Mark-Recapture and Exploitation Rate Estimation

Introduction

Creel surveys are often paired with tagging studies to estimate either angler population size or seasonal exploitation rate. tidycreel provides three estimators for these purposes:

estimate_angler_n() — closed-population mark-recapture for total angler count (Chapman, Petersen, or Schnabel)
estimate_mr_harvest() — total harvest derived from a mark-recapture population estimate
estimate_exploitation_rate() — seasonal exploitation rate from tagged-fish recoveries (Pollock et al. 1994)

All three return creel_estimates objects compatible with the standard tidycreel output ecosystem (print(), autoplot(), write_estimates()).

Goal	Estimator
Estimate total anglers from a tag-and-resight study	`estimate_angler_n()`
Convert angler population estimate to total harvest	`estimate_mr_harvest()`
Estimate fraction of population harvested from tagged-fish recoveries	`estimate_exploitation_rate()`

Angler Population Size

Chapman estimator (default)

The Chapman estimator is a bias-corrected Petersen estimator recommended when the recapture count is small. With $M$ tagged anglers released, $n$ anglers checked in the second sample, and $m$ recaptures:

$\hat{N} = \frac{(M+1)(n+1)}{(m+1)} - 1$

Variance:

$\widehat{\text{Var}}(\hat{N}) = \frac{(M+1)(n+1)(M-m)(n-m)}{(m+1)^2(m+2)}$

library(tidycreel)

result_chapman <- estimate_angler_n(M = 200L, n = 50L, m = 10L)
print(result_chapman)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: mark-recapture-chapman
#> Variance: chapman
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 6
#>   parameter estimate    se ci_lower ci_upper     n
#>   <chr>        <dbl> <dbl>    <dbl>    <dbl> <int>
#> 1 N_hat         931.  232.     477.    1385.    10

The n column in the estimates tibble records the recapture count (m), which determines precision.

Petersen estimator

The unadjusted Lincoln–Petersen estimator:

$\hat{N} = \frac{M \cdot n}{m}$

tidycreel enforces a minimum of $m \geq 7$ recaptures; below this threshold the Petersen estimator carries large positive bias and Chapman should be used instead.

result_petersen <- estimate_angler_n(M = 200L, n = 50L, m = 10L, method = "petersen")
print(result_petersen)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: mark-recapture-petersen
#> Variance: petersen
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 6
#>   parameter estimate    se ci_lower ci_upper     n
#>   <chr>        <dbl> <dbl>    <dbl>    <dbl> <int>
#> 1 N_hat         1000  283.     446.    1554.    10

Schnabel estimator (multi-occasion)

For $K \geq 2$ successive sampling occasions, the Schnabel estimator pools information across occasions:

$\hat{N} = \frac{\sum_{k=1}^{K} M_k n_k}{\sum_{k=1}^{K} m_k}$

where $M_k$ is the cumulative count of marked-at-large anglers before occasion $k$ (so $M_1 = 0$ ), $n_k$ is the catch on occasion $k$ , and $m_k$ is the number of recaptures.

Confidence intervals use the Poisson distribution when $\sum m_k < 50$ and a normal approximation on $1/\hat{N}$ otherwise.

result_schnabel <- estimate_angler_n(
  M      = c(0L, 47L, 91L, 131L),
  n      = c(50L, 50L, 50L, 50L),
  m      = c(0L,  4L,  6L,  8L),
  method = "schnabel"
)
print(result_schnabel)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: mark-recapture-schnabel
#> Variance: delta
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 6
#>   parameter estimate    se ci_lower ci_upper     n
#>   <chr>        <dbl> <dbl>    <dbl>    <dbl> <int>
#> 1 N_hat         747.  176.     498.     1345    18

Four sampling occasions yield 18 total recaptures here, so the Poisson CI branch is used automatically.

Total Harvest from Mark-Recapture

Once angler population size is estimated, estimate_mr_harvest() scales it by a known harvest rate to obtain total harvest. The harvest rate is typically derived from creel interview data — the proportion of interviewed anglers who kept fish.

$\hat{H} = \hat{N} \times r, \quad SE(\hat{H}) = r \times SE(\hat{N})$

# harvest_rate derived from creel interviews: 35% of anglers harvested fish
harvest <- estimate_mr_harvest(angler_n = result_chapman, harvest_rate = 0.35)
print(harvest)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: mark-recapture-harvest
#> Variance: delta
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 5
#>   parameter     estimate    se ci_lower ci_upper
#>   <chr>            <dbl> <dbl>    <dbl>    <dbl>
#> 1 total_harvest     326.  81.1     167.     485.

The delta method propagates uncertainty in $\hat{N}$ only. Uncertainty in the harvest rate itself is not propagated in this release; if harvest rate uncertainty is substantial, re-run across plausible bounds as a sensitivity check.

Exploitation Rate

estimate_exploitation_rate() implements the Pollock et al. (1994) moment estimator. Rather than counting anglers, it estimates the fraction of a tagged cohort that was harvested during the season. The inputs come from two sources:

Tagging study: $T$ fish tagged and released at season start; $m$ tagged fish recovered among $n$ fish inspected in the creel.
Creel survey: $C$ total estimated harvest (e.g., from estimate_total_catch()) with standard error $SE_C$ .

