Incoprea Research · Supplied Round 3 · 30 October 2026

A supplied Allee model: a threshold exists only within the bistable regime

Evidence status: Passed conditional deterministic-model calculation after correcting the parameter regime and positive equilibrium; no population observation or extinction probability.

In the supplied mate-limitation model, starting just below a threshold leads toward zero, while starting just above it leads toward a positive equilibrium. At the demonstration parameters, the threshold is approximately 4.586 model population units and the positive equilibrium is 65.414. The source’s description “near K,” with K=100, is imprecise, and the threshold does not exist across the entire supplied mortality range.

The posted claim and the model

The Allee hypothesis connects persistence of sparse populations with mate encounter and mortality. The supplied deterministic equation is dN/dt=rN[N/(N+h)](1−N/K)−dN. The factor N/(N+h) reduces reproductive contribution at low abundance; logistic crowding limits it at high abundance.

Here r is a growth-rate scale, d is mortality, h controls the abundance scale of mate limitation, and K is the crowding parameter. The script fixes r=1, h=10 and K=100. Its d/r axis is mortality relative to the growth scale, not a directly observed mate-encounter rate. The threshold’s value also depends on h and K, so d/r alone does not specify a universal population threshold.

The demonstrated bistability

For d=0.3, positive equilibria satisfy N/(N+10)×(1−N/100)=0.3. Rearrangement gives N²−70N+300=0, with roots approximately 4.586187 and 65.413813.

The lower root is unstable: just below it, per-capita growth is negative; just above it, per-capita growth is positive. The upper root is stable. Zero is also stable for these parameters. This is the model’s bistable regime, where the initial abundance matters.

The trajectory integration uses forward Euler with step 0.01. The near-zero result is not an observed extinction event or an extinction probability: N is continuous, and a small positive deterministic abundance is not a fractional count of a real surviving individual.

Where the threshold disappears

A positive threshold requires the reproductive term to exceed mortality somewhere. For fixed h and K, the maximum of [N/(N+h)](1−N/K) sets the relevant upper limit on d/r. At the supplied h/K=0.1, that limit is approximately 0.536675.

Below that limit there can be two positive roots; at the merger there is a double root, and above it no positive equilibrium remains. The supplied mortality grid extends from 0.05 to 0.9. Its later rows therefore contain None for the threshold. The correct statement is that the lower threshold rises within the regime where it exists, not that every mortality setting has a bistable threshold.

This correction requires no new ecological model: it follows from the same per-capita growth function used in the supplied root search. The code locates the lower root with a scan and bisection; its narrative reference to an “analytic threshold” should not imply that it used a closed-form solver.

What this calculation supports

The result is an internal demonstration that this chosen mate-limitation equation permits abundance-dependent outcomes. It does not estimate thresholds for a species, isolate a causal mortality mechanism in nature, or show that every sparse population behaves this way. Random births and deaths, sex structure, spatial distribution, migration and environmental change are absent, and no numerical time-step convergence analysis is supplied.

The primary study of competitive dynamics with an Allee effect provides established context for analysis of alternative population states. It does not validate the supplied parameter values or make this exact functional form universal.

Provenance

The supplied DeepSeek-assisted Round 3 code, result file and numeric series reproduced byte-for-byte in isolation. Codex audited the equilibrium interpretation and range of validity and edited this publication. No new population dataset, parameter sweep or experiment was performed. The original hypothesis remains unchanged through Incoprea Research. This is a conditional model check, not external peer review or a new biological discovery.