Mark-Recapture Estimates For Open Populations

A. Fundamental concepts

• Jolly (1965) and Seber (1965) submitted idea separately.

• Marked population (M_i) is unknown because some may have died or emigrated. The basic problem is to estimate the unknown parameter M_i .

• Use information obtained later in a study to improve estimates obtained early in the study.

• This works for open populations to estimate number still alive in the marked population.

B. Notation for the Jolly-Seber model:

S_i = a discrete sample (i) is taken at point S_i in time.

Ø(phi)_i = death + emigration rate = probability that an individual alive at time S_i is still there at S_i + 1.

B_i = birth + immigration rate = number of individuals in the population at S_i + 1 that were not there at S_i.

N_i = size of the population at S_i.

M_i = subpopulation marked by the experimenter previous to S (generally the M_i’s are subject to Ø_i‘s).

P_i = probability of any individual in N_i occurring in the sample at S_i.

n_i = number in the sample at S_i (all n_i returned).

m_i = number of the marked subpopulation in sample at S_i.

z_i = number of marked individuals that were not in sample at S_i (=M_i - m_i) that were recaptured in samples after S_i.

r_i = number of n_i recaptured after S_i.

C. Example:

• From Krebs, C. J. 1978. Ecology: The Experimental Analysis of Distributions and Abundance. Appendix I.

Capture history for groundsquirrels:

Matrix summary of individual captures:

From the matrix we can calculate Z_i and r_i

• e.g., for sample period 3:

Z₃ = [5 + 2 + 2] + [18 + 8 + 4] = 39

r₃ = 33 + 13 + 8

so:

^{^}M₃ = [(Z_i * R_i) / r_i ] + m_i = [(39 * 164) /54] + 37 = 155.4

and

^{^}N₃ = (n₃ * ^{^}M₃) / m₃ = (169 * 155.4) / 37 = 709.8

• For sample period 4:

Z₄ = [2 + 2] + [8 + 4] + [13 + 8] = 37

r₄ = 30 + 20 = 50

^{^}M₄ = [(38 * 202) /50] + 56 = 205.5

^{^}N₄ = (209 * 205.5) / 56 = 767

We can now estimate survival and birth rate for period 3:

^{^}Ø₃ = M₄ / (M₃ + n₃ - m₃) = 205.5 / (155.4 + 169 - 37) = 0.72

^{^}B₃ = ^{^}N₄ - ( ^{^}N₄ * ^{^}Ø₃) = 767 - (709.8 * 0.72) = 250.5

D. Assumptions

1. Population is geographically closed.

2. Every animal present in the population (marked and unmarked) has the same probability of capture (p_i) in the ith sample.

3. Every marked animal present in the population immediately after the ith sample has the same probability of survival (Ø_i) until the (i + 1)th sampling time.

4. Marks are not lost or overlooked.

5. All samples are instantaneous (i.e., demographic closure is satisfied during each sampling occasion) and each release is made immediately after the sample.

Return to Lecture Notes IV. Estimation of Population Parameters - I.

Updated 31 July 1996