a large effort
to keep its production rate up when it otherwise would decline sharply for
constant $k$. That is, the society's increased efforts to extract the
mineral will cause $k$ to decrease with time rather than be constant. This
will cause the production-rate curve to be asymmetrically skewed toward
large times; most nearly depleted United States minerals have such skewed
production-rate curves. Also, gradual substitution of one mineral for
another (e.g., oil for coal) could cause $k$ to change with time.

\subsubsection{Logistic Equation}

Eq. (1) given above for the production rate $P(t)$ as a function of the
amount $Q(t)$ yet to be extracted at time t can be solved to yield the
widely-used logistic equation:
\begin{equation}
Q(t)=\frac{1}{2}Q_{\infty }\left[ 1-\tanh \left( \frac{t-t_{1/2}}{2\tau }%
\right) \right] \text{ ,}  \tag{Eq. (2)}
\end{equation}

where $\tau =1/k$ is the time constant and $t_{1/2}$ is the time at which
the mineral is one-half depleted. Label $t_{1/2}$ as the ``half date''. Then
\begin{equation*}
P\left( t\right) =\frac{Q_{\infty }}{4\tau }\left[ 1-\tanh ^{2}\left( \frac{%
t-t_{1/2}}{2\tau }\right) \right] 
\end{equation*}

\subsubsection{Variable Decay-Rate Model}

One can complicate the simple model developed above by assuming that the
decay rate, $k(t)$, is a function of time. Then
\begin{equation}
P(t)=-\frac{dQ\left( t\right) }{dt}=k\left( t\right) \frac{Q\left( t\right) 
}{Q_{\infty }}\left[ Q_{\infty }-Q\left( t\right) \right] \text{ ,} 
\tag{Eq. (3)}
\end{equation}
which has the solution
\begin{equation}
Q(t)=\frac{1}{2}Q_{\infty }\left[ 1-\tanh \left( \frac{g\left( t\right)
-g\left( t_{1/2}\right) }{2\tau }\right) \right] \text{ ,}  \tag{Eq. (4)}
\end{equation}
where
\begin{equation*}
g\left( t\right) \equiv \int^{t}k\left( t\right) dt\text{ .}
\end{equation*}
This approach could be used to give the large-time skewing that often occurs
in nearly depleted production data.

\subsubsection{Generalized Verhulst Equation}

A more complicated, but still analytically solvable, case that contains Eqs.
(2) and (4) above as special cases is the Verhulst equation, defined by:
\begin{equation}
P(t)=-\frac{dQ\left( t\right) }{dt}=\frac{k\left( t\right) Q\left( t\right) 
}{n}\left[ 1-\left\{ \frac{Q\left( t\right) }{Q_{\infty }}\right\} ^{n}%
\right] \text{ ,}  \tag{Eq. (5)}
\end{equation}
where $n\equiv ``$asymmetry parameter''. It has the solution
\begin{eqnarray}
Q\left( t\right)  &=&\frac{Q_{\infty }}{\left[ 1+\left( 2^{n}-1\right) \exp
\left\{ g(t)-g(t_{1/2})\right\} \right] ^{1/n}}  \TCItag{Eq. (6)} \\
&&\overrightarrow{\tau =\text{constant}}\text{ }\frac{Q_{\infty }}{\left[
1+\left( 2^{n}-1\right) \exp \left( \frac{t-t_{1/2}}{2\tau }\right) \right]
^{1/n}}\text{ .}  \notag
\end{eqnarray}
Then
\begin{equation*}
P(t)=\frac{Q_{\infty }}{n\tau }\frac{\left( 2^{n}-1\right) \exp \left( \frac{%
t-t_{1/2}}{2\tau }\right) }{\left[ 1+\left( 2^{n}-1\right) \exp \left( \frac{%
t-t_{1/2}}{2\tau }\right) \right] ^{1/n}}\text{ .}
\end{equation*}

Note that this appoach assumes that $P(t)$ is linear in $Q(t)$ for large
times but is nonlinear in $\left[ Q_{\infty }-Q(t)\right] $ for small times.

For $n=1$ Eq. (6) can be shown to be the same as Eq. (4), which is the same
as Eq. (2) when $k(t)$ is constant in $t$. However, even with $k=$ constant,
Eq. (6) contains a large-time skewing if $n>1$. (The generalized Verhulst
curve is skewed toward short times for $0\geq n>1$, is symmetrical for $n=1$
and is skewed toward large times for $n>1$.) We choose Eq. (6) in fitting
the nearly-depleted skewed data rather than the Eq. (4) because it is
mathematically simpler.

\subsubsection{Gompertz Equation}

Another special case of the Verhulst equation [Eq. (6)] is when $k=$constant
and $n=0$. This yields the Gompertz equation, which is given by the
equations:
\begin{equation}
P(t)=-\frac{dQ\left( t\right) }{dt}=\frac{1}{\tau }Q(t)\ln \left[ \frac{Q(t)%
}{Q_{\infty }}\right]   \tag{Eq. (7)}
\end{equation}
and
\begin{equation}
Q(t)=Q_{\infty }\left( \frac{1}{2}\right) ^{\exp \left[ \left(
t-t_{1/2}\right) /\tau \right] }\text{ .}  \tag{Eq. (8)}
\end{equation}
This curve is skewed toward short times. Then
\begin{equation*}
P(t)=\ln 2\frac{Q_{\infty }}{\tau }\exp \left( \frac{t-t_{1/2}}{\tau }%
\right) \left( \frac{1}{2}\right) ^{^{\exp \left[ \left( t-t_{1/2}\right)
/\tau \right] }}\text{ .}
\end{equation*}

\subsubsection{Error Function Equation}

There are at least two other symmetric peak functions that often occur in
nature. One is the Gaussian function, which is given by the equations:
\begin{equation}
P(t)=-\frac{dQ\left( t\right) }{dt}=\frac{Q_{\infty }}{\tau \sqrt{\pi }}\exp %
\left[ -\left( \frac{t-t_{1/2}}{\tau }\right) ^{2}\right]   \tag{Eq. (9)}
\end{equation}
and
\begin{equation}
Q(t)=\frac{1}{2}Q_{\infty }\left[ 1-\func{erf}\left( \frac{t-t_{1/2}}{\tau }%
\right) \right] \text{ .}  \tag{Eq. (10)}
\end{equation}

\subsubsection{Lorentzian Equation}

Another symmetric peak function that often occurs in nature is the
Lorentzian function, which is given by the equations:
\begin{equation}
P(t)=-\frac{dQ\left( t\right) }{dt}=\frac{Q_{\infty }}{\pi \tau }\left[ 
\frac{1}{1+\left( \frac{t-t_{1/2}}{\tau }\right) ^{2}}\right]   \tag{Eq.
(11)}
\end{equation}
and
\begin{equation}
Q(t)=\frac{1}{\pi }Q_{\infty }\cot ^{-1}\left( \frac{t-t_{1/2}}{\tau }%
\right) \text{ .}  \tag{Eq. (12)}
\end{equation}
One can show that this is a situation in which $P(t)$ is proportional to $%
\left[ Q_{\infty }-Q(t)\right] ^{2}$ for short times and to $Q^{2}(t)$ for
long times, but is \textbf{not} proportional to $Q^{2}(t)\left[ Q_{\infty
}-Q(t)\right] ^{2}$ for intermediate times.

\bigskip 

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