LAB 2POPULATION ESTIMATION Part 1Closed Models

1
LAB 2--POPULATION ESTIMATIONPart 1Closed Models

• Readings
• Krebs. 1989. Ecological Methodology. Chapter 2
• White et al. 1981. Capture-Recapture and
  Removal Methods
• Sutherland. 1996. Ecological census techniques
• (1995)

2

• How do we determine abundance?

Is an absolute density needed
Relative Abundance Indices
No
3

• How do we determine abundance?

Is an absolute density needed
Relative Abundance Indices
No
Yes
Need data on individuals?
Yes
Mark- Recapture techniques
4

• How do we determine abundance?

Is an absolute density needed
Relative Abundance Indices
No
Yes
Need data on individuals?
Yes
Mark- Recapture techniques
No
Organisms Mobile?
No
Quadrat counts
5

• How do we determine abundance?

Is an absolute density needed
Relative Abundance Indices
No
Yes
Change-in-ratio methods
Need data on individuals?
Yes
Mark- Recapture techniques
Yes
No
Is exploitation age- or sex- selective
Yes
Yes
Organisms Mobile?
Is population being exploited?
No
No
No
Quadrat counts
Catch-effort methods
6
Mark-recapture

• ratio of marked to unmarked animals during the
  recapture period represents the proportion of
  marked animals to the total population
• (1995)

7
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• (1995)

8
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• We catch and mark 10 animals on day 1
• (1995)

9
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• We catch and mark 10 animals on day 1
• We then trap a second day and find we have caught
  10 animals, two of which are marked
• (1995)

10
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• We catch and mark 10 animals on day 1
• We then trap a second day and find we have caught
  10 animals, two of which are marked
• The number of recaptures is thus 2/10 or 20.
• (1995)

11
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• We catch and mark 10 animals on day 1
• We then trap a second day and find we have caught
  10 animals, two of which are marked
• The number of recaptures is thus 2/10 or 20
• We then assume that the total number of marked
  animals (10)20 of the population
• (1995)

12
Mark-recapture Lincoln-Peterson

• Simplest form is a Lincoln/Petersen estimate
• We catch and mark 10 animals on day 1
• We then trap a second day and find we have caught
  10 animals, two of which are marked
• The number of recaptures is thus 2/10 or 20
• We then assume that the total number of marked
  animals (10)20 of the population
• therefore the population estimate is 50.
• (1995)

13
Mark-recapture Lincoln-Peterson

• Mathematically
• N/MC/R and N/1010/2
• NCM/R1010/250
• where Npopulation estimate--the Petersen
  estimate
• Mnumber of marked animals
• Cnumber of recaptures
• Rnumber of marked recaptures

14
Mark-recapture Lincoln-Peterson

• Or in general

where Npopulation estimate Mnumber of
marked animals Cnumber of recaptures Rnumber
of marked recaptures
15
Mark-recapture Lincoln-Peterson

• Uses
• Estimate population size

16
Mark-recapture Lincoln-Peterson

• Uses
• Estimate population size
• Determine rate of exploitation

17
Mark-recapture Lincoln-Peterson

• Uses
• Estimate population size
• Determine rate of exploitation
• Determine discrete survivorship rates

18
Mark-recapture Lincoln-Peterson

• Uses
• Estimate population size
• Determine rate of exploitation
• Determine discrete survivorship rates
• Determine recruitment rates

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Mark-recapture Lincoln-Peterson

• Uses
• Estimate population size
• Determine rate of exploitation
• Determine discrete survivorship rates
• Determine recruitment rates
• Uses 2,3 4 require running the method more than
  once (multiple)

20
Mark-recapture Lincoln-Peterson

• Problems
• Calculation of confidence intervals
• Statistical bias due to small sample size
• Technique bias

21
Mark-recapture Lincoln-Peterson

• Problem 1
• Calculation of confidence intervals
• are not normally distributed

22
Mark-recapture Lincoln-Peterson

• Problem 1
• Calculation of confidence intervals
• are not normally distributed
• But, reciprocal of , (1/ ), is

23
Mark-recapture Lincoln-Peterson

• Problem 1
• Calculation of confidence intervals

24
Mark-recapture Lincoln-Peterson

• Problem 1
• Calculation of confidence intervals
• where

25
Mark-recapture Lincoln-Peterson

• Problem 2
• Statistical bias
• if R 7 we have high bias

26
Mark-recapture Lincoln-Peterson

• Problem 3
• Technique bias
• Can be most serious and invalidate assumptions

27
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• No Additions (Birth or Immigration)
• No Subtractions (Death or Emigration)

28
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• If trap shy?

