Population Size Estimation via Mark-Release-Recapture: Lincoln-Petersen Index & Bailey’s Correction
Determining animal population density cannot rely on total censuses for vagile organisms. The mark-release-recapture (MRR) framework assumes that the proportion of marked animals in a second sample equals the proportion of marked animals in the total population:
- Lincoln-Petersen Equation:
$$frac{m}{n} = frac{M}{N} implies hat{N} = frac{M cdot n}{m}$$
where $M$ is the number of individuals marked and released in Sample 1, $n$ is the total number of individuals captured in Sample 2, and $m$ is the number of previously marked recaptures in Sample 2. - Bailey’s Correction for Small Recaptures: The standard Lincoln-Petersen estimator is positively biased when sample sizes or recapture rates are low ($m < 10$). Bailey (1951) introduced the nearly unbiased estimator: $$hat{N}_{text{Bailey}} = frac{M(n + 1)}{m + 1}$$
- Variance & Confidence Limits:
$$operatorname{Var}(hat{N}) = frac{M^2 cdot (n + 1)(n – m)}{(m + 1)^2 (m + 2)}$$
The 95% Confidence Interval is given by: $hat{N} pm 1.96 cdot sqrt{operatorname{Var}(hat{N})}$.
- Field Trial (e.g., Tenebrionid Beetles / Snails): Pitfall traps with non-toxic collection funnels; quick-drying, waterproof, non-odorous cellulose paint markers (white or fluorescent red); fine point brushes; sorting trays.
- Bench Simulation Alternative: Large opaque mixing bin containing 500-1000 identical white plastic beads/beans representing the unknown population; permanent colored marking pens; sampling scoop.
- Sample 1 Capture & Tagging: Draw a random sample of $M$ individuals (e.g., $M = 50$ beetles from pitfall traps or beads from the container). Apply a discrete dot of waterproof paint onto the dorsal elytron or bead surface. Record the exact number $M$.
- Release & Random Mixing: Release all marked individuals back into their habitat/bin. Allow sufficient time ($Delta t = 24-48text{ h}$ in the field, or thorough hand-stirring in simulation) to ensure completely homogeneous random redistribution throughout the population.
- Sample 2 Recapture: Take a second random sample without replacement. Count the total number of individuals captured ($n$, e.g., $n = 45$).
- Recapture Tally: Inspect each individual in Sample 2 for paint marks. Record the number of marked recaptures ($m$).
- Statistical Calculation: Compute $hat{N}$ using both the standard Lincoln-Petersen formula and Bailey’s modification. Calculate variance and 95% confidence intervals.
Given Data: $M = 60$, $n = 50$, $m = 12$.
Lincoln-Petersen Estimate: $hat{N} = frac{60 times 50}{12} = 250text{ individuals}$.
Bailey’s Corrected Estimate: $hat{N}_{text{Bailey}} = frac{60 times (50 + 1)}{12 + 1} = frac{60 times 51}{13} = 235.4 approx 235text{ individuals}$.
Variance: $operatorname{Var}(hat{N}) = frac{3600 times 51 times 38}{169 times 14} = frac{6,976,800}{2366} approx 2948.7$.
Standard Error: $text{SE} = sqrt{2948.7} = 54.3$.
95% CI: $235 pm (1.96 times 54.3) = [128.6, 341.4]$.
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