# Lab 4 Estimating Mortality_ Part II by tyndale

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Lab 1: Estimating Abundance
MSCI 458/558 – Fisheries Science

Background
Sources for this laboratory include:
 King, M. 1995. Fisheries Biology, Assessment, and Management. Blackwell Science Ltd., Oxford.
 Jennings, S., M.J. Kaiser, and J.D. Reynolds. 2001. Marine Fisheries Ecology. Blackwell Science Ltd.,
Oxford.

If you want to manage a fishery, one of the first things you’ll want to know is some estimate of population size.
This week we will explore a number of techniques to estimate stock abundance. These include partial counts, the
swept area method, mark-recapture methods, depletion methods, and egg production methods. But first of all, since
we are estimating stock abundance, what is a stock?

1: What is a stock and do you expect the biological definition of a stock and the working, or management,
definition of a stock to be the same? Explain.

Partial Counts Method

In the Partial Counts Method, we count the number of individuals in small parts (sampling units) of the whole
population. If we have an estimate for what proportion of the total population each sampling unit represents, we can
calculate the total population. Sampling units are randomly selected. This assumes that the individuals in the stock
are randomly distributed at the scale you are sampling. Alternatively, if a distribution bias is known, the sampling

Figure 1 represents a 15,600 m2 area around a sandbank to be surveyed for sea cucumber abundance. Each square
grid is 100 m2, and represents the area that can be checked by divers in one survey. The depth contours are shown.

2. Design a sampling scheme to estimate sea cucumber abundance based on 10 surveys, or sampling units. Describe
your design, including which specific grids you will sample and how they were chosen. Do this for 2 assumptions:
a. sea cucumbers are randomly distributed
b. sea cucumbers are distributed non-randomly with a known depth bias (preference for depths between 5 and
10 m -- you may assume that approximately 58 of the 156 grids on the map fall within this depth zone)
a.

b.
Figure 1. Sampling grid for sea cucumbers

A       B       C       D       E       F       G       H        I       J      K       L       M

1

2

3

4

5

6

7

8
5m

9

10                                                                                    10 m

11

12

question 4. You may use the blank space on the next page for both question 3 and 4, as needed.

3. Now look at the figure on the next page (from King, 1995) with the actual sea cucumber distributions. Using
your sampling design from the previous question that assumes a random distribution, calculate
a. the mean abundance per cell
b. the standard deviation of the mean abundance per cell
c. the standard error of the mean abundance per cell (SE = st. dev. / sq. root of n)
d. a 95% confidence interval for the mean abundance per cell (you may recall from statistics that the 95% C.I.
= mean abundance  1.96*SE, but if sample size is below 30, you should substitute the 1.96 with a
different coefficient from appendix C on pg 299 of your text)
e. an estimate and 95% C.I. for the total abundance
4. Now, use you sampling design from Question 2 that assumes a known biased distribution to compute the mean
and 95% C.I. for the total abundance. This isn't quite as straight forward as the last question, but I’ll bet you can
figure it out. Try this without using your book at all – in fact you may find it easier to just reason through it. Show
Swept Area Method

The swept area method is essentially a partial counts method that is modified for towed gear (i.e. trawls). You
calculate the abundance within a given sub-area and then multiply it up to account for the abundance in the total area
of interest. The following question is from King (1995). You should be able to do this without even looking at the
book, but if you get really stuck, go ahead and use it.

5. A research vessel completes a standard trawl at 41 stations on a fish stock distributed over an area of 360 km 2.
The trawl net, which has an effective fishing width of 20 m, was towed at a velocity of 8 km/h for 20 minutes at
each station. From this, you should be able to calculate the area sampled per trawl. The mean catch per trawl was
64 kg, and the SE was 18. Assuming that vulnerability of the fish to the trawl net is 50 percent (v=0.5), use the
swept area method to estimate the total stock size with a 95 % C.I. (assume the fish are randomly distributed). Show
Mark - Recapture Methods

This technique works well for closed populations or populations in defined areas (estuaries, reefs, etc.). A
subsample of the population is caught, tagged, and released. A similar capture effort is repeated later, and some of
the previously tagged fish are recaptured, along with many that are not tagged. The stock size can be estimated by
assuming that the ratio of tagged fish (T) in the stock (N) is equal to the ratio of recaptured tagged fish (R) in the
catch (C).

