FA-65801 / Ecological population dynamics / Open access
Rosenzweig-MacArthur equilibrium and enrichment stability: predator equilibrium · case 01
Predator equilibrium is understated for slow handlers.
ROOT CAUSE
The attack rate is missing from the saturation factor.
VERIFIED REPAIR
Restore the predator equilibrium rule: `ps = r / a * (1 - ns / k) * (1 + a * h * ns)`.
Unsuccessful approach: Omitting h ignores handling time in the saturation factor.
Case contract
Type II predator-prey: N* = m/(a(e-hm)) requires e>hm and N*<K else ['predator-extinct', [K, 0]]; P* = (r/a)(1-N*/K)(1+ahN*); hump = (K-1/(ah))/2 for h>0 else -1; stable if N*>hump, 'hopf' if equal, else 'cycles'; values rounded 6; None for invalid parameters.
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(r, k, a, h, e, m):
if min(r, k, a, e, m) <= 0 or h < 0:
return None
if e <= h * m:
return ['predator-extinct', [float(k), 0.0]]
ns = m / (a * (e - h * m))
if ns >= k:
return ['predator-extinct', [float(k), 0.0]]
ps = r / a * (1 - ns / k) * (1 + h * ns)
hump = (k - 1 / (a * h)) / 2 if h > 0 else -1.0
if ns > hump:
kind = 'stable'
elif ns == hump:
kind = 'hopf'
else:
kind = 'cycles'
return [kind, [round(ns, 6), round(ps, 6)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('regression: just below enrichment threshold',
(1.0, 9.94, 1.0, 0.5, 1.5, 2.0),
['stable', [4.0, 1.792757]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])]]
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: exact hopf point | ['hopf', [4.0, 1.8]] | ['hopf', [4.0, 1.8]] | Passed |
| regression: enriched cycles | ['cycles', [4.0, 2.7]] | ['cycles', [4.0, 2.7]] | Passed |
| regression: poor habitat stable | ['stable', [4.0, 1.5]] | ['stable', [4.0, 1.5]] | Passed |
| control: inefficient predator | ['predator-extinct', [50.0, 0.0]] | ['predator-extinct', [50.0, 0.0]] | Passed |
| control: predator cannot persist capacity | ['predator-extinct', [3.0, 0.0]] | ['predator-extinct', [3.0, 0.0]] | Passed |
| control: invalid rate | None | None | Passed |
| regression: lynx hare like | ['stable', [11.111111, 63.012346]] | ['stable', [11.111111, 21.728395]] | Failed |
SHA-256 / 07debd4ae9b1303a01a8d2156b08f7ed99c27e725bb84eee821df5cb15e454dd
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(r, k, a, h, e, m):
if min(r, k, a, e, m) <= 0 or h < 0:
return None
if e <= h * m:
return ['predator-extinct', [float(k), 0.0]]
ns = m / (a * (e - h * m))
if ns >= k:
return ['predator-extinct', [float(k), 0.0]]
ps = r / a * (1 - ns / k) * (1 + a * ns)
hump = (k - 1 / (a * h)) / 2 if h > 0 else -1.0
if ns > hump:
kind = 'stable'
elif ns == hump:
kind = 'hopf'
else:
kind = 'cycles'
return [kind, [round(ns, 6), round(ps, 6)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('regression: just below enrichment threshold',
(1.0, 9.94, 1.0, 0.5, 1.5, 2.0),
['stable', [4.0, 1.792757]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])]]
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: exact hopf point | ['hopf', [4.0, 3.0]] | ['hopf', [4.0, 1.8]] | Failed |
| regression: enriched cycles | ['cycles', [4.0, 4.5]] | ['cycles', [4.0, 2.7]] | Failed |
| regression: poor habitat stable | ['stable', [4.0, 2.5]] | ['stable', [4.0, 1.5]] | Failed |