29
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• If trap shy--R decreases

30
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• If trap shy
• If trap happy?

31
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• If trap shy
• If trap happy--R increases

32
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality

33
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• Incr mortality of marked animals?

34
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• Incr mortality of marked animals--decr. R

35
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• Incr mortality of marked animals-
• Decr mortality of marked animals?

36
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• No tag loss

37
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• No tag loss
• decreases R, thus?

38
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• No tag loss
• Marked and unmarked animals mix

39
Mark-recapture Lincoln-Peterson

• Assumptions
• Closed population
• Equal probability of capture
• Equal mortality
• No tag loss
• Marked and unmarked animals mix
• All marks are identified and reported correctly

40
Mark-recapture Lincoln-Peterson

• How to decrease bias
• Different capture techniques for each phase

41
Mark-recapture Lincoln-Peterson

• How to decrease bias
• Different capture techniques for each phase
• Different types of marks

42
Mark-recapture Lincoln-Peterson

• How to decrease bias
• Different capture techniques for each phase
• Different types of marks
• Short time period

43
Mark-recapture Lincoln-Peterson

• Example
• M7 C12 R4

44
Mark-recapture Lincoln-Peterson

• Example

M7 C12 R4
45
Mark-recapture Lincoln-Peterson

• Example

M7 C12 R4
46
Mark-recapture Lincoln-Peterson

• Example

M7 C12 R4
47
Mark-recapture Lincoln-Peterson

• Example

M7 C12 R4
48
Mark-recapture Lincoln-Peterson

• Is this a good estimate?

49
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation

50
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance

51
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance

52
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI

53
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI

54
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI

55
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
• Positive interval 0.050505 0.038107 0.088612

56
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
• Positive interval 0.050505 0.038107 0.088612
• Negative interval 0.050505 - 0.038107 0.012398

57
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
• Positive interval 0.050505 0.038107 0.088612
• Negative interval 0.050505 - 0.038107 0.012398
• Do these seem small, why?

58
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
  and taking the reciprocal (since we were using 1/
  )

59
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
  and taking the reciprocal (since we were using 1/
  )

60
Mark-recapture Lincoln-Peterson

• 95 Confidence Interval calculation
• Calculate the variance, then we can estimate CI
  and taking the reciprocal (since we were using 1/
  )

61
Mark-recapture Multiple Peterson

• A series of estimates based on single Petersen

62
Mark-recapture Multiple Peterson

• A series of estimates based on single Petersen
• Allows comparison of daily estimates

63
Mark-recapture Multiple Peterson

• A series of estimates based on single Petersen
• Allows comparison of daily estimates
• Similar assumptions

64
Mark-recapture Multiple Peterson

• Set up data in a table

65
Mark-recapture Multiple Peterson

• Since

66
Mark-recapture Multiple Peterson

• Since
• and

67
Mark-recapture Multiple Peterson

• General form

68
Mark-recapture Multiple Peterson

• Confidence intervals

69
Mark-recapture Multiple Peterson

• But the variance is now

70
Mark-recapture Multiple Peterson

• Example
• 1 died following tagging

71
Mark-recapture Multiple Peterson

• Calculating for EACH occasion

72
Mark-recapture Multiple Peterson

• Calculating for EACH occasion

73
Mark-recapture Multiple Peterson

• Calculating for EACH occasion

74
Mark-recapture Multiple Peterson

• Calculating for EACH occasion

75
Mark-recapture Multiple Peterson

• Calculating variance for t2

76
Mark-recapture Multiple Peterson

• Calculating variance for t2

77
Mark-recapture Multiple Peterson

• Calculating variance for t2

78
Mark-recapture Multiple Peterson

• Calculating variance for t2

79
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2

80
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2

81
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2
• Positive interval 0.028571 0.024402 0.052973

