Meet two real peppered-moth forms. Follow survivors, trace inherited copies into offspring, and replay a population under different conditions. Compare the teaching model with what researchers actually measured.
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The adults stay themselves. A new generation receives copies from its parents. When inherited differences affect survival and reproduction, those copies can become more or less common over generations. That process is natural selection. Chance changes the counts too.
If both moth forms have the same survival chance, must their proportions stay exactly the same?
Equal chances remove this systematic survival advantage. They do not remove random sampling from a finite population.
The pale, speckled typica and dark carbonaria belong to the same species: the peppered moth, Biston betularia. The photographs show two real individuals. Our population uses those forms to explore inherited differences; the photographed moths’ genotypes were not measured here.
Choose a survival rule, then try Survival. Each adult keeps its identity, two letters and wing-form category. A changed rule affects its chance of surviving this event; it does not repaint the adult. These simple probabilities are authored teaching choices, not a forest forecast.
Make the next generation. Each new individual receives one modeled copy from each of two surviving parents. Some parents contribute several times; some contribute none. Follow one child’s record to see where its copies came from. Keep the old adults and their offspring separate.
The starting 40 adults include 10 CC, 20 Cc and 10 cc. Both CC and Cc are dark-form categories, so 30 of 40 look dark. But only 40 of the 80 copies are C. A visible category is not the same thing as an allele frequency.
Equal survival probabilities remove the systematic survival advantage at this modeled locus. They do not remove chance from survival or parent sampling. A finite run can still gain or lose C copies. Changes caused by this sampling are genetic drift; drift can also operate alongside selection.
One source study followed live moths over short morning windows. Another placed artificial moth-like targets in woodland for up to 48 hours. Neither observed our imaginary generations. Compare their measurements with the model without turning one kind of evidence into another.
Inherited differences can affect expected contribution to future generations through survival and reproduction. “Fitness” in this context does not mean strongest, smartest or morally best. A trait’s contribution depends on conditions. A form favored by one survival rule need not be favored by another, and a chance result is not a universal ranking.
This diploid teaching model uses C and c at one locus: CC and Cc are carbonaria-like; cc is typica-like. The carbonaria allele is dominant for this adult classification. Dominance does not give a C copy a greater transmission probability from a Cc parent: either tracked copy is chosen with probability 1/2. The intermediate insularia forms found in nature are omitted.
Each adult receives a stored survival draw. Surviving parents are sampled uniformly, with replacement, from separate female and male pools; one copy is sampled from each selected parent. New individuals receive alternating pool labels, an authored fixed-sex-ratio rule rather than a model of sex determination. If either pool has no survivors, this run stops under its mating rule. The computer must not invent parents to keep it going.
Pale-favored survival uses 0.60 for dark-form adults and 0.90 for pale-form adults. Equal probabilities use 0.80 for both. Dark-favored reverses the first contrast. These are event probabilities chosen to make the mechanism visible; they are neither source estimates nor lifetime reproductive fitness. Changing an animation speed or revisiting an event must not resample its recorded outcome.
For an ideal randomly mating population with C frequency p, let wD and wP denote the chosen dark and pale event-survival probabilities. Starting genotype proportions are p², 2p(1−p), (1−p)². The mean survival is W = wD[p² + 2p(1−p)] + wP(1−p)², and the ideal survivor-copy frequency is p′ = wD p / W. At p = 1/2 and wD = 0.6, wP = 0.9, this gives W = 0.675 and p′ = 4/9. The ideal next dark-form fraction is 56/81. These calculations do not force a finite run to hit those values.
Let pF and pM be C-copy fractions in the two surviving parent pools. Under this model’s independent sampling, P(CC) = pF pM; P(Cc) = pF(1−pM) + (1−pF)pM; P(cc) = (1−pF)(1−pM). Expected offspring C frequency is (pF + pM)/2. Its conditional variance is [pF(1−pF) + pM(1−pM)]/(4N). The familiar p(1−p)/(2N) form requires equal pool frequencies. This concerns a distribution of possible runs, not a promise about every small-versus-large comparison.
