NRES 488 HOMEWORK 2

SOLUTIONS

  1. Our estimate of population size is the ratio of the estimated size of the marked population to the proportion of the population that was marked. Our estimate of the proportion marked is :

Our estimate of the size of the marked population is:

.

For the data below we need to calculate the statistics for the eighth capture occasion.

Time of Capture
Time of last capture / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11
1 / 15 / 1 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
2 / 15 / 0 / 1 / 0 / 0 / 0 / 0 / 0 / 0
3 / 37 / 2 / 0 / 0 / 0 / 0 / 0 / 0
4 / 61 / 4 / 1 / 1 / 0 / 0 / 0 / Z8
5 / 75 / 3 / 2 / 0 / 0 / 0
6 / 77 / 4 / 0 / 0 / 0
7 / 69 / 0 / 0 / 0
8 / 8 / 1 / 0 / R8
9 / 14 / 0
10 / 19
Total marked (mt) / 0 / 15 / 16 / 37 / 64 / 79 / 81 / 76 / 8 / 15 / 19
Total unmarked (ut) / 22 / 26 / 32 / 45 / 25 / 22 / 26 / 15 / 11 / 12 / 3
Total caught (nt) / 22 / 41 / 48 / 82 / 89 / 101 / 107 / 91 / 19 / 27 / 22
Total relaeased (st) / 21 / 41 / 46 / 82 / 88 / 99 / 106 / 90 / 19 / 26 / 22

m8 = 76

n8 = 91

s8 + 1 = 91

Z8 = 0

R8 + 1 = 10

  1. For closed capture models M0 represents no variation in p (capture probability), Mb represents behavioural response to capture (capture probabilities differ from recapture probabilities), Mt represents temporal variation in capture probability and Mh represents heterogeneity among individuals in capture probability. Combinations of subscripts reflect multiple sources of variation; for example Mth represents heterogeneity and temporal variation.

The table of model results is below.

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Delta AICc Model

Model AICc AICc Weight Likelihood #Par Deviance

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{pi,p(g*t)=c(g*t),N)} Mht} 327.372 0.00 0.68500 1.0000 19.000 278.040

{pi,p(g*t),c(g*t),N} Mbht 330.312 2.94 0.15749 0.2299 30.000 258.153

{pi,p(t),c(t),N)} Mbt 330.312 2.94 0.15749 0.2299 30.000 258.153

{pi,p(t)=c(t),N)} Mt 352.540 25.17 0.00000 0.0000 18.000 305.265

{pi,p(g)=c(g),N)} Mh 369.645 42.27 0.00000 0.0000 4.0000 350.847

{pi,p(.)=c(.),N)} M0 379.603 52.23 0.00000 0.0000 2.0000 364.826

{pi,p(.),c(.),N)} Mb 381.550 54.18 0.00000 0.0000 3.0000 364.764

{pi,p(g),c(g),N)} Mbh 381.550 54.18 0.00000 0.0000 3.0000 364.764

If we examine the parameter estimates, however, we find that population size was not estimated for models Mbht and Mbt. Many of the ps were not estimated for Mht. The model results table without these models looks like:

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Delta AICc Model

Model AICc AICc Weight Likelihood #Par Deviance

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{pi,p(t)=c(t),N)} Mt 352.540 25.17 0.00000 0.0000 18.000 305.265

{pi,p(g)=c(g),N)} Mh 369.645 21.66 0.00002 0.0000 4.0000 350.847

{pi,p(.)=c(.),N)} M0 379.603 31.62 0.00000 0.0000 2.0000 364.826

{pi,p(.),c(.),N)} Mb 381.550 33.57 0.00000 0.0000 3.0000 364.764

{pi,p(g),c(g),N)} Mbh 381.550 33.57 0.00000 0.0000 3.0000 364.764

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Mt is, thus the best performing model. Most ps were estimated and the estimate of population size seems reasonable. This model and the others except Mh all produced estimates of population size of about 95. Mh produced an estimate of population size of 135 with a 95% confidence interval of 95-256.

We know that the true population size = 135. Thus, none of the “well performing models produced estimates of population whose CI included the true population size. Hence, their estimates were biased. The one model that produce a reasonable population estimate had very low precision. These problems indicate that our models were unable to adequately model capture probability, thereby creating poor estimates of population size.

Parameter estimates and PIMs are shown below.

M0

N = 96.258769

SE = 6.8786054

95% CI Lower = 86.603277

95 % CI Higher = 114.70669


Mb

N = 93.89

SE = 10.82

95% CI Lower = 81.99

95 % CI Higher = 129.42


Mt

N = 95.08

SE = 6.61

95% CI Lower = 85.86

95 % CI Higher = 112.92


Mh

N = 135.48

SE = 36.60

95% CI Lower = 95.59

95 % CI Higher = 256.61


Mbh

N = 93.89

SE = 10.82

95% CI Lower = 81.99

95 % CI Higher = 129.42320


Mbt

N = 76.0

SE = 0.30 E-004

95% CI Lower = 76.0

95 % CI Higher = 76.0


Mht

het, p (t) = c (t), N

N = 97.62

SE = 7.32

95 % CI LOWER = 87.33

95% CI HIGHER = 117.24

Mbht

N = 76.0

SE = 0.0

95% CI Lower = 76.0

95 % CI Higher = 76.0