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\textbf{College of the Holy Cross, Fall Semester 2019} \\
\textbf{MONT 104N -- Modeling the Environment} \\
\textbf{Solutions for the Final Exam, December 20}

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\vskip 10pt
\noindent
I.  The table below shows estimates collected in 
the EDGAR database created by the European Commission and
the Netherlands Environmental Assessment Agency of the amounts of \emph{carbon
dioxide} emissions from burning of fossil fuels in 2015 and 2016 by country.  
The units are megatonnes $= 10^6$ metric tonnes.  The populations are in units of 
millions $ = 10^6$ of people:
$$\begin{array}{cccc}
 {\bf Country} & {\bf CO2\ 2015} & {\bf CO2\ 2016} & {\bf 2015\ Population}\\
           {\rm China} & 10.642 & 10.433 & 1367\\
           {\rm United\ States} & 5.712 & 5.012 & 321 \\
           {\rm India} & 2.455 & 2.533 & 1252 \\
           {\rm Russia} & 1.761 & 1.662 & 142 \\
           {\rm Japan} & 1.253 & 1.240 & 127 \\
           {\rm Germany} & .778 & .776 & 81\\
           {\rm Whole\ world} & 36.061 & 35.753 & 7256
\end{array}$$
\begin{enumerate}
\item[A.]  1 metric tonne is $1000$ kg and $1\ {\rm kg} \doteq 2.205$ lb.  What was the 
equivalent amount of carbon dioxide emissions for India in 2016, in units of pounds.  
\item[] {\it Answer:}  
$$(2.533 \times 10^6) \text{ tonnes } \cdot (2205) \text{ lb/tonne } \doteq 5.585 \times 10^9\text{ lb }$$
\item[B.]  Suppose that the U.S. carbon dioxide emissions were decreasing \emph{exponentially}.
What would be the exponential model fitting the two data points you have exactly?  Take
$t = 0$ to correspond to the year 2015.  What would your model predict for U.S. 
emissions in the year 2020? 
\item[] {\it Answer:}  We are looking for a function of the form $USCO2(t) = c a^t$
for which $USCO2(0) = 5.712$ (from the 2015 figure) and $USCO2(1) = 5.012$ (from the 2016 figure).
The first equation says $c = 5.712$ and then $5.012 = 5.172\cdot a^1$ implies $a \doteq .8775$.  
Hence, the model has the form $USCO2(t) \doteq 5.712 (.8775)^t$ (units are megatonnes $CO_2$).  
The prediction for 2020 is $USCO2(5) \doteq 2.972$ megatonnes $CO_2$.  
\item[C.]  Now, let's look at this data from another perspective.  Which countries
here had \emph{per capita} $CO_2$ emissions greater than the world per capita emissions in 2015, and which 
are less than the world average?
\item[] {\it Answer:}  The world per capita figure is 
$$\frac{(36.61 \times 10^6)\text{ tonnes }}{(7256 \times 10^6) \text{ people }} \doteq .005 \text{ tonnes per person } 
\doteq 5 \text{ kg per person }.$$
The country by country averages are as follows (note that I'm not putting in the $\times 10^6$ in the numerators
and denominators because they cancel):
$$\begin{array}{cc}
           {\bf Country} & \text{\bf 2015 CO2 per capita}\\
           {\rm China} & 10.642/1367 \doteq .0078 \text{ tonnes per person } \\
           {\rm United\ States} & 5.712/321 \doteq .018 \text{ tonnes per person } \\
           {\rm India} & 2.455/1252 \doteq .002 \text{ tonnes per person }\\
           {\rm Russia} & 1.761/142 \doteq .012 \text{ tonnes per person }\\
           {\rm Japan} & 1.253/127 \doteq .0098 \text{ tonnes per person }\\
           {\rm Germany} & .778/81 \doteq .0096 \text{ tonnes per person }\\
           {\rm Whole\ world} & 36.061/7256 \doteq .005 \text{ tonnes per person } 
\end{array}$$  
So all of these countries except India have per capita emissions larger than the world 
per capita figure.  Of course, the explanation is that all of these
except India are largely industrialized societies, so $CO_2$ emissions are higher than the world average.
\end{enumerate}
\vskip 10pt
\noindent
II.  The following graph, constructed by Makiko Sato and James Hansen at the Columbia
University Climate Science, Awareness and Solutions project, 
shows the change in \emph{global average sea level} over the period 1900 to 2017 (i.e. the present).  
It's also a very nice example of how the basic modeling techniques we have discussed
can be adapted to deal with dynamic, changing situations.
\begin{center}
\includegraphics[width=4.5in,height=3in]{GlobalMeanSeaLevel.jpg}
\end{center}
The vertical axis is the \emph{sea level anomaly}---the \emph{difference} between
the observed levels in later years and the average level in 1900.  
\begin{enumerate}
\item[A.] The information included in the graph shows that Sato and Hansen have
constructed three separate linear models over three different time periods: one covering the 
period 1900 - 1930, the second covering the period 1930 - 1992 and the last one 
covering the period 1993 to 2017. The estimated slopes are given in units of mm/yr.
 Find the equations of the three linear models
(the equations of the three black lines in the graph) for average mean sea level $L(t)$, using
$t = $ actual year in each case.  Use the values $0$ for $L(1900)$, $18$ for $L(1930)$, and 
$104.8$ for $L(1993)$.  
\item[] {\it Answer:}  The equations are (output in units of mm):
$$L(t) = \begin{cases}
.6 (t - 1900) = .6 t - 1140, & 1900 \le t \le 1930 \\
1.4(t - 1930) + 18 = 1.4t - 2684, & 1930 \le t \le 1992 \\
3.1(t - 1993) + 104.8 = 3.1t - 6073.5, & 1992 \le t \le 2017
\end{cases}$$
(Note: the problem said to use $t = $ the actual year in each case, so you need the point-slope
form of the equation of the line with $(t - \text{starting year})$ each time.  Also, the negative values for the 
intercepts just mean that if the lines were
extended to the left back to the year 0, they would be intersecting the vertical axis way below 
0.  However, that is essentially a silly thing to do here, since it's extrapolating
so far beyond the range of times the data came from.)
\item[B.] Why do you suppose Sato and Hansen chose to fit three separate linear models
rather than a single exponential or power law model?
\item[] {\it Answer:}  One reasonable explanation is that the rate of increase
seems to be quite nearly constant over each of the three time intervals separately.  
So three linear models are a better description than a single exponential or power law.
\item[C.] What is the ``take-away'' message from this data and this graphic?  How would you
summarize the conclusion(s) we should draw from it? 
\item[] {\it Answer:}  One reasonable take-away message would be that sea-levels have been
increasing at a faster and faster rate over the period from 1900 to 2017, and that
there have been essentially three different phases or stages in that increase.  
\end{enumerate}