82
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2
• Positive interval 0.028571 0.024402 0.052973
• Negative interval 0.028571 - 0.024402 0.004169

83
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2
• Positive interval 0.028571 0.024402 0.052973
• Negative interval 0.028571 - 0.024402 0.004169

84
Mark-recapture Multiple Peterson

• Calculating 95 CI for t2
• Positive interval 0.028571 0.024402 0.052973
• Negative interval 0.028571 - 0.024402 0.004169

85
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Why?

86
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Increased time?

87
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Increased time more immigr, emigr., mortality

88
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Multiple Petersen increases precision and
  decreases variability, why?

89
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Multiple Petersen increases precision and
  decreases variability
• Increased sample size

90
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Multiple Petersen increases precision and
  decreases variability
• Multiple Petersen allows better data
  interpretation

91
Mark-recapture Single vs Multiple Peterson

• Multiple Petersen increases potential bias
• Multiple Petersen increases precision and
  decreases variability
• Multiple Petersen allows better data
  interpretation
• E.g. trap-shy or -happy animals

92
Mark-recapture Schnabel estimate

• multiple recapture method that produces one
  population estimate
• Allows for time varying capture probability on
  each occasion

93
Mark-recapture Schnabel estimate

• Formula

94
Mark-recapture Schnabel estimate

• Formula

95
Mark-recapture Schnabel estimate

• Table

96
Mark-recapture Schnabel estimate

• Example

97
Mark-recapture Schnabel estimate

• Example

98
Mark-recapture Schnabel estimate

• Calculating CIs
• What is the variance?

99
Mark-recapture Schnabel estimate

• Calculating CIs
• What is the variance?

100
Mark-recapture Schnabel estimate

• Calculating CIs

101
Mark-recapture Schnabel estimate

• Calculating Cis
• and after taking reciprocals

102
Mark-recapture Schnabel estimate

• Assumptions
• Same as Petersen
• Advantage
• Easier to test for violations of assumptions
• Regression plots

103
Removal Methods

• Zippin Method
• Electrofishing data
• Assumptions
• Population is closed
• Constant capture probability
• Uses Maximum Likelihood Estimator

104
Regression MethodCatch Per Unit Effort (CPUE)

• Tabulate the number of animals caught on each
  sample
• uj removed on occasion j
• Tabulate the total number removed before each
  sample
• Mj u1 u2 .uj-1

105
Regression Method
j
uj
Mj
0
260
1
2
260
141
97
3
401
4
50
498
106
Regression MethodCatch Per Unit Effort (CPUE)

• Under the assumption of constant capture
  probability, relationship between uj and mj is
  linear

107
Linear Regression Model
Remember apN b-p
108
POPULATION ESTIMATIONPart 2--the Jolly-Seber
estimator

• Readings
• Krebs. 1989. Ecological Methodology. Chapter 2
• Hayek and Buzas. 1997. Surveying natural
  populations.
• Sutherland. 1996. Ecological census techniques
• (1995)

109
Mark-recapture Jolly-Seber estimate

• A method that accounts for mortality and
  immigration

110
Mark-recapture Jolly-Seber estimate

• Assumptions
• open population

111
Mark-recapture Jolly-Seber estimate

• Assumptions
• open population
• 3 or more sampling periods

112
Mark-recapture Jolly-Seber estimate

• Assumptions
• open population
• 3 or more sampling periods
• marks must be identifiable to sampling occasion

113
Mark-recapture Jolly-Seber estimate

• Concept
• All animals in first sample are unmarked

114
Mark-recapture Jolly-Seber estimate

• Concept
• All animals in first sample are unmarked
• In subsequent samples the total catch can be
  subdivided into two fractions marked and
  unmarked animals.

