Mark recapture lecture 2: Jolly-Seber Confidence intervals And a wee note on bias

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Mark recapture lecture 2: Jolly-Seber Confidence intervals And a wee note on bias . Jolly-Seber For an OPEN population Repeatedly sampled Information on when an individual was last marked. LPB Colony size. Year. Open populations.

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Mark recapture lecture 2:

• Jolly-Seber
• Confidence intervals
• And a wee note on bias

Jolly-Seber

• For an OPEN population
• Repeatedly sampled
• Information on when an individual was last marked

LPB Colony size

Year

Open populations

Individuals enter or leave the population between surveys

Survey 2

Survey 1

Catch nt animals

Check if each animal is marked

Total unmarked (ut) Total marked (mt)

Mark all with

code for this time

period

Release St (equals nt if no handling mortality)

NO

YES

Question:

What is formula for proportion marked?

Jolly-Seber

Remember Petersen (biased):

N= C M

R

Problem: We don’t know how many marked in population (M)

Sample 1: mark 21 animals

Sample 2: mark 41 animals

Sample 3: mark 46 animals

How many marked at beginning of sample 4?

Not 21+41+46=108, as some will have died or emigrated

Time 1

Time 2

Time 3

Mark 3, but

1 of these emigrates

Mark 2 more, no loss of marked animals

Mark 3 more, but 1 marked animal dies

Time 4

How many marked animals are alive and present in the population at time 4?

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

Time 4

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

6 marked at end of time 4 (S4)

Time 4

Time 5

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

6 marked at end of time 4 (S4)

Time 4

Time 5

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1

Time 4

Time 5

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1

Time 4

Time 5

Time 6

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1+1

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1+1

= Recaptures after sample 4 (Z4=1)

x

factor accounting for animals missed or lost from population

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1+1

= Recaptures after sample 4 (Z4=1)

x

factor accounting for animals missed or lost from population (S4 / R4) = 6/2 = 3

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4), recaptured (R4)=1+1

= Z4 * S4 = 1* 6 = 3

R42

Marked animals in sample 4 (m4) = 3

+ Marked animals not in sample 4 (=3)

=Total number of marked animals in population

(M4 = 6)

Biased formula for number of marked animals in population:

Mt= mt + Zt * St

Rt

Unbiased formula for number of marked animals in population:

Mt= mt + Zt * (St+ 1)

(Rt + 1)

Jolly-Seber

Remember Petersen (biased):

N= C M

R

Rearrange to:

N = M

(R/C)

Number marked in population

Proportion marked

in sample

Jolly-Seber

Nt = Mt

(?)

Number marked in population (t)

Proportion marked

in sample t

Catch nt animals

Check if each animal is marked

Total unmarked (ut) Total marked (mt)

Mark all with

code for this time

period

Release St (equals nt if no handling mortality)

NO

YES

Jolly-Seber

Nt = Mt

(?)

mt

nt

mt +1

nt + 1

? =

? (unbiased) =

Number marked in population

Proportion marked

in sample

Question:

m5 = 21

S5 = 9

R5 = 4

Z5 = 10

n5 = 43

What is N?

M5 = 21 + 10*(9+1) = 21+20 = 41

(4+1)

Proportion marked in sample

= (21+1) = 0.5

(43+1)

N5 = 41/0.5 = 82

Y

Petersen

Is the ratio of

R/C > 0.10?

Binomial

Is the number of recaptures, R > 50?

Schnabel

Y

Normal

Schumacher-

Eschmeyer

Poisson

Jolly-Seber: complex lognormal assumed,

See Krebs p 47

Step 2: Calculate CI for either R or R/C (as appropriate)

-see formulae in Krebs

Step 3: Insert upper and lower bound for R or R/C into the formula for estimating population size to obtain CI

For example, if CI for R/C is (0.083, 0.177), to calculate CI for N by Petersen:

N=M/ 0.083 (upper bound)

N=M/ 0.177 (lower bound)