| control: inefficient predator | ['predator-extinct', [50.0, 0.0]] | ['predator-extinct', [50.0, 0.0]] | Passed |
| control: predator cannot persist capacity | ['predator-extinct', [3.0, 0.0]] | ['predator-extinct', [3.0, 0.0]] | Passed |
| control: invalid rate | None | None | Passed |
| regression: lynx hare like | ['stable', [11.111111, 30.419753]] | ['stable', [11.111111, 21.728395]] | Failed |
SHA-256 / 6820dd8ba70cf4c54ccb78cb6ce2218d4e2edc8b048472a39304c44ad8576dd4
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(r, k, a, h, e, m):
if min(r, k, a, e, m) <= 0 or h < 0:
return None
if e <= h * m:
return ['predator-extinct', [float(k), 0.0]]
ns = m / (a * (e - h * m))
if ns >= k:
return ['predator-extinct', [float(k), 0.0]]
ps = r / a * (1 - ns / k) * (1 + a * h * ns)
hump = (k - 1 / (a * h)) / 2 if h > 0 else -1.0
if ns > hump:
kind = 'stable'
elif ns == hump:
kind = 'hopf'
else:
kind = 'cycles'
return [kind, [round(ns, 6), round(ps, 6)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('regression: just below enrichment threshold',
(1.0, 9.94, 1.0, 0.5, 1.5, 2.0),
['stable', [4.0, 1.792757]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: type I predator', (0.8, 30.0, 0.2, 0.0, 0.5, 0.4), ['stable', [4.0, 3.466667]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: exact hopf point', (1.0, 10.0, 1.0, 0.5, 1.5, 2.0), ['hopf', [4.0, 1.8]]),
('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]]),
('control: borderline conversion', (1.0, 20.0, 0.5, 0.5, 0.9, 1.6), ['predator-extinct', [20.0, 0.0]])],
[('regression: enriched cycles', (1.0, 40.0, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 2.7]]),
('regression: poor habitat stable', (1.0, 8.0, 1.0, 0.5, 1.5, 2.0), ['stable', [4.0, 1.5]]),
('regression: just above hump', (1.0, 10.06, 1.0, 0.5, 1.5, 2.0), ['cycles', [4.0, 1.807157]]),
('control: inefficient predator', (1.0, 50.0, 1.0, 0.5, 1.0, 2.0), ['predator-extinct', [50.0, 0.0]]),
('control: predator cannot persist capacity',
(1.0, 3.0, 1.0, 0.5, 1.5, 2.0),
['predator-extinct', [3.0, 0.0]]),
('control: invalid rate', (0.0, 10.0, 1.0, 0.5, 1.5, 2.0), None),
('regression: lynx hare like', (1.2, 60.0, 0.05, 0.2, 0.6, 0.3), ['stable', [11.111111, 21.728395]])]]
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: exact hopf point | ['hopf', [4.0, 1.8]] | ['hopf', [4.0, 1.8]] | Passed |
| regression: enriched cycles | ['cycles', [4.0, 2.7]] | ['cycles', [4.0, 2.7]] | Passed |
| regression: poor habitat stable | ['stable', [4.0, 1.5]] | ['stable', [4.0, 1.5]] | Passed |
| control: inefficient predator | ['predator-extinct', [50.0, 0.0]] | ['predator-extinct', [50.0, 0.0]] | Passed |
| control: predator cannot persist capacity | ['predator-extinct', [3.0, 0.0]] | ['predator-extinct', [3.0, 0.0]] | Passed |
| control: invalid rate | None | None | Passed |
| regression: lynx hare like | ['stable', [11.111111, 21.728395]] | ['stable', [11.111111, 21.728395]] | Passed |
SHA-256 / 00564b49a251e8a75c81d2d35b9c6d46df2f350d2ec1e48c36210f76dc2ab5f3
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 2026-09-29T14:47:37.274914+00:00.
Case digest / 8e3b46647971e6e802b4c2a78070b6e71b3a1f2ce88231efb53e58df3201703c