Mutation changes DNA sequence; inherited changes can introduce variants. Mutation and selection are different mechanisms. A variant does not appear because a population anticipates what would help. Mutation processes may nevertheless be biased or environmentally influenced. With no mutation or migration in our closed model, an absent allele cannot reappear merely through selection or ordinary copying. This is a model boundary, not a claim that all real populations stay fixed forever.
Luria and Delbrück compared patterns of virus-resistant bacteria among independently grown cultures. Their 1943 results supported inherited changes arising independently of the selecting virus in that case. This separates the origin of a variant from its later detection under selection. It is not a moth mutation-rate measurement or a claim that environments never affect mutation processes. No culture experiment is proposed here.
Suppose all 40 adults survive, exactly one of the 20 female parents is Cc, and all other parents are cc. An offspring has probability 1/40 of receiving C through its female parent and zero through its male parent. The chance that none of 40 offspring receives C is (39/40)⁴⁰, about 0.3632. This is conditional on these specific pools and sampling rules. It is not a universal probability that a new variant disappears.
Eacock and colleagues documented background-related color change in peppered-moth caterpillars, including responses when their eyes were obscured. Their study distinguishes larval plasticity from genetically determined adult wing-pattern forms. Thus “these modeled adults keep their category” must not become “individual organisms can never change appearance.”
Cook and colleagues reported Majerus’s 4,864 released moths across six years. Moths absent four hours after sunrise were presumed eaten; approximately 26% of disappearing individuals were directly observed being eaten by birds. In 2006, 28/34 carbonaria remained, compared with 629/774 typica. This one raw yearly comparison points the other way from the overall evidence. The source’s combined analysis accounts for different overall predation among years; our model does not convert its short-term estimate into lifetime fitness.
Walton and Stevens placed 250 targets resembling each form, with printed wings and edible bodies, and checked them at 6, 24 and 48 hours. The original workbook codes bird-predation events as 1, despite naming the column “Censored”; 0 includes final survivors and other outcomes. A non-avian or otherwise excluded outcome leaves the risk set without becoming an observed bird-predation event. It does not establish survival beyond that time.
The product-limit estimate multiplies (1 − dᵢ/nᵢ) over observed event times, where nᵢ is the number still at risk and dᵢ the bird-predation events. Events are processed before censoring at the same recorded time. The data give 48-hour estimates about 0.6446 for typica-like and 0.4343 for carbonaria-like targets: a difference of about 21.03 percentage points. This is not simply final survivors divided by 250. We do not repeat the paper’s internally inconsistent reported test-statistic/p-value combination or label this as a new complete statistical analysis.
The peppered-moth studies connect appearance, background and predation. A trait can have a different advantage when its surroundings change. Our reversed survival rule is a counterfactual comparison, not a fitted reconstruction of a polluted woodland.
Ask: living organisms or artificial targets? Hours or generations? Observed counts or estimated probabilities? A clear denominator and a description of what was measured often reveal more than a smooth-looking graph.
A caterpillar changing appearance and an inherited adult form becoming more common are both biologically interesting. Identifying the mechanism and timescale helps avoid treating every change as the same process.
Create cards F1–F4 and M1–M4. In each pool write CC, Cc, Cc, cc, one genotype per card. Add a dark pattern to CC/Cc and a speckled pattern to cc if you like; keep letters readable. These are paper individuals, not real animals or anyone’s family traits.
Use this paper-only rule: dark-form cards survive on die results 1–3; pale-form cards on 1–5. Write what you expect and why. These probabilities are 1/2 and 5/6, deliberately different from the app’s 0.6 and 0.9. Neither pair is a measured moth parameter.
Roll once for each adult and record its ID, roll and outcome. Leave surviving cards in their own parent pool; keep non-survivors in a separate record. Never change an adult’s letters. If either pool has no survivor, stop and record that outcome instead of inventing a parent.
Mix surviving cards face down separately in each pool. Select one from each without looking, then return each after recording its ID. Flip once per parent: heads copies its first letter, tails its second. Write both copied letters on a fresh child card and record the two parent IDs. A parent can be selected again.
