Chance between generations
A population of N diploid individuals contains 2N copies of an autosomal locus. If K copies carry allele A, its frequency is p = K/(2N). In the Wright–Fisher model, each new generation is formed by independently sampling copies from the previous one, with replacement.
The assumptions are random reproduction, nonoverlapping generations and no selection, mutation or migration. N represents an ideal population; it does not estimate the effective size of a real species.
The same start, different paths
Each curve represents the history of one simulated population. In each generation, the copies forming the next generation are sampled using the current frequency of A as its probability. The sampling outcome determines the new frequency and therefore the probability used in the next generation.
The five trajectories repeat this process from the same initial conditions, with different random draws. Sampling tends to cause larger frequency changes in small populations. No individual trajectory is meant to represent the average outcome.
The dashed line marks p₀. Mean frequency across all possible populations remains at that value, although each population may move away from it. With N = 1 and K₀ = 1, one generation can yield 0, 1 or 2 copies of A with probabilities 1/4, 1/2 and 1/4.
Repeating the experiment under the same initial conditions produces five new paths, equally valid. No repetition is more representative than another: the object of study is the set of possible paths, not any one of them.
The loss of diversity
H = 2p(1−p) is the probability that two independently sampled copies, with replacement, differ. It is not the observed percentage of heterozygous individuals: this is why Nei (1973) proposed calling it gene diversity, a name that does not presuppose random mating. Its expectation is computed across all possible populations, not by averaging five paths.
That formula yields two readings worth keeping apart. E[H_t] is the expected diversity: a value of H at generation t, between 0 and 0.5. The ratio E[H_G]/H₀ is the diversity retained: the fraction of the initial diversity that survives G generations, given as a percentage. In conservation genetics this second form is the usual way to express the effect of drift. If only one allele is present initially, H₀ = 0 and the ratio is undefined. Although the expectation decreases, diversity in an individual population can temporarily increase.
What a bottleneck changes
Size falls to Nᵦ during generations s,…,s+d−1 and returns to N₀ at s+d. The transition into s already samples 2Nᵦ copies. If the bottleneck ends at G, recovery falls outside the displayed period.
The reduced population loses expected diversity more quickly. Restoring individuals reduces future drift but does not restore lost alleles. The relevant comparison is against the same initial population holding N₀ constant: the gap between the two expectations is the cost of the bottleneck.
A single small generation at the start also illustrates idealized founding. It does not model colonization or sampling individuals without replacement.
Worked example: constant population size
1. Substitute the parameters. Choose N₀ = 10 individuals, K₀ = 10 copies of A and G = 3 generations, without a bottleneck. There are 20 copies in each generation.
2. Sample the first generation. Substituting into the binomial transition gives:
This means 20 draws, each with a 50% probability of A. Each draw amounts to taking a random number u, uniform from 0 inclusive to 1 exclusive: if u < 0.5 the copy is A; otherwise it is the other allele. K₁ is the total number of A copies drawn.
3. Use the outcome to continue. Suppose the result is K₁ = 12 copies of A. Then p₁ = 12/20 = 0.6: the next 20 draws use 60%, rather than the initial 50%. If subsequent draws give K₂ = 11 and K₃ = 9, the trajectory is:
These counts are one possible realization, chosen to explain the procedure: the formula does not prescribe them, and another repetition of the experiment would give others. The remaining four trajectories also start at p₀ = 0.5 and use their own random draws.
4. Calculate expected and retained diversity. Here the theoretical formula has three equal factors; the ratio to H₀ follows from the same result:
On average, 85.74% of the initial diversity is retained. This percentage is an expectation across possible populations; it is not calculated by averaging the five curves and need not describe the example trajectory.
The same example with a bottleneck
1. Change the size of one generation. Keep N₀ = 10, K₀ = 10 and G = 3; enable the bottleneck with Nᵦ = 2, start s = 2 and duration d = 1. The size schedule becomes:
2. Enter the bottleneck. The first transition is unchanged. If it produced K₁ = 12, the parental frequency is 12/20 = 0.6. Only 2Nᵦ = 4 copies are sampled to form generation 2:
If the result is K₂ = 3 copies of A, the frequency becomes p₂ = 3/4 = 0.75. The probability comes from the parental population; the number of draws comes from the size of the new generation.
3. Restore population size. Generation 3 contains 20 copies again, but sampling uses the bottleneck frequency, 3/4:
One possible outcome is K₃ = 14 and p₃ = 14/20 = 0.7. That trajectory would be (0.5, 0.6, 0.75, 0.7). Restoring N₀ does not force the frequency to return to its initial value.
4. Recalculate expected and retained diversity. Since the entire bottleneck lies within the G generations, there are G−d transitions at size N₀ and d at size Nᵦ:
The theoretical result falls from 85.74% without a bottleneck to 67.69% with one. Both predictions depend only on the sizes and on p₀: repeating the experiment changes the random draws and the paths, not the expectation.
Allele loss and fixation
At p = 0, A has been lost; at p = 1, A has become fixed. Without mutation or migration, these states are absorbing: frequency stays at that endpoint even if population size changes. Between 0 and 1, both alleles remain.
The eventual fixation probability of A is p₀ if this neutral model continues with bounded positive sizes. It refers to an unlimited horizon; it does not state that A is already fixed at G or determine the fate of an individual trajectory.
All of the above describes neutral drift. Genotype counts, inference of Nₑ and effects of selection, mutation or migration require other models and data.