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\centerline{MONT 104N -- Modeling the Environment}
\centerline{Chapter 7 Project}
\centerline{November 18 and 20, 2019}
\vskip 10pt
\noindent 
\subsection*{Background Information}
This project is inspired by a similar project developed for the book
{\it Quantitative Reasoning and the Environment} by 
Greg Langkamp and Joseph Hull (Prentice Hall, 2007), but 
uses different data.

Environmental scientists often try to estimate populations of plant or animal species and understand
to what extent they can be used as resources by humans without being depleted.  For instance, 
populations of wild fish and other marine creatures around the world have been a major source of food for 
humans for many years.  Yet there is evidence that many of them have been overfished and
there is fear some of them may be headed toward extinction.  
For example, following 500 years of fishing, by the summer of 1992, the biomass of northern cod 
observed in Atlantic waters off the coast of Newfoundland had fallen to an estimated $1\%$ of its
previous levels.  As a result, the federal government of Canada declared a moratorium on cod
fishing, hoping to give the cod populations time to recover.  The economic and social impacts on the human
population of Newfoundland were severe, since many of the people in the area derived 
their entire livelihood from cod fishing and others depended on cod fishermen as customers for
their businesses.  An estimated $35,000$ people lost their jobs 
and the whole society of Newfoundland has not really recovered to this day.  As of around 2010,
there were some encouraging signs that the cod fishery might be recovering, but the effects of other 
factors such as changes in ocean temperature and loss of populations of the food species that cod eat 
have kept the ultimate fate of the north Atlantic cod fishery uncertain.  Similar decreases have been
observed in New England cod populations more recently and the U.S. government has
instituted more and more stringent fishing limits to try to avert collapses here as well.  The 
fishing industries of Maine and Massachusetts are under similar pressures.  

In this project, you will study various models of a fishery including effects from fishing by humans.
Let $P(n)$ represent the total mass of mature Pacific halibut in units of $10^6$ kg.  We will model
the wild halibut biomass \emph{without any fishing} by the following logistic difference equation:
\begin{equation}
\label{eq:Halibut}
P(n+1) = (1.71)\cdot P(n) - (.00875)\cdot (P(n))^2.
\end{equation}
Here $r = .71$ and $r/M = .00875$.
\subsection*{Questions}
\begin{enumerate}
\item[(A)] {\it ``Greedy'' harvesting.}  Suppose the halibut stock started out at $95\%$ percent of the carrying 
\index{harvesting!greedy}
capacity according to the model \eqref{eq:Halibut}.   But in one massive fishing effort, the halibut
biomass is reduced all the way down to $1\times 10^6$ kg (say all within one year).  If no 
further fishing is allowed until stocks recover to $95\%$ of the carrying capacity, how 
long will that take, according to the model?  Estimate by computing a solution
of the difference equation \eqref{eq:Halibut}.  What would be the \emph{average fish amount
taken per year} if this process of massive fishing followed by fallow time to allow recovery to $95\%$ of the carrying capacity
was done repeatedly over a long period?
\item[(B)]  {\it Constant harvesting.}   One way to make use of a resource like
the halibut fishery that is less drastic than the ``greedy'' approach in (A) is to take some constant amount of
the fish biomass every year for human use. \vskip 5pt
\begin{enumerate}
\item[(1)]  Suppose that everything remains as in \eqref{eq:Halibut}
above, but some constant amount $h$ (in $10^6$ kg)
of halibut biomass is removed each year via fishing.\footnote{Think $h$ means a ``harvesting level,'' hence
the notation.}  What modified difference equation models
this situation?  (Think about the derivation of the logistic difference equation and take the fishing
amount $h$ into account.)
\item[(2)]  Investigate the solutions of your constant harvesting difference equation from part
(1) if the fishing term is each of these values:  $h = 5, 10, 14,  20$, one at a time.  
Choose enough different $P(0)$ values for each so that you think you see the whole picture and then
describe what is happening in words.  In particular, for each $h$
how many different equilibrium solutions are there?  Where are they located?  How
do they change as $h$ increases?  Are they
stable or unstable?  
\item[(3)] The situation here is often described by saying that a harvesting level $h > 0$ introduces
a \emph{threshold value} for the population.  If $P(0)$ is greater than the threshold, the population
increases to a positive equilibrium, but if $P(0)$ is less than the threshold, the population crashes.
What are the threshold values for the $h$ that you considered in part (2)?
\index{threshold value}
\item[(4)]  By rewriting your difference equation from part (1) in the form 
$$P(n+1) - P(n) = \cdots ,$$
what is the \emph{maximum value} of $h$ for which the equation still has a stable equilibrium?
(Hint: this question can be answered by means of algebra alone if you think about it the right way.)
\item[(5)]  What should it mean to say that a fishing level $h$ is \emph{sustainable}?

What is the maximum sustainable constant fishing level?  Does the answer depend on what the initial value $P(0)$ at the
start of the fishing intervention is?
\item[(5)] What would be the \emph{average fish amount
taken per year} if constant harvesting at the maximum sustainable level is done repeatedly over a long period?
\end{enumerate}
\item[(C)] {\it Proportional harvesting.}   Instead of taking a constant amount of fish, we
could also take a constant proportion of whatever fish biomass is present.  \vskip 5pt
\begin{enumerate}
\item[(1)]  Next, suppose that everything remains as in \eqref{eq:Halibut}
above, but instead of a constant amount, suppose that a constant \emph{proportion} $p$
of the halibut biomass (whatever it is) is removed each year via fishing.\footnote{Think $0 < p < 1$ with $p = 1$ meaning
all of the fish are removed.}  
What modified difference equation models
this situation?  (Think about the derivation of the logistic difference equation
 and take the proportion removed by fishing into account.)
\item[(2)]  Investigate the solutions of your constant harvesting differential equation from part
(1) if the fishing term is each of these values:  $p = 0.1, 0.3, 0.5, 0.8$, one at a time.  
Choose enough different $P(0)$ values so that you think you see the whole picture and then
describe what is happening in words.  In particular, for each value of $p$
how many different equilibrium solutions are there?  Where are they located?  How
do they change as $h$ increases?  Are they stable or unstable?  
\item[(3)]  By rewriting your difference equation from part (1) in the form 
$$P(n+1) - P(n) = \cdots ,$$
what is the $p$ for which the halibut population starts to ``crash'' for all $P(0)$?
(This question can be answered by means of algebra alone if you think about it the right way. And the answer
should make biological sense too!)
\item[(4)]  What should it mean to say that a fishing proportion $p$ is \emph{sustainable}?
What values of $p$ are sustainable?  Does the answer depend on what the initial value $P(0)$ at the
start of the fishing intervention is?
\item[(5)] What would be the \emph{average fish amount
taken per year} if proportional harvesting at the level $p = .3$ is done repeatedly over a long period?
\end{enumerate}
\end{enumerate}
\subsection*{Assignment} Compare the strategies in parts (A), (B), (C) from the point of view of their effect on the
halibut fishery and the average amounts taken per year.   If you were going to recommend one, which
would it be?  Explain how you are making your determination.
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