Unstratified

$\hat{u} = \frac{C \cdot m}{T \cdot n}, \quad \widehat{\text{Var}}(\hat{u}) \approx \left(\frac{C}{T}\right)^2 \frac{p(1-p)}{n} + p^2 \frac{SE_C^2}{T^2}$

where $p = m/n$ .

result_expl <- estimate_exploitation_rate(
  T    = 200L,
  C    = 450.0,
  se_C = 42.0,
  n    = 180L,
  m    = 15L
)
print(result_expl)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: Exploitation Rate (Mark-Recapture)
#> Variance: delta
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 8
#>   estimate     se ci_lower ci_upper     n     T     C     m
#>      <dbl>  <dbl>    <dbl>    <dbl> <int> <int> <dbl> <int>
#> 1    0.188 0.0495   0.0904    0.285   180   200   450    15

Stratified (T-weighted)

When the survey spans multiple strata (day types, access areas), supply a data frame with one row per stratum. The T-weighted aggregate is automatically appended as an .overall row:

$\hat{u}_{overall} = \frac{\sum_h T_h \hat{u}_h}{\sum_h T_h}, \quad \widehat{\text{Var}}(\hat{u}_{overall}) = \frac{\sum_h T_h^2 \,\widehat{\text{Var}}(\hat{u}_h)}{(\sum_h T_h)^2}$

strata_df <- data.frame(
  stratum = c("weekday", "weekend"),
  T_h     = c(120L, 80L),
  C_h     = c(280.0, 170.0),
  se_C_h  = c(28.0, 22.0),
  n_h     = c(110L, 70L),
  m_h     = c(9L, 6L)
)

result_strat <- estimate_exploitation_rate(strata = strata_df, by = "stratum")
print(result_strat)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: Exploitation Rate (Mark-Recapture)
#> Variance: delta
#> Confidence level: 95%
#> Grouped by: stratum
#> 
#> # A tibble: 3 × 9
#>   stratum  estimate     se ci_lower ci_upper     n     T     C     m
#>   <chr>       <dbl>  <dbl>    <dbl>    <dbl> <int> <int> <dbl> <int>
#> 1 weekday     0.191 0.0639   0.0657    0.316   110   120   280     9
#> 2 weekend     0.182 0.0749   0.0353    0.329    70    80   170     6
#> 3 .overall    0.187 0.0487   0.0920    0.283   180   200   450    15

Pass aggregate = FALSE to suppress the .overall row.

Reporting rate adjustment

If not all harvested tagged fish are reported, supply the reporting rate $\lambda \in (0, 1]$ :

$\hat{u}_{adj} = \frac{\hat{u}}{\lambda}$

result_adj <- estimate_exploitation_rate(
  T              = 200L,
  C              = 450.0,
  se_C           = 42.0,
  n              = 180L,
  m              = 15L,
  reporting_rate = 0.80
)
print(result_adj)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: Exploitation Rate (Mark-Recapture)
#> Variance: delta
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 8
#>   estimate     se ci_lower ci_upper     n     T     C     m
#>      <dbl>  <dbl>    <dbl>    <dbl> <int> <int> <dbl> <int>
#> 1    0.234 0.0619    0.113    0.356   180   200   450    15

Uncertainty in $\lambda$ is not propagated; treat reporting rate accuracy as a separate sensitivity analysis.

Choosing an Estimator

Scenario	Recommended estimator
Single recapture event, small $m$ ( $< 7$ )	`estimate_angler_n(method = "chapman")`
Single recapture event, $m \geq 7$	`"chapman"` or `"petersen"` (Chapman preferred)
Multiple survey occasions with cumulative marking	`estimate_angler_n(method = "schnabel")`
Converting population estimate to total harvest	`estimate_mr_harvest()`
Fraction of tagged cohort harvested (season-level)	`estimate_exploitation_rate()`

Assumptions

All three estimators share the standard closed-population assumptions:

Closure — the population is closed between marking and recapture; no births, deaths, emigration, or immigration.
Equal catchability — all individuals have equal probability of being captured or inspected on each occasion.
Mark retention — tags are not lost or overlooked at recapture.
Random mixing — marked individuals are randomly distributed in the population before the second sample.

For estimate_exploitation_rate() two additional caveats apply:

Natural mortality between tagging and the creel survey is not corrected for; if natural mortality is non-trivial, the estimator overestimates exploitation.
The reporting rate is treated as known without error (see adjustment section above).

References

Hansen, M. J., & Van Kirk, R. W. (2018). A mark-recapture-based approach for estimating angler harvest. North American Journal of Fisheries Management, 38(2), 400–410. https://doi.org/10.1002/nafm.10038

Pollock, K. H., Jones, C. M., & Brown, T. L. (1994). Angler Survey Methods and Their Applications in Fisheries Management. AFS Special Publication 25. American Fisheries Society.

Jones, C. M., & Pollock, K. H. (2012). Recreational survey methods: estimating effort, harvest, and abundance. In A. V. Zale et al. (Eds.), Fisheries Techniques (3rd ed., Ch. 19). American Fisheries Society.