Lincoln-Petersen Calculators

0 calculators tagged with “Lincoln-Petersen”

The Lincoln-Petersen estimator is the most widely used mark-recapture method for estimating wildlife population size: N̂ = (M × C) / R, where M = number of individuals marked and released, C = total number captured in the second sample, and R = number of marked individuals in the second sample. The Chapman modification (N̂ = [(M+1)(C+1)/(R+1)] − 1) gives less biased estimates for small samples. The method assumes: closed population (no births, deaths, immigration, emigration between samples); random mixing of marked individuals; equal capture probability for marked and unmarked individuals; no loss of marks.

All Calculators

No calculators found for this topic.

Lincoln-Petersen Formula

N̂ = (M × C) / R

M = marked + released; C = total second-sample size; R = marked in second sample. Chapman modified (less biased for small R): N̂ = [(M+1)(C+1)/(R+1)] − 1. Example: tag 200 fish (M); catch 150 fish (C); 30 are tagged (R). N̂ = (200 × 150)/30 = 1,000 fish.

Confidence Interval

Variance: var(N̂) = M²C(C−R) / (R+1)²(R+2) (Chapman). 95% CI: N̂ ± 1.96 × √var(N̂).

Assumptions and Violations

  • Closed population: no births, deaths, immigration, emigration between samples
  • Random mixing: marked animals distribute randomly — no home range fidelity affecting recapture
  • Equal catchability: marked and unmarked have same probability of capture — violations cause bias
  • Mark retention: tags must not fall off; marking must not affect behavior or survival

Alternative Methods

Multiple mark-recapture: Cormack-Jolly-Seber (open populations, survival estimation). Huggins model (heterogeneous capture probability). Program MARK (White & Burnham).

Glossary

Lincoln-Petersen Estimator
N̂ = MC/R; M = marked released; C = second sample size; R = marked in second sample; assumes closed population and random mixing; Chapman modification used for small samples.
Chapman Modification
N̂ = [(M+1)(C+1)/(R+1)] − 1; less biased than basic Lincoln-Petersen for small samples; always use when R < 10; avoids infinite estimates when R = 0.
Trap-Happiness / Trap-Shyness
Behavioral responses to marking affecting recapture probability; trap-happy: N̂ underestimated (R too high); trap-shy: N̂ overestimated (R too low); detected by comparing capture rates of marked vs. unmarked.

Frequently Asked Questions

The Lincoln-Petersen estimator estimates population size from two sampling events: First sample: capture M individuals; mark distinctively; release. Wait: allow marked individuals to mix thoroughly with unmarked population. Second sample: capture C individuals; count R that are marked. Estimate: N̂ = MC/R. Logic: if marked animals mix randomly, the proportion marked in the second sample (R/C) equals the proportion marked in the whole population (M/N̂). Solving for N̂: N̂ = MC/R. Example: mark 100 butterflies; capture 80 in second sample; 20 are marked: N̂ = 100×80/20 = 400 butterflies.

The Chapman estimator: N̂ = [(M+1)(C+1)/(R+1)] − 1. Preferred over the basic Lincoln-Petersen when: small samples (especially when R < 10); R = 0 (basic formula gives N̂ = infinity; Chapman gives a finite estimate). The Chapman estimator is nearly unbiased when (M+C) > N (a condition often violated in practice). When R is large (> 20) and the basic assumptions are well met: the basic formula and Chapman give essentially identical estimates — use either. For any R < 10: always use Chapman. For very small samples (M or C < 30): mark-recapture estimates are unreliable regardless of formula.

Closed population: test with open-population models; if birth/death/migration cannot be excluded, use Jolly-Seber open-population model instead. Equal catchability: trap-happiness (marked animals more likely to be recaptured → R too high → N̂ underestimated); trap-shyness (marked avoid traps → R too low → N̂ overestimated). Tests: compare capture rates of marked vs. unmarked animals; program MARK tests for capture heterogeneity. Random mixing: animals with restricted home ranges (territories, site fidelity) may not mix → spatial stratification of sampling helps. Mark retention: physical marks should be permanent or decay-resistant; test mark retention rate separately.

Fish stock assessment: tag fish with numbered spaghetti tags or PIT tags; recapture in commercial fishing vessels or electrofishing surveys; estimate population size → set harvest quotas. Deer and elk management: ear tags or radio collars; hunters recapture in harvest; population estimates guide season length and bag limits. Bird banding: federal bird banding laboratories maintain recapture databases; bird populations estimated across large areas. Invertebrates: small insects (butterflies, beetles) marked with paint; amphibians (frogs) marked by toe-clipping or visible implant elastomers; fish larva marked with alizarin dye (marks otoliths). Photo-identification: whale fluke patterns, shark fins, dolphin dorsal fins as natural marks → photographic mark-recapture.