This document discusses virtual population analysis, a method for estimating fish population sizes over time using known variables like catch data, fishing mortality rates, and natural mortality rates. It presents equations that allow calculation of the number of fish surviving after a given number of years based on the total mortality rate. For example, if the mortality rate is 20% per year, the number of fish surviving after 1 year from an initial population of 1000 would be 800. The document works through an example of using catch data from multiple years to estimate population sizes and fishing mortality rates for those years.

1. Virtual Population Analysis
Fish and Fisheries 2014
Dr Magnus Johnson

2. m
f
N
z
Which of these factors do we know?
t
t-2 t-1

3. If we have 1000 fish and we have a mortality rate of 20% (m = 0.2)
How many are left after 1 year?
What figure do you get if mortality is constant throughout the
year. Try calculating the same thing but using monthly mortality.
Z
t tN N e
  1
Using natural logs we can calculate the number of fish remaining
after a year. . . Or 2 years . . Or three years etc

4. Zt
N  N e
t  1
tNumber of fish
surviving after 1
year
Zt
t tN  N e 1
Number of fish
that were
around the
previous year
3
Z N N e 
3 1
A really useful aspect of logs
is that we can work out fish
surviving after X years simply
by multiplying Z
NB! Zt is not the same as Zt !!!!

5. Z
Z
t tSurvivors N N e
   1
t t t tDeaths N N N N e
     1
 Z 
t N e 1

6.  Z 
t N e 1
This number is the
number of fish that die
due to fishing and natural
mortality.
Z = F + M
Total mortality = Fishing mortality + Natural mortality
F

Z
Proportion of fish dying due to fishing mortality
(i.e. Catch)

7. F
*
 Z 
t N e  1 
t
Z
t
Number of fish
dying in a year
t C
Proportion
due to
fishing
Calculate the number of fish from a thousand that would be caught if F = 0.5
and M = 0.1

8. F
* t C
 Z 
t N e  1 
t
Z
t
1 3 
3
3
3
3
Z e
F
Z
C
N
 

Number of fish
that must be
around in year 3
in order to yield
a particular
catch.
Assume that M = 0.2, that there is a terminal value of F = 0.6 and that we had a
catch of 80 fish in year t

9. 80
3
C
1 
e
 Z 3 
 0.6 / 0.8  1

 0.8  
e
3
3
Nt
= 194 fish
3
F
Z
N

This is the number of fish that must
have been around this year to result
in a catch of 80 fish.

10. We now know that 194 fish survived and from that we had a catch of 80
fish this year. We also know that we had a catch of 90 fish the previous
year and that M = 0.2.
N  N e Z 2
. .but we don’t know Z2
2 3
2 1 2  3
F
C  2

 2
2
Z Z N e e
Z
 2 1
3
1 2  2
C    ≡
F
 2
N e
Z  2 2
Z
F
2
Simplifies to: C
What numbers can we slot into this equation?
2
2
Z N e
Z
2  2 1 Z Z e e

11.  2 1
3
F
 F M N e
F M
2
2
2 

C
194 1
F
 F  e
F
90 0.2
0.2
2
2 2 

F2 M N N e  
( )
2 3
Z = F + M
What tool in
excel can we use
to find F?
Now we can use the
fact that we know F2
to calculate the
population of fish N2

12. Problems?
• We are using catch figures
• We don’t know M
• We don’t know the terminal F
• We are assuming M to be constant
• We are treating the population as a closed
system
• Can we use it to predict anything?