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Unformatted text preview: Fish
Stock Natural Mortality (m) Nt = N 0e− mt = N 0e− M
One of the most difficult parameters to measure
Important parameter, but may be minor influence
Mortality relationship between number and time
Fishing Mortality (f)
Leslie Model U t = U 0 + qΣCt
Delury Model U t = U 0e − qΣft
log(U t ) = log(U 0 ) + qΣft Total mortality =  Nt = N 0e − qf − mt = N 0e − Zt POPULATION STATISTICS  many standard relationships exist which are used extensively in
working with biological populations. We will generally follow the notation used by
Ricker, though there will be exceptions.
First imagine a population. The first step in working with a biological population is
defining a statistical population. There are often assumptions made about the
population, depending on the model used.
Typical assumptions made are;
1) All fish are equally susceptible to the fishing gear.
2) The population is closed (no immigration or emigration) or the number of immigrants
equals the number of emigrants.
3) Stable age structure  the pattern observed for various cohorts at one time is the same as
for one cohort over time.
BASIC POPULATION RELATIONSHIPS (numbers of individuals)
Nt = the number of individuals in the population at time t
if the fish are randomly distributed and the nets are set at random the catch in the net will
average some fraction of the total population (Nt ). q = fraction of total population caught by 1 application of effort.
called the catchability coefficient
this is often a weak link in calculations because the assumptions are often not met. (ie.
fishing is not random) and the value may not be a constant (ie. seasonal variation, etc.) CPUEt = Ut = the average catch in 1 unit of effort at time t
CPUE> = Ut = q*Nt = Ct
ft where;
Ct = the total catch at time t
ft = the total effort at time t EXST025  Biological Population Statistics INSTANTANEOUS RATES  there are many applications and advantages in using
instantaneous rates. It describes losses and gains which are proportional instead of
linear.
MORTALITY is usually described as an instantaneous rate.
eg. suppose 50% of a population dies off in a time interval of 1 (the time starts at t" and
ends at t# )
define
dt = ˜ t = t#  t" N! as the population size at time 0, for example 1000 individuals
then N! = 1000 Nt as the population size after “t" intervals.
r is the rate of loss, for example 0.50 or 50% (negative since it is a loss)
r = 0.50 this results in a loss of 500 in the first time interval.
as a linear equation;
N" = N!  500*t = 1000  500*1 = 500
N# = N!  500*t = 1000  500*2 = 0
so as a linear equation there is a 50% loss in the first time period, but a 100% loss in the
second. if there is a constant 50% reduction each time we expect the losses to go 1000, 500, 250, 125, 62, 31, 16, 8, 4, 2, 1, 0. to follow this pattern we need an exponential equation.
Nt = N! ert
the r in this equation is NOT the same as the previous r, the r in this equation is an
instantaneous rate. Page 2 EXST025  Biological Population Statistics To calculate instantaneous rates it we can use
Nt
N = ert we know that N" /N! = 0.50 for our example, and if t = 1 (for only 1 time interval), then
ert = er = 0.50,
so
N
r = ln(0.50) = ln( N!t ) taking logs we get
r = ln(0.50) = 0.693147
The value or r = 0.693147 can be used in the equation Nt = N! ert to solve for Nt at any time t
eg. N! = N! e0.693147*0 = 1000*1 = 1000
N" = N! e0.693147*1 = 1000*0.5 = 500
N# = N! e0.693147*2 = 1000*0.25 = 250 then at time (t) Nt
0
1000
.
1
500
.
2
250
.
3
125
.
4
62 5
.
5
31 25
.
6
15 625
.
7
7 8125
.
8
3 90625
.
9
1 953125
.
10
0 976562
.
11
0 488281
.
12
0 244141
. Page 3 EXST025  Biological Population Statistics NOTE that the whole expression “ert " acts like a proportion such that
Nt
N! = ert the proportion “ert " is the surviving portion and is called S
S = ert
A = 1  ert = the nonsurviving portion
NOTE: The two times do not have to be time 0 and some other time, they can be any two
times, for example t# and t" = ˜ t. Then
Nt# = Nt" er(t# t" ) = Nt" er(˜t) Previously we defined CPUE (catch per unit effort) as Ut = qNt which is fine for 1 application of a unit of effort, however, q is also a proportion of the catch, so as soon as the application of effort starts to
reduce the population we must consider q as an instantaneous rate.
q is not a function of time, it is a function of effort (f).
Then,
Nt = N! eqf> Page 4 EXST025  Biological Population Statistics Page 5 EXPONENTIAL GROWTH (AND DECAY) MODEL  NATURAL MORTALITY
Nt = N! ert
where;
t is time (in units which correspond to units of r)
N! is number at time zero
Nt is number at time t
r is the rate of growth in units of N
t
(eg. indiv.
day = 1
day , otherwise it could be in pounds
kg
week , month , etc.) the direction of the curve is determined by
+ r growth
 r mortality or decay (usually indicated as “m" instead of “r")
in theory
r = births deaths = b  d
where;
b is the rate of birth (per individual)
d is the rate of death
so the change in the number of individuals (N) is given by ˜ N = bN  dN the “intrinsic" rate of growth is the maximum possible rate of growth ( in the absence of
deaths) rmax = b , where d = 0 (or an absolute minimum) Instantaneous rates can be easily converted to different units of time suppose growth = 10% per year
annual rate
b = 0.1
monthly rate b = 0.008333
daily rate
b = 0.000274
all give the same increase t=1
t = 12
t = 365 EXST025  Biological Population Statistics Page 6 The exponential DECAY model is asymptotic on 0.0.
Yi = "! e"" Xi
In order to get the model to be asymptotic on something other than 0.0, we must add a
constant.