Repeat until there are eight new cards. Alternate F and M labels for their next-generation pools; this is a simplified pool rule, not real sex determination. Count C copies out of 16 and dark-form cards out of eight separately. Keep the original adults and the new cohort in distinct rows.
Restart from the original eight cards. This time both forms survive on 1–4, so both have probability 2/3. Repeat the same copying procedure if both pools have survivors. Compare the counts and, if desired, repeat several runs. Equal probabilities need not produce identical counts.
Write one sentence about inherited copies, one about chance and one model limitation. Include your original prediction and actual counts—even if surprising. A single short run does not prove a statistical effect. No animal observation, bait or outdoor experiment is needed.
Which adults contributed copies to your new cohort—and how could the counts change even when both forms had the same survival chance?
Original tabletop sampling model, not a tested classroom protocol or animal experiment. Fixed cohort size, two parent pools and simplified probabilities leave out real moth ecology. No special supplies, cutting, pins, food bait or personal genetic information are needed.
Existing inherited differences contributed differently to new generations. Variation was already present. Survival and reproduction changed which copies reached new offspring.
No; we must also follow inherited copies into offspring. Survival allows a possible contribution. The parent and copy record shows the contribution that actually happened.
Chance can change allele frequencies even without a survival advantage. Finite survival and reproduction involve sampling. This can change frequencies: genetic drift.
Its origin and its later spread are separate questions. A mutation can introduce variation. Conditions and chance affect what happens to its descendants.
One half: 40 of the 80 copies. The ten CC adults contain 20 C copies; the twenty Cc adults contain another 20. Visible form and copy frequency differ.
A yearly sample can differ from the overall evidence. Keep the observed row and its denominators. The study combined years while accounting for differences in overall predation.
500 calibrated paper-wing targets with edible bodies, followed over 48 hours. This physical field experiment used artificial targets. Its results are separate from the live-moth study and our generation model.
Yes; individual plasticity and inherited population change are different mechanisms. Peppered-moth caterpillars have documented color plasticity. This differs from the inherited adult wing-form frequencies modeled here.
Tables 1–2, methods and discussion. 4,864 released moths; most presumed predation was not directly witnessed. CC BY 3.0.
Cook et al. (2012) · live peppered-moth experimentFigures 1–3 and methods. Artificial paper wings and edible bodies, not 500 living moths. CC BY 4.0; identify source-derived replots separately.
Walton & Stevens (2018) · camouflage and artificial preyArtificial Predation Experiment sheet; reason fields confirm event coding. Preserve row provenance and distinguish events from censored outcomes. CC BY 4.0.
Walton & Stevens · original supplementary dataOriginal introduction, experiments and discussion; no animal procedure is proposed as a home activity. CC BY 4.0; no source figure is copied here.
Eacock et al. (2019) · caterpillar color plasticityOriginal introduction, fluctuation reasoning, discussion and summary. Bacterial case, not a moth mutation rate. Read-only citation: no source PDF pages or figures are redistributed.
Luria & Delbrück (1943) · origin versus selectionSource-owned explanation; it does not supply our moth survival parameters.
NHGRI · genetic driftUse within the relevant biological context; no universal mutation rate or moth sequence is asserted.
NHGRI · mutationPhotograph taken July 8, 2010, CC BY 2.0. This individual is not an experimental genotype record; identify any display crop.
Ben Sale · dark peppered-moth photographPhotograph taken June 21, 2011, CC BY 2.0. The title’s [1931] is not the photograph year; identify any display crop.
Ben Sale · pale peppered-moth photographCredit photographer and source, retain license link, and identify changes. Photograph reuse does not imply endorsement.
CC BY 2.0 · photograph licenseCredit the source and identify any extraction or replot; preserve original labels on original figures.
CC BY 3.0 · Cook source licenseCredit Walton & Stevens and specify the original figure or supplementary data plus any modifications.
CC BY 4.0 · Walton source licenseIndependent subject review is pending.
Read the sources and model assumptions
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