ReNunney/docs/NUNNEY_ANALYSIS.md

Nunney Analysis

Updated: 2026-04-11

Purpose

This note gives a compact in-repo analysis of Nunney's main claims, equations, and reported results, and how the current replication effort interprets them.

It is not a complete paper summary. Its job is to make the scientific and implementation targets explicit enough that code and results can be reviewed against them.

Primary paper:

• nunney_cost_of_substitution_anz40-185.pdf

Related internal notes:

• COST_OF_SUBSTITUTION.md
• TRACK1_BASELINE.md
• PAPER_CODE_ALIGNMENT.md

Core Claim

Nunney's central claim is that the rate of environmental change a population can tolerate depends on the cost of repeated adaptive substitutions, and that this cost can be decomposed into:

• a fixed component, and
• a per-locus component.

The paper presents simulation evidence that both components depend on mutation supply, summarized by M = 2Ku, and that extinction occurs when the environment changes too quickly for the population to keep pace.

Model Structure

The paper describes four interacting components:

1. constant environmental change
2. genotype-dependent survival relative to the moving optimum
3. density-dependent female fecundity
4. Mendelian transmission with mutation

The adaptive problem is staged so that one substitution is needed every T generations at each selected locus. Smaller T means faster change and thus a more demanding environment.

Main Equations

Growth/Fecundity

Nunney uses:

R = 2 exp(r)

and:

f = 2 exp(r * (1 - (N/K)^(1/r)))

Interpretation:

• R is the density-independent net reproductive rate.
• f is density-dependent female fecundity.
• fecundity is genotype-independent; selection enters through survival.

Fitness / Offspring Survival

The key selection equation is:

w_i = exp(-(r/n) * Σ_j (Av_ij - t/T)^2)

where:

• Av_ij is the mean allelic value at locus j in genotype i
• t/T is the moving optimum on the allele-value scale
• n is the number of loci

Interpretation:

• survival declines as genotype means lag the moving optimum,
• the factor r/n scales selection intensity across different numbers of loci,
• and the Gaussian form makes tracking lag the central state variable.

Mutation Supply

The paper uses u as the mutation-rate parameter and M = 2Ku as a derived population-level mutation-supply quantity for comparing treatments.

Important consequence for replication:

• u is the paper-native input,
• M is a derived comparison variable,
• and the simulation must expose mutation across both diploid strands for the M = 2Ku interpretation to make sense.

Threshold Claim

Nunney's reported threshold is not a mathematically defined extinction probability threshold. It is a simulation-search heuristic:

• find the lowest T with no extinctions in 20 runs,
• search from below,
• then require no extinction at 1.02T, 1.05T, and 1.10T,
• with extra retesting in borderline cases.

This matters because the paper's "threshold" mixes:

• biological persistence,
• stochastic variation,
• and the search protocol itself.

That is acceptable for Track 1 replication, but it is one of the main reasons Track 2 exists.

Claimed Result Structure

The paper's most important reported pattern is that threshold cost can be regressed on number of loci:

C = C0 + n C1

where:

• C0 is the fixed cost component,
• C1 is the per-locus cost component,
• and both are analyzed as functions of mutation supply M.

Figure 1 and Table 1 are therefore not just descriptive outputs; they are the main statistical structure the replication must recover if the implementation is faithful.

What Must Be Reproduced

A credible Track 1 replication should reproduce, or clearly fail to reproduce, all of the following:

• the paper's parameter framing in terms of u, K, R, T, and derived M
• the threshold-search behavior over repeated stochastic runs
• the locus-sweep regression structure C = C0 + n C1
• the directional effect of mutation supply on fixed and per-locus cost
• the extinction/non-extinction boundary under the published search rule

Key Ambiguities In The Paper

Several implementation details are underdetermined by the paper text and must be treated explicitly as reconstruction choices:

• exact generation update order
• exact stochastic law for realized births
• exact mutation operator over the allele set
• exact practical allele-state truncation for finite runs
• exact sex realization rule
• exact extinction condition in code

These do not make replication impossible, but they mean "faithful replication" is always conditional on a documented reconstruction policy.

Current Replication Reading

The present Track 1 implementation uses the following interpretation:

• integer allele states with finite truncation tied to the run horizon
• lottery polygyny with one male sampled per female reproductive event
• births drawn stochastically from fecundity
• offspring survival governed by w_i
• extinction on zero population or absence of one sex
• explicit reporting of f, mean w, f*w, mutation supply, allele tracking, and extinction timing

This is intended to stay as close as possible to the paper while making the reconstruction auditable.

Main Scientific Risks

The main ways the replication could still diverge from the paper are:

1. the wrong stochastic realization of fecundity and survival
2. an off-by-one time alignment in t versus offspring evaluation
3. mutation semantics that do not match the paper's effective M = 2Ku treatment
4. a threshold-search implementation that is formally similar to Nunney's but operationally too permissive or too strict

These are exactly the points where current diagnostics and paper-scale runs should be examined.

How This Effort Differs From Nunney

Nunney's paper presents the biological argument through a simulation workflow. This repo separates that into layers:

• simulation kernel
• threshold search
• analysis and reporting
• dataset generation
• extinction fitting
• orchestration

That separation does not change the Track 1 target; it makes it inspectable.

Current Bottom Line

The project should treat Nunney's paper as making three distinct deliverables necessary:

1. a faithful historical reconstruction of the published simulation and search rule
2. a clear statement of where the paper is underdetermined
3. a modern replacement path that keeps the biological question while replacing the threshold heuristic and performance limitations

Track 1 in renunney now addresses the first two. Track 2 remains the next major scientific and engineering step.