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\noindent
III.  Suppose that a endangered population of salamanders in a protected wetland is
\emph{decreasing} at a net rate of $15\%$ per year from births and deaths, but
human-reared salamanders are being reintroduced into the habitat at a rate
of $40$ individuals per year.
\vskip 10pt
\begin{enumerate}
\item[A.]  Write a difference equation that models the change in the
salamander population from each year to the next.
\item[] {\it Answer:}  Letting $P(n)$ be the number of salamanders in year $n$,
the difference equation is 
$$P(n+1) = .85 P(n) + 40$$
\item[B.]  Using an initial value $P(0) = 100$, determine the populations
in years $1,2,3,4,5$ according to the model you stated in part A and
record the values in the following table (round any decimal values to the nearest
whole number)
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
P(n)  & 100 & 125 & 146 & 164 & 180 & 193  \\
\hline
\end{array}
$$
\item[C.] What happens to the population in the long run?  Does
it tend to a definite value as $n$ increases?  If so, what is it?  If not, why not?
\item[] {\it Answer:} This difference equation from part A has an equilibrium solution of
$$P(n) = \frac{40}{.15} \doteq 267,$$
since $P(n+1) - P(n) = -.15 P(n) + 40 = 0$ when $P(n) = \frac{40}{.15}$.  
Since $.85 < 1$, this is a stable equilibrium, and the population is tending
to that value as $n$ increases.  (This can be seen by writing down the general 
solution using the formulas for affine first order equations.)
\end{enumerate}