T/N = R/C
Therefore, the stock abundance can be calculated as:

N = TC/R
The SE for this estimate is

T 2 C (C  R)
SE 
R3
This technique is called the Petersen method, and is by far the simplest version of mark-recapture studies. Many
alternative models exist and can account for open vs. closed populations, single or repeated samplings and taggings,
etc. All of them are subject to restrictive assumptions, however. For the Petersen method, we assume it is a closed
population, that tag retention is perfect, that tagging does not increase mortality, that tagged fish mix randomly with
non-tagged fish, and that all fish have the same catchability.

6.   The figures below are modified from King (1995). All the fish represent the total stock, and the black fish
represent 31 tagged fish from a previous capture. The fish in the small box represent the fish caught in a
sampling catch (C). Fish are considered “caught” if you judge more than half of a fish to be inside the box.
a. Using Fig. 6-1 on the next page, calculate the stock abundance and 95% C.I. Input your
parameters and formulas into the Excel spreadsheet. You will need these formulas for the rest of
this question.
b. Download this lab exercise from the class webpage. On the computer, you can drag the small box
around and place it randomly on other sections of the figure to “catch” another sample. Take 3
more random samples from Fig. 6-1 and on your spreadsheet, calculate the abundance and 95%
C.I. for each. Are you comfortable about the accuracy and precision of this method? Explain. (in
space below)
c. Now, take 4 random samples from Fig. 6-2 and calculate the abundance and 95% C.I. for each.
Are you as happy with the accuracy/precision this time? What happened? (in space below).
Figure 6-1

Figure 6-2
Depletion Method

For a fished population, the short term rate of reduction in abundance is determined by catch rate and population
size. The rate of depletion, relative to fishing effort, can be used to estimate abundance. This technique works when
a small, isolated, closed population can be fished intensively in a short time interval, such that depletion is evident.

A closed population has some unknown abundance, N∞. We don’t know it, but it exists. After some period of time,
t, if you remove (fish out) some number of individuals, you will be left with the original abundance minus the
cumulative catch ((∑Ct)).

Nt = N∞ - ∑Ct

If you remove fish on a number of occasions, the population will continue to decline. If you can document the rate
of population decline relative to fishing effort, you should be able to back-calculate the original abundance that you
started with. These removals must be made over a short time period of time so that we can discount births, deaths,
and immigration/emigration.

Catch rate is assumed to be proportional to effort,
so

CPUEt = qNt

Where CPUE is catch per unit effort and q is
“catchability.”

Substituting this into the prior formula, we get

CPUEt = qN∞ - q∑Ct
(are you taking my word for that, or can you
manipulate the formulas yourself?)

This is the formula for the regression line for the graph of CPUE vs. ∑Ct. The slope of the line is q, the y-intercept
is qN∞, and the x-intercept is N∞ (see fig 3.12 from King, 1995).

7.   (from your text) A depletion experiment using traps on an isolated 24 km2 stock of crabs was run over 4 weeks.
The number of crabs caught and the number of traps used per week are shown in the following table. Fill in the
missing cells and estimate the catchability coefficient and the initial exploitable stock size. Please fill in the
blanks below, but you will want to do these calculations on the spreadsheet and you will need to use Excel to
graph the data (CPUE vs. ∑Ct) and fit a regression line with a formula for the line in order to identify q and N∞.

Week       # Traps       Catch        Accumulated catch prior to time interval (∑Ct)        CPUEt
1         140         2274                               0
2         183         2376
3         235         2734
4         204         1836

q=
N∞ =
Egg Production Model

This method estimates stock abundance by the number of eggs determined from egg surveys. This works well for
species which spawn predictably on defined spawning grounds. It is used for such important species as anchovies
and mackerel. It works best for synchronous spawners where all individuals collectively release eggs within a
discrete period. Intensive plankton net sampling can be done to estimate egg abundance. You also need good
samples to estimate the size of spawners, the proportion of the population spawning, and fecundity.

A basic version of these models is:

B = P / (F * proportion of stock spawning)

Where B = stock biomass, P = egg production (total eggs produced), and F = fecundity (eggs per unit weight of
female).

8.   What would be the spawning stock biomass (B) for the following species? You will need to calculate the mean
fecundity per female (F) in eggs/g, and the proportion of the population that spawns (lays eggs). You will need
to use all the information given below. Show your work, as always.

Annual egg production = 1.9 x 1012 eggs
Mean fecundity of female = 6.9 x 104 eggs
Mean weight of female = 324 g
% population that is female (by weight) = 46.6%
Proportion of females that spawned this year = 89%

When done, turn in this packet to me and email me a copy of your Excel spreadsheet.

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