Yi = "# "! e"" Xi
This model is now NONLINEAR. It will be discussed later. CALCULATION OF HALFLIFE OR DOUBLING TIMES
Nt = N! ert
then, to calculate the time (t " ) at which N! is reduced by
#
N!
2 1
2 = N! e = e "
# rt "
# rt "
# log( " ) = rt "
#
#
and t " =
# log( " )
#
r = 0.6931472
r = 0.6931472
r doubling times are given by 2N! = N! ert‡# and t‡# = 0.6931472
r so halving and doubling take the same time, and the sign on the instantaneous rate
determines which occurs EXST025  Biological Population Statistics Page 7 DERIVATION of the Exponential Growth model
r is a proportionality constant  this is important
Y = bX Y proportional to X assume that the change in N in one time unit is proportional to the number at the start of
that time unit
eg.
˜N
˜t = + rN where,
˜ N is the change in N
˜ t is 1 time unit
r is a proportionality constant
N is the number at the start of the time period
then;
˜t
˜N = 1
rN integrate over time (one unit, such as one year)
t = 1
r ' ˜N
N = 1
r ln(N) + c = t ln(N) + ln(c') = rt
N c' = ert
N = c'' ert
where, solving for c at t = 0
N! = c ert! = c er 0 = c = N!
c'' = N when t = 0, or N! (for biological application)
eg.
Nt = c ert = N! ert
for any 1 interval of time we can also use Nt = Nt1 er EXST025  Biological Population Statistics Page 8 as a STATISTICAL MODEL let
Nt = N! ert %>
log(Nt ) = log(N! ) + rt + log(%> )
a semilog model in simple linear regression form,
Y = b! + b" X
where
Y = ln(Nt )
X=t
"! = intercept = log(N! )
"" = r = slope
Note: assume that log(%> ) is distributed NID (0,5 # )
all assumptions apply to transformed model
original model has multiplicative error term OTHERWISE: NONLINEAR where Nt = N! ert + %>
Shape of the Curves;
when r p exponential growth curve r p exponential decay or mortality curve
1000 Mortality over time 900
800
700 N t 600
Natural mortality 500
400 Fishing mortality
300
200
Total mortality
100
0
0 5 10 15 20 25 30 Time as a statistical model  “X" (the independent variable) does not have to be “time". We will see
other uses. EXST025  Biological Population Statistics Page 9 TOTAL MORTALITY
Assume that natural mortality is a function of time only and is given by
Nt = N! emt
Assume that the fishing mortality is a function of only effort and is given by
Nt = N! eqft
As instantaneous rates these can be easily combined into a single model
Nt = N! eqft +mt
where both “q" and “m" would be negative numbers (for mortality)
Very often the time unit will be 1 year so we often see this simplified as
Nt = N! eqf+m
The total fishing mortality (qf) is also often represented as F and the total natural mortality
as M.
where;
F = qf
M = mt
then Nt = N! eF+M or Nt = N! e(F+M)t This can be expressed as a simplified total mortality
where;
Z = F+M
or
Z> = qf> + m> or Zt = qf> + mt then
Nt = N! eZ or Nt = N ! e Z > EXST7025 : Biological Population Statistics
Mortality James Geaghan
Page 1 Cohort of Flies (S. nudiseta). From Poole, RW. 1974. An Introduction to Quantitative Ecology. McGrawHill, NY.
Days
Number
log(number)
Predicted Y
Residuals
Standardized
exp(Pred)
0
100
4.61
4.71
0.1
0.24
110.65
2.5
75
4.32
4.64
0.32
0.74
103.11
7.5
64
4.16
4.49
0.34
0.79
89.53
12.5
64
4.16
4.35
0.19
0.46
77.74
17.5
64
4.16
4.21
0.05
0.12
67.5
22.5
64
4.16
4.07
0.09
0.21
58.61
27.5
48
3.87
3.93
0.06
0.14
50.89
32.5
46
3.83
3.79
0.04
0.09
44.19
37.5
44
3.78
3.65
0.14
0.32
38.37
42.5
42
3.74
3.51
0.23
0.54
33.31
47.5
34
3.53
3.36
0.16
0.38
28.93
52.5
34
3.53
3.22
0.3
0.71
25.12
57.5
32
3.47
3.08
0.38
0.9
21.81
62.5
28
3.33
2.94
0.39
0.92
18.94
67.5
20
3
2.8
0.2
0.46
16.44
72.5
20
3
2.66
0.34
0.79
14.28
77.5
18
2.89
2.52
0.37
0.87
12.4
82.5
14
2.64
2.38
0.26
0.62
10.76
87.5
6
1.79
2.23
0.44
1.04
9.35
92.5
2
0.69
2.09
1.4
3.28
8.11 Regression Statistics (from Excel) Multiple R 0.89 R Square 0.8 Adjusted R2 0.79 Standard Error 0.43 Observations 20 Analysis of Variance df Sum of Squares Mean Square F Significance F Regression 1 13.08 13.08 71.64 1.08803E07 Residual 18 3.29 0.18 Total 19 16.36 Coefficients Standard Error Intercept anitlog Intercept x1 t Statistic Pvalue Lower 95% Upper 95% 110.65 75.94 160.77 4.71 0.18 26.39 1.9584E16 4.33 5.08 0.03 0 8.46 7.18267E08 0.04 0.02 EXST7025 : Biological Population Statistics
Mortality James Geaghan
Page 2 120 Mortality of cohort of flies Number 100
80
60
40 92.5 87.5 82.5 77.5 72.5 67.5 62.5 57.5 52.5 47.5 42.5 37.5 32.5 27.5 22.5 17.5 12.5 7.5 0 0 2.5 20 Days 0.40 Residual 0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60 0 10 20 30 40 50 60 70 80 90 100 Days ...
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.
 Spring '08
 Geaghan,J

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