FA-65506 / Ecological population dynamics / Open access
Two-sample mark-recapture abundance: chapman numerator · case 01
Chapman estimates are biased low, especially with few recaptures.
ROOT CAUSE
The +1 corrections are dropped from the numerator but kept in the denominator.
VERIFIED REPAIR
Restore the chapman numerator rule: `est = (marked + 1) * (caught + 1) /`.
Unsuccessful approach: Correcting only the marked count is still inconsistent.
Case contract
Recaptures must be in [0, min(marked, caught)] else None; 'lp' = M*C/R ([None, None] when R=0, no variance); Chapman = (M+1)(C+1)/(R+1)-1 with variance (M+1)(C+1)(M-R)(C-R)/((R+1)^2 (R+2)); return [estimate rounded 2, variance rounded 2 or None].
Why this case matters
Population projections set harvest quotas, conservation status and pest-control timing; a wrong update order, boundary or rate conversion silently changes management advice.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(marked, caught, recaptured, method):
if recaptured < 0 or recaptured > min(marked, caught):
return None
if method == 'lp':
if recaptured == 0:
return [None, None]
return [round(marked * caught / recaptured, 2), None]
est = marked * caught / (recaptured + 1) - 1
var = (marked + 1) * (caught + 1) * (marked - recaptured) * (caught - recaptured) / ((recaptured + 1) ** 2 * (recaptured + 2))
return [round(est, 2), round(var, 2)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None)],
[('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84])]]
for label, args, expected in fixtures[N - 1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression: trout chapman | [570.43, 10077.1] | [580.95, 10077.1] | Failed |
| control: trout petersen | [600.0, None] | [600.0, None] | Passed |
| regression: zero recaptures chapman | [1999.0, 2091000.0] | [2090.0, 2091000.0] | Failed |
| control: zero recaptures petersen | [None, None] | [None, None] | Passed |
| regression: all recaptured | [28.03, 0.0] | [30.0, 0.0] | Failed |
| control: recaptures exceed catch | None | None | Passed |
| control: recaptures exceed marked | None | None | Passed |
SHA-256 / 1abf51a95a8d5d88923c717a0f360568049b8b5ed2ae0f233b86c7bba5708ee9
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(marked, caught, recaptured, method):
if recaptured < 0 or recaptured > min(marked, caught):
return None
if method == 'lp':
if recaptured == 0:
return [None, None]
return [round(marked * caught / recaptured, 2), None]
est = (marked + 1) * caught / (recaptured + 1) - 1
var = (marked + 1) * (caught + 1) * (marked - recaptured) * (caught - recaptured) / ((recaptured + 1) ** 2 * (recaptured + 2))
return [round(est, 2), round(var, 2)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None)],
[('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84])]]
for label, args, expected in fixtures[N - 1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression: trout chapman | [575.19, 10077.1] | [580.95, 10077.1] | Failed |
| control: trout petersen | [600.0, None] | [600.0, None] | Passed |
| regression: zero recaptures chapman | [2039.0, 2091000.0] | [2090.0, 2091000.0] | Failed |
| control: zero recaptures petersen | [None, None] | [None, None] | Passed |
| regression: all recaptured | [29.0, 0.0] | [30.0, 0.0] | Failed |
| control: recaptures exceed catch | None | None | Passed |
| control: recaptures exceed marked | None | None | Passed |
SHA-256 / 4395add82f069b6d94f84c020fbc0ae819fde990072c0d0536e37f49b0b9c056
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(marked, caught, recaptured, method):
if recaptured < 0 or recaptured > min(marked, caught):
return None
if method == 'lp':
if recaptured == 0:
return [None, None]
return [round(marked * caught / recaptured, 2), None]
est = (marked + 1) * (caught + 1) / (recaptured + 1) - 1
var = (marked + 1) * (caught + 1) * (marked - recaptured) * (caught - recaptured) / ((recaptured + 1) ** 2 * (recaptured + 2))
return [round(est, 2), round(var, 2)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None)],
[('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84]),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: trout chapman', (120, 100, 20, 'chapman'), [580.95, 10077.1]),
('control: trout petersen', (120, 100, 20, 'lp'), [600.0, None]),
('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: negative recapture', (20, 20, -1, 'lp'), None)],
[('regression: zero recaptures chapman', (50, 40, 0, 'chapman'), [2090.0, 2091000.0]),
('control: zero recaptures petersen', (50, 40, 0, 'lp'), [None, None]),
('regression: all recaptured', (30, 30, 30, 'chapman'), [30.0, 0.0]),
('control: recaptures exceed catch', (80, 10, 12, 'chapman'), None),
('control: recaptures exceed marked', (10, 80, 12, 'lp'), None),
('regression: small sample', (15, 12, 3, 'chapman'), [51.0, 280.8]),
('regression: butterflies', (200, 150, 45, 'chapman'), [658.8, 4966.84])]]
for label, args, expected in fixtures[N - 1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression: trout chapman | [580.95, 10077.1] | [580.95, 10077.1] | Passed |
| control: trout petersen | [600.0, None] | [600.0, None] | Passed |
| regression: zero recaptures chapman | [2090.0, 2091000.0] | [2090.0, 2091000.0] | Passed |
| control: zero recaptures petersen | [None, None] | [None, None] | Passed |
| regression: all recaptured | [30.0, 0.0] | [30.0, 0.0] | Passed |
| control: recaptures exceed catch | None | None | Passed |
| control: recaptures exceed marked | None | None | Passed |
SHA-256 / eb949029b4e8b92cf268ff70ea80f30fca023ba48366f42df708644d1fa8174c
Verification & scope
Deterministic bounded teaching model with a stipulated contract; not a validated scientific or public-health modelling library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:47:34.655526+00:00.
Case digest / caa097eefdd94ce722f2f1c82144da0ef54e17c15870184f075973c9351b39a6