**Project
#3 Fishing in George's Bank**

**Introduction**: This model is concerned with the New England fishing
industry, especially George's Bank, which has fallen on desperate times
of late. The model covers most of the 20th century as there is a multitude
of data available. It is based upon the age specific Leslie model.

Here, we assume we are looking at the haddock population of the north Atlantic, particularly George's Bank. This is a critical area, for in the early and mid 20th century, New England fishing produced consistently over 80% of the United States fish. Also,in recent times, the diet of Americans has slowly moved away from red meat, for health reasons, and has included more fish.

**Background** matrix algebra (chapter 1 of Kolman or Leon) and the
basic Leslie population model (pp. 305-6 Kolman)

In this model, we assume that haddock live to 10 years of age, and may
have offspring as early as 3 years old. The Leslie model for this situation,
with a time period of **1 **year, is of the form

**(1) x**(k+1)**
= **L** x**(k)** x**(0**)
**given
k=0,1,2,3...

where **L** is a 10 x 10 matrix which is initially assumed to have
values

Equation (1) may be solved iteratively to give

**(2)
x**(k)** = **L^{k}** x(**0**) **k=1,2,3,...

**PART ONE**: If the haddock population in** 1900** is (**55,0,0,0,0,0,0,0,0,0) ^{t}**
(units are millions of pounds), what will the population be like in 1910?
1920? 1930? 1950? 1995?2000? Based upon what you have seen, do you believe
the population to be

If you believe it to be unstable, pick two consecutive populations where
the ratios of all components (cohorts) are approximately *constant*.
What is that constant? What does it tell you about the long term behavior
of the system?

**PART TWO:**

Suppose pollution has the effect of **lowering** each birth rate
by 10% of the value given above and each survival coefficient by 15% beginning
in 1950. What effect does this have on the population in 1990 as compared
to having *no* pollution effects?

**PART THREE**: **Harvesting**.

Beginning in 1925, assume that fish 3 years old and older are caught
at a rate such that each year, 35% of those groups are taken. Fish under
3 years old may *not* be taken (the distinction is based upon size).

**a**. What is the matrix form of the model now? ( modification of
equation (1), above)

Did you assume the fish were harvested *before* or *after*
the annual birth process occured?

Please generalize to show a model reflecting **any** harvesting rate
which you may denote by** **h. Thus above, h = .20.

**b**. What does the population **now** look like for 1930, 1950,
1995, 2000? Is this a good strategy?

**c**. Can you come up with an alternate strategy (i.e. a different
value for** **h) which will result in a **stable** long term population?
(you will want to experiment here; if you can set up your worksheet with
h as a single parameter which can then be quickly changed, all other equations
can remain the same and the effect of changes quickly seen)

**PART FOUR**: In an attempt to become more efficient, in the 1940s,
finer nets were introduced in hopes of catching more fish with the same
effort. However in doing so, many millions of young fish (1-3 years old)
were caught, killed and disposed of overboard. What does the mathematical
model say about the wisdom of such a move?

To explore this, revise your model from **Part Three a** to include
the same harvesting rate for fish in the 1-2 and 2-3 age groups (instead
of 0 as before)

Population dynamics aside for the moment, authorities argued that finer nets cost more, were more work to maintain, and took more horsepower to drag. Despite this, they were often used over a 30 year period.

To make the comparison valid, generate populations for the same years as Part Three.

**PART FIVE: **Fisheries consultants have suggested using **two**
nets, each of area half that of the nets normally used, but of different
mesh fineness. This would mean that one net would have mesh fine enough
to catch all fish from age 3 to 5 while the second net would be coarser
and only catch fish from age 5 and up.

a. how would the __matrix model__ be revised to include this change?

b. could such a scheme be used to generate a better **overall** harvesting
strategy? (meaning a better yield **and** a stable long term population?)

**References**:

*The New England Fishing Industry* by Donald White. The Harvard
University Press 1954

*An Introduction to Population Ecology* by G. Evelyn Hutchinson.
The Yale University Press 1979.

*Mathematical Ecology* by E.C.Pielou. John Wiley and Sons 1978.