115
Mark-recapture Jolly-Seber estimate

• Concept
• All animals in first sample are unmarked
• In subsequent samples the total catch can be
  subdivided into two fractions marked and
  unmarked animals
• For marked individuals we ask When was this
  marked individual last captured?

116
Mark-recapture Jolly-Seber estimate

• Concept
• All animals in first sample are unmarked
• In subsequent samples the total catch can be
  subdivided into two fractions marked and
  unmarked animals
• For marked individuals we ask When was this
  marked individual last captured?
• Most of the marked animals caught will have been
  last caught at the previous sampling and
  consequently will appear along the subdiagonal

117
Mark-recapture Jolly-Seber estimate

• Step 1- Setting up the J-S table
• mt of marked individuals caught in sample t

118
Mark-recapture Jolly-Seber estimate

• Step 1- Setting up the J-S table
• mt of marked individuals caught in sample t
• ut of unmarked individuals caught in sample t

119
Mark-recapture Jolly-Seber estimate

• Step 1- Setting up the J-S table
• mt of marked individuals caught in sample t
• ut of unmarked individuals caught in sample t
• nttotal of individuals caught in sample
  t (mtut)

120
Mark-recapture Jolly-Seber estimate

• Step 1- Setting up the J-S table
• mt of marked individuals caught in sample t
• ut of unmarked individuals caught in sample t
• nttotal of individuals caught in sample
  t (mtut)
• sttotal of individuals released after sample t
  (nt-deaths or removals)

121
Mark-recapture Jolly-Seber estimate

• Step 1- Setting up the J-S table

122
Jolly-Seber estimate

• Step 2- Calculate Rt
• Rt of the st indiv released at sample t and
  caught again in some later sample (e.g., for t3,
  R33)

123
Jolly-Seber estimate

• Step 3- Calculate Zt
• Ztof individuals marked before sample t, not
  caught in sample t, but caught in some other
  sample after sample t (e.g., for t3, Z35)

124
Jolly-Seber estimate

• Step 4- Calculate the proportion of animals
  marked
• For example, using time3, m32 and n37

125
Jolly-Seber estimate

• Step 4- Calculate the proportion of animals
  marked
• For example, using time3, m32 and n37

126
Jolly-Seber estimate

• Step 5- Calculate the size of the marked
  population
• 2 components of the marked population
• marked animals actually caught
• marked animals present but not captured in sample
  t

127
Jolly-Seber estimate

• Step 5- Calculate the size of the marked
  population
• for example, using time3, m32, R33, Z35, and
  S37

128
Jolly-Seber estimate

• Step 5- Calculate the size of the marked
  population
• for example, using time3, m32, R33, Z35, and
  S37

129
Jolly-Seber estimate

• Step 6- Calculate
• ratio of the size of the marked population (step
  5) to the proportion of animals marked (step 4)

130
Jolly-Seber estimate

• Step 6- Calculate
• ratio of the size of the marked population (step
  5) to the proportion of animals marked (step 4)

131
Jolly-Seber estimate

• Can Calculate Apparent Survival
• Apparent because mortality is confounded with
  emigration (cant get separate estimates for each)

132
Jolly-Seber estimate

• Assumptions
• Equal probability of capture for marked and
  unmarked individuals

133
Jolly-Seber estimate

• Assumptions
• Equal probability of capture for marked and
  unmarked individuals
• Equal mortality for marked and unmarked
  individuals

134
Jolly-Seber estimate

• Assumptions
• Equal probability of capture for marked and
  unmarked individuals
• Equal mortality for marked and unmarked
  individuals
• No tag loss

135
Jolly-Seber estimate

• Assumptions
• Equal probability of capture for marked and
  unmarked individuals
• Equal mortality for marked and unmarked
  individuals
• No tag loss
• All marks are identified and reported correctly

136
Jolly-Seber estimate

• Assumptions
• Equal probability of capture for marked and
  unmarked individuals
• Equal mortality for marked and unmarked
  individuals
• No tag loss
• All marks are identified and reported correctly
• Sampling time is negligible relative to the time
  between samples