\vskip 10pt
\noindent
IV.  Answer {\it any three} of the following briefly.  If you submit answers for
all four, only the best three will be counted.
\begin{enumerate}
\item[A.]  If you are fitting an \emph{exponential}
model to a data set $(x_i,y_i)$ ``by hand'' (i.e. not using the shortcuts available
in a Google spreadsheet) 
you would start by transforming the data to a new form $(X_i,Y_i)$.  What is that form
in terms of the $x_i$ and $y_i$?.  If the best fit
regression line for the \emph{transformed data} is $Y = m X + b$, what is the corresponding
exponential model?  (You may use the exponential function with base 10 as we discussed in class, 
or any other exponential function.)
\item[] {\it Answer:}  You would start out by computing the log-transformed data points 
$(X_i,Y_i) = (x_i,\log_{10}(y_i))$.  If the regression line for the log-transformed
data is $Y = mX + b$, then 
$\log_{10}(y) = mx + b$, so 
$$y = 10^{mx + b} = (10^b)\cdot (10^m)^x$$
That is, in the exponential function $y = c\cdot a^x$, $a = 10^m$ and $c = 10^b$.  
\item[B.] Give the difference equation that would model a population that (in the absence of interactions
with humans) would undergo \emph{logistic growth}
with a net growth rate $15\%$ per year when the population is much less than 
the carrying capacity $M = 1000$ of the habitat.   Assume in addition that humans
are doing constant harvesting of $h = 50$ individuals per year.  
\item[] {\it Answer:}  As in the Chapter 7 project, the difference equation is
$$P(n+1) = (1.15)\cdot P(n) - \left(\frac{.15}{1000}\right)\cdot (P(n))^2 - 50.$$
(The first two terms on the right are from the standard form of the logistic equation,
the last is the harvesting term.) 
\item[C.] (5) How do you determine \emph{equilibrium value(s)} of a first order difference equation 
$Q(n+1) = A Q(n) + B$ (where $A,B$ are constants)?
\item[] {\it Answer:}  You would subtract $Q(n)$ from both sides:
$Q(n+1) - Q(n) = (A - 1)Q(n) + B$, then set the right side to zero and solve for 
$Q(n)$:  $Q(n) = \frac{B}{1 - A}$ (or equivalently, $\frac{-B}{A - 1}$.  This
is the equilibrium value.  (Note: this is exactly what we did in III C above.)
\item[D.] What feature of the solutions of the Lotka-Volterra equations is considered to be a
confirmation that this model is capturing an important aspect of real-world predator-prey
interactions?
\item[] {\it Answer:}  It's the fact that \emph{oscillations} in the predator and prey populations
are observed, as in such interactions in the natural world.
\end{enumerate}
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\noindent
{\it Essay} 
\vskip 10pt
In general terms, what is a mathematical model?  
Describe in general terms what they are, 
how they are constructed, and how they are used.  Give examples of three
different types of mathematical models we have studied this semester.  
Even if mathematical models don't capture {\it every feature} of a real world
situation, why is it still important to develop them and understand the
information we get from them?  For instance, what conclusions about use
of natural resources did we derive by looking at logistic models with various 
types of harvesting in the Chapter Project from Chapter 7?  As another example, 
how are mathematical models important in understanding our choices of which 
energy sources to use?  Why is it important to understand 
how radioactive substances decay?  What type(s) of model(s) that we discussed would 
apply to describe that process?  What are some of the issues involved with
using radioactive decay to generate electricity--that is, why is this not
a ``no-brainer'' as a solution to the problem of $CO_2$ buildup in the atmosphere
from fossil fuel burning? 
\vskip 10pt
\noindent
{\it Model Response:}  A mathematical model of something is a function, graph, an
equation or system of equations, etc. constructed within the ``mathematical 
world'' in order to study, or even make predictions about, the behavior of a 
real-world system.  Constructing mathematical models relies on a process of
abstraction, by which some aspects of the real-world system under consideration
are not included.  The predictions produced from these simplified versions
of reality through use of mathematical tools are
then compared with real-world data and further iterations of model construction
and testing often ensue.  

We have studied various types of models using linear, exponential, 
and power functions.  We also studied single difference equations (affine and logistic
equations especially) and
systems of coupled difference equations (such as the Lotka-Volterra
predator-prey equations) as models.  In many cases, a mathematical
model may be the {\it only} way to study a real world system where we cannot do
controlled experiments.  This is because it is often impossible or impractical 
to manipulate a natural environment for the purposes of seeing how it
behaves under certain circumstances.  Even when some
things are ``left out'' of a model, if enough of the properties of the
real-world system are captured, we can still get some insight or information
from a model.  Careful comparison with the real-world system might be 
needed to validate those insights or that information, though.   

For instance, in looking at logistic equations with constant harvesting in 
the Chapter 7 project, we saw that it is possible to introduce ``thresholds'' 
at unstable equilbrium values in systems.    This means that there can be
harvesting levels that produce population crashes if the initial population
is too small, while initial populations that are large enough yield growth
toward a stable equilibrium as in the usual logistic case (with small enough
$r$ parameter values).  This behavior was not present
before the human intervention through the harvesting and knowing
the population dynamics can work that way is an important part of sustainable 
management of natural resources.  

Various models we studied are important in understanding our choices of 
energy sources.  In the Chapter 3 and 4 projects, we studied how major
components of our economy are based on burning of fossil fuels (petroleum,
coal, and natural gas).  But this is producing steadily rising atmospheric
$CO_2$ levels that are contributing to climate change, sea level rise, and
other undesirable effects.  Nuclear power is sometimes considered to be
a solution to some of these issues because generating electricity by
using radioactive decay to produce steam for turbines does not involve
any burning of fossil fuels (at least not directly).  However, this 
source of energy has its own problems, which are revealed by considering
the properties of exponential decay processes.  There's always a possibility
of catastrophic damage to a nuclear reactor and release of radioactivity
into the environment, which would case damage to human and other populations.
Even if there is no major disaster of that sort, the waste products of
nuclear power remain radioactive for extended periods and hence are also 
dangerous for humans and other life forms.  Safe long-term disposal methods for 
those wastes have not been developed as yet and this is another obstacle
to using nuclear power for electricity generation on